AND THE ATIYAH-SINGER INDEX THEOREM by Peter B Gilkey

INVARIANCE THEORY, THE HEAT EQUATION,AND THEATIYAH-SINGER INDEX THEOREMby Peter B. GilkeyElectronic reprint, copyright 1996, Peter B. GilkeyBook originally published on paper by Publish or Perish Inc., USA, 1984Library of Congress Catalog Card Number 84-061166ISBN 0-914098-20-9

INTRODUCTIONThis book treats the Atiyah-Singer index theorem using heat equationmethods. The heat equation gives a local formula for the index of anyelliptic complex. We use invariance theory to identify the integrand ofthe index theorem for the four classical elliptic complexes with the invariantsof the heat equation. Since the twisted signature complex providesa suciently rich family of examples, this approach yields a proof of theAtiyah-Singer theorem in complete generality. We also use heat equationmethods to discuss Lefschetz xed point formulas, the Gauss-Bonnet theoremfor a manifold with smooth boundary, and the twisted eta invariant.We shall not include a discussion of the signature theorem for manifoldswith boundary.The rst chapter reviews results from analysis. Sections 1.1 through 1.7represent standard elliptic material. Sections 1.8 through 1.10 contain thematerial necessary to discuss Lefschetz xed point formulas and other topics.Invariance theory and dierential geometry provide the necessary link betweenthe analytic formulation of the index theorem given by heat equationmethods and the topological formulation of the index theorem contained inthe Atiyah-Singer theorem. Sections 2.1 through 2.3 are a review of characteristicclasses from the point of view of dierential forms. Section 2.4gives an invariant-theoretic characterization of the Euler form whichisusedto give a heat equation proof of the Gauss-Bonnet theorem. Sections 2.5and 2.6 discuss the Pontrjagin forms of the tangent bundle and the Chernforms of the coecient bundle using invariance theory.The third chapter combines the results of the rst two chapters to provethe Atiyah-Singer theorem for the four classical elliptic complexes. We rstpresent a heat equation proof of the Hirzebruch signature theorem. Thetwisted spin complex provides a unied way of discussing the signature,Dolbeault, and de Rham complexes. In sections 3.2{3.4, we discuss thehalf-spin representations, the spin complex, and derive a formula for the^A genus. We then discuss the Riemann-Roch formula for an almost complexmanifold in section 3.5 using the SPIN c complex. In sections 3.6{3.7 wegive a second derivation of the Riemann-Roch formula for holomorphicKaehler manifods using a more direct approach. In the nal two sectionswe derive the Atiyah-Singer theorem in its full generality.

viIntroductionThe nal chapter is devoted to more specialized topics. Sections 4.1{4.2deal with elliptic boundary value problems and derive the Gauss-Bonnettheorem for manifolds with boundary. In sections 4.3{4.4 we discuss thetwisted eta invariant on a manifold without boundary and we derive theAtiyah-Patodi-Singer twisted index formula. Section 4.5 gives a brief discussionof Lefschetz xed point formulas using heat equation methods. Insection 4.6 we use the eta invariant to calculate the K-theory of sphericalspace forms. In section 4.7, we discuss Singer's conjecture for the Eulerform and related questions. In section 4.8, we discuss the local formulasfor the invariants of the heat equation which have been derived by severalauthors, and in section 4.9 we apply these results to questions of spectralgeometry.The bibliography at the end of this book is not intended to be exhaustivebut rather to provide the reader with a list of a few of the basic paperswhich have appeared. We refer the reader to the bibliography of Bergerand Berard for a more complete list of works on spectral geometry.This book is organized into four chapters. Each chapter is divided intoa number of sections. Each Lemma or Theorem is indexed according tothis subdivision. Thus, for example, Lemma 1.2.3 is the third Lemma ofsection 2 of Chapter 1.

CONTENTSIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .vChapter 1. Pseudo-Dierential OperatorsIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1. Fourier Transform, Schwartz Class, and Sobolev Spaces . . . . . 21.2. Pseudo-Dierential Operators on R m . . . . . . . . . . . . . . . 111.3. Ellipticity and Pseudo-Dierential Operators on Manifolds . . . 231.4. Fredholm Operators and the Index of a Fredholm Operator . . . 311.5 Elliptic Complexes, The Hodge Decomposition Theorem,and Poincare Duality . . . . . . . . . . . . . . . . . . . . . . 371.6. The Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . 421.7. Local Formula for the Index of an Elliptic Operator . . . . . . . 501.8. Lefschetz Fixed Point Theorems . . . . . . . . . . . . . . . . . . 621.9. Elliptic Boundary Value Problems . . . . . . . . . . . . . . . . . 701.10. Eta and Zeta Functions . . . . . . . . . . . . . . . . . . . . . . . 78Chapter 2. Characteristic ClassesIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872.1. Characteristic Classes of a Complex Bundle . . . . . . . . . . . . 892.2 Characteristic Classes of a Real Vector Bundle.Pontrjagin and Euler Classes . . . . . . . . . . . . . . . . . . 982.3. Characteristic Classes of Complex Projective Space . . . . . . 1042.4. The Gauss-Bonnet Theorem . . . . . . . . . . . . . . . . . . . 1172.5 Invariance Theory and the Pontrjagin Classesof the Tangent Bundle . . . . . . . . . . . . . . . . . . . . 1232.6 Invariance Theory and Mixed Characteristic Classesof the Tangent Space and of a Coecient Bundle . . . . . . 141

viiiContentsChapter 3. The Index TheoremIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1473.1. The Hirzebruch Signature Formula . . . . . . . . . . . . . . . 1483.2. Spinors and their Representations . . . . . . . . . . . . . . . . 1593.3. Spin Structures on Vector Bundles . . . . . . . . . . . . . . . . 1653.4. The Spin Complex . . . . . . . . . . . . . . . . . . . . . . . . 1763.5. The Riemann-Roch Theorem for Almost Complex Manifolds . 1803.6. A Review of Kaehler Geometry . . . . . . . . . . . . . . . . . 1933.7 An Axiomatic Characterization of the Characteristic Formsfor Holomorphic Manifolds with Kaehler Metrics . . . . . . 2043.8. The Chern Isomorphism and Bott Periodicity . . . . . . . . . . 2153.9. The Atiyah-Singer Index Theorem . . . . . . . . . . . . . . . . 224Chapter 4. Generalized Index Theorems and Special TopicsIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2414.1. The de Rham Complex for Manifolds with Boundary . . . . . 2434.2. The Gauss-Bonnet Theorem for Manifolds with Boundary . . . 2504.3. The Regularity at s =0of the Eta Invariant . . . . . . . . . . 2584.4. The Eta Invariant with Coecients in a Locally Flat Bundle . 2704.5. Lefschetz Fixed Point Formulas . . . . . . . . . . . . . . . . . 2844.6. The Eta Invariant and the K-Theory of Spherical Space Forms 2954.7. Singer's Conjecture for the Euler Form . . . . . . . . . . . . . 3074.8. Local Formulas for the Invariants of the Heat Equation . . . . 3144.9. Spectral Geometry . . . . . . . . . . . . . . . . . . . . . . . . 331Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

CHAPTER 1PSEUDO-DIFFERENTIAL OPERATORSIntroductionIn the rst chapter, we develop the analysis needed to dene the indexof an elliptic operator and to compute the index using heat equation methods.Sections 1.1 and 1.2 are brief reviews of Sobolev spaces and pseudodierentialoperators on Euclidean spaces. In section 1.3, we transfer thesenotions to compact Riemannian manifolds using partition of unity arguments.In section 1.4 we review the facts concerning Fredholm operatorsneeded in section 1.5 to prove the Hodge decomposition theorem and todiscuss the spectral theory of self-adjoint elliptic operators. In section 1.6we introduce the heat equation and in section 1.7 we derive the local formulafor the index of an elliptic operator using heat equation methods.Section 1.8 generalizes the results of section 1.7 to nd a local formulafor the Lefschetz number of an elliptic complex. In section 1.9, we discussthe index of an elliptic operator on a manifold with boundary and insection 1.10, we discuss the zeta and eta invariants.Sections 1.1 and 1.4 review basic facts we need, whereas sections 1.8through 1.10 treat advanced topics which may be omitted from a rstreading. We have attempted to keep this chapter self-contained and toassume nothing beyond a rst course in analysis. An exception is thede Rham theorem in section 1.5 which is used as an example.A number of people have contributed to the mathematical ideas whichare contained in the rst chapter. We were introduced to the analysis ofsections 1.1 through 1.7 by a course taught by L. Nirenberg. Much of theorganization in these sections is modeled on his course. The idea of usingthe heat equation or the zeta function to compute the index of an ellipticoperator seems to be due to R. Bott. The functional calculus used in thestudy of the heat equation contained in section 1.7 is due to R. Seeley asare the analytic facts on the zeta and eta functions of section 1.10.The approach to Lefschetz xed point theorems contained in section 1.8is due to T. Kotake for the case of isolated xed points and to S. C. Leeand the author in the general case. The analytic facts for boundary valueproblems discussed in section 1.9 are due to P. Greiner and R. Seeley.

1.1. Fourier Transform, Schwartz Class,And Sobolev Spaces.The Sobolev spaces and Fourier transform provide the basic tools weshall need in our study of elliptic partial dierential operators. Let x =(x 1 ...x m ) 2 R m : If x y 2 R m , we dene:x y = x 1 y 1 + + x m y m and jxj =(x x) 1=2as the Euclicean dot product and length. Let =( 1 ... m )beamultiindex.The j are non-negative integers. We dene:jj = 1 + + m ! = 1 !... m ! x = x 11 ...x m m :Finally, we dene:d x = @ 1 @ m...and Dx =(;i) jj d x@x 1 @x mas a convenient notation for multiple partial dierentiation. The extrafactors of (;i) dening Dx are present to simplify later formulas. If f(x) isa smooth complex valued Xfunction, then Taylor's theorem takes the form:f(x) = d x f(x 0) (x ; x 0) + O(jx ; x 0 j n+1 ):!jjnThe Schwartz class S is the set of all smooth complex valued functionsf on R m such that for all there are constants C such thatjx D x fjC :This is equivalent to assuming there exist estimates of the form:jD x fjC n(1 + jxj) ;nfor all (n ). The functions in S have all their derivatives decreasing fasterat 1 than the inverse of any polynomial.For the remainder of Chapter 1, we let dx, dy, d, etc., denote Lebesguemeasure on R m with an additional normalizing factor of (2) ;m=2 . Withthis normalization, the integral of the Gaussian distribution becomes:Ze ; 1 2 jxj2 dx =1:We absorb the normalizing constant into the measure in order to simplifythe formulas of the Fourier transform. If C 1 0 (R m ) denotes the set ofsmooth functions of compact support on R m , then this is a subset of S .Since C 0 (R m ) is dense in L 2 (R m ), S is dense in L 2 (R m ).We dene the convolution product of two elements of S by:(f g)(x) =Zf(x ; y)g(y) dy =Zf(y)g(x ; y) dy:This denes an associative and commutative multiplication.there is no identity, there do exist approximate identities:Although

Fourier Transform, Schwartz Class 3Lemma 1.1.1. Let f 2S with R f(x) dx =1. Dene f u (x) =u ;m f( x u ).Then for any g 2S, f u g converges uniformly to g as u ! 0.Proof: R Choose C so jf(x)j dx C and jg(x)j C. Because the rstderivatives of g are uniformly bounded, g is uniformly continuous. Let" > 0 and choose > 0 so jx ; yj implies jg(x) ; g(y)jR". Becausefu (x) dx =1,we compute:jf u g(x) ; g(x)j = Z f u (y)fg(x ; y) ; g(x)g dyZjf u (y)fg(x ; y) ; g(x)gj dy:We decompose this integral into two pieces. If jyj we bound it by C".The integral for jyj can be bounded by:2CZjyjjf u (y)j dy =2CZjyj=ujf(y)j dy:This converges to zero as u ! 0 so we can bound this by C" if u < u(").This completes the proof.A similar convolution smoothing can be applied to approximate any elementofLp arbitrarily well in the L p norm by a smooth function of compactsupport.We dene the Fourier transform ^f() by:^f() =Ze ;ix f(x) dx for f 2S:For the moment 2 R m when we consider operators on manifolds, it willbe natural to regard as an element of the ber of the cotangent space.By integrating by parts and using Lebesgue dominated convergence, wecompute:dD f ^f()g =(;1) jj fx fg and ^f() = fD d x fg:This implies ^f 2S so Fourier transform denes a map S !S.We compute the Fourier transform R of the Gaussian distribution.f 0 (x) = exp(; 1 2 jxj2 ), then f 0 2S and f 0 (x) dx =1. We compute:Z^f 0 () = e ;ix e ; 1 2 jxj2 dxZ= e ; 1 2 jj2 e ;(x+i)(x+i)=2 dx:Let

4 1.1. Fourier Transform, Schwartz ClassWe make a change of variables to replace x + i by x and to shift thecontour in C m back ; to the original contour R m . This shows the integral is1and ^f 0 () = exp ; 1 jj2 so the function f2 0 is its own Fourier transform.In fact, the Fourier transform is bijective and the Fourier inversion formulagives the inverse expressing f in terms of ^f by:Zf(x) =e ix ^f() d = ^f(;x):We dene T (f) = ^f(;x) =R eix ^f() d as a linear map from S !S. Wemust show that T (f) =f to prove the Fourier inversion formula.Suppose rst f(0) = 0. We expand:f(x) =Z 10ddt ff(tx)g dt = X jx jZ 10@f@x j(tx) dt = X jx j g jwhere the g j are smooth. Let 2 C 1 0 (R m ) be identically 1 near x = 0.Then we decompose:f(x) =f(x)+(1; )f(x) = X jx j g j + X jx jxj (1 ; )fjxj 2Since g j has compact support, it is in S . Since is identically 1 nearx = 0, x j (1 ; )f=jxj 2 2 S . Thus we can decompose f = P x j h j forh j 2S. We Fourier transform this identity to conclude::^f = X jd fx j h j g = X ji @^h j@ j:Since this is in divergence form, T (f)(0) = R ^f() d =0=f(0).More generally, let f 2 S be arbitrary. We decompose f = f(0)f 0 +(f ; f(0)f 0 ) for f 0 = exp(; 1 2 jxj2 ). Since ^f 0 = f 0 is an even function,T (f 0 )=f 0 so that T (f)(0) = f(0)f 0 (0)+T (f ;f(0)f 0 )=f(0)f 0 (0) = f(0)since (f ; f(0)f 0 )(0) = 0. This shows T (f)(0) = f(0) in general.We use the linear structure on R m to complete the proof of the Fourierinversion formula. Let x 0 2 R m be xed. We let g(x) =f(x + x 0 ) then:f(x 0 )=g(0) = T (g)(0) ==ZZe ;ix f(x + x 0 ) dx de ;ix e ix 0 f(x) dx d= T (f)(x 0 ):

And Sobolev Spaces 5This shows the Fourier transform denes a bijective map S !S. If weuse the constants C = sup x2R m jx Dx fj to dene a Frechet structureon S , then the Fourier transform is a homeomorphism of topological vectorspaces. It is not dicult to show C 1 0 (R m ) is a dense subset of S in thistopology. We can use either pointwise multiplication or convolution todene a multiplication on S and make S into a ring. The Fourier transforminterchanges these two ring structures. We compute:^f ^g ===ZZZe ;ix f(x)e ;iy g(y) dx dye ;i(x;y) f(x ; y)e ;iy g(y) dx dye ;ix f(x ; y)g(y) dx dy:The integral is absolutely convergent so we may interchange the order ofintegration to compute ^f ^g = (f d g). If we replace f by ^f and g by ^gwe see (f g)(;x) = ( ^f d ^g) using the Fourier inversion formula. We nowtake the Fourier transform and use the Fourier inversion formula to see(fd g)(;) =(^f ^g)(;) so that (f d g)= ^f ^g.The nal property we shall R need of the Fourier transform is related tothe L 2 inner product (fg)= f(x)g(x) dx. We compute:( ^fg)=Zf(x)e ;ix g() dx d ==(f ^g(;x)):Zf(x)e ;ix g() d dxIf we replace g by ^g then ( ^f^g) =(f ^g(;x))=(fg) so the Fourier transformis an isometry with respect to the L 2 inner product. Since S is densein L 2 , it extends to a unitary map L 2 (R m ) ! L 2 (R m ). We summarizethese properties of the Fourier transform as follows:Lemma 1.1.2. The Fourier transform is a homeomorphism S !S suchthat:R (a) f(x) = e ix ^f() d =R e i(x;y) f(y) dy d (Fourier inversion formula)R (b) Dx f(x) = e ix ^f() d and ^f() =R e ;ix Dx f(x) dxd(c) ^f ^g = (f g) and ^f ^g = (f d g)(d) The Fourier transform extends to a unitary map of L 2 (R m ) ! L 2 (R m )such that (fg)=(^f^g). (Plancherel theorem).We note that without the normalizing constant of (2) ;m=2 in the denitionof the measures dx and d there would be various normalizing constantsappearing in these identities. It is property (b) which will be of the

6 1.1. Fourier Transform, Schwartz Classmost interest to us since it will enable us to interchange dierentiation andmultiplication.We dene the Sobolev space H s (R m ) to measure L 2 derivatives. If s isa real number and f 2S, we dene:jfj 2 s =Z(1 + jj 2 ) s j ^f()j 2 d:The Sobolev space H s (R m ) is the completion of S with respect to the normj s . The Plancherel theorem shows H 0 (R m ) is isomorphic to L 2 (R m ). Moregenerally, H s (R m ) is isomorphic to L 2 with the measure (1 + jj 2 ) s=2 d.Replacing (1 + jj 2 ) s by (1 + jj) 2s in the denition of j s gives rise to anequivalent norm since there exist positive constants c i such that:c 1 (1 + jj 2 ) s (1 + jj) 2s c 2 (1 + jj 2 ) s :In some sense, the subscript \s " counts the number of L 2 derivatives. Ifs = n is a positive integer, there Xexist positive constants c 1 , c 2 so:c 1 (1 + jj 2 ) n j j 2 c 2 (1 + jj 2 ) n :jjnThis implies that we could deneX Z Xjfj 2 n = j ^fj 2 d =jjnjjnZjD x fj2 dxas an equivalent norm for H n (R m ). With this interpretation in mind, it isnot surprising that when we extend D x to H s , that jj L 2 derivatives arelost.Lemma 1.1.3. D x extends to dene a continuous map D x : H s ! H s;jj .Proof: Henceforth we will use C to denote a generic constant. C candepend upon certain auxiliary parameters which will usually be supressedin the interests of notational clarity. In this proof, for example, C dependson (s ) but not of course upon f. The estimate:j j 2 (1 + jj 2 ) s;jj C(1 + jj 2 ) simplies that:jD x fj2 s; =Zj ^f()j 2 (1 + jj 2 ) s;jj d Cjfj 2 sfor f 2 S . Since H s is the closure of S in the norm j s , this completes theproof.

And Sobolev Spaces 7If k is a non-We can also use the sup norm to measure derivatives.negative integer, we dene:jfj 1k = supx2R mXjjkjDx fj for f 2S:The completion of S with respect to this norm is a subset of C k (R m )(the continuous functions on R m with continuous partial derivatives upto order k). The next lemma relates the two norms j s and j 1k . It willplay an important role in showing the weak solutions we will construct todierential equations are in fact smooth.Lemma 1.1.4. Let k be a non-negative integer and let s > k + m 2 . Iff 2 H s , then f is C k and there is an estimate jfj 1k Cjfj s . (SobolevLemma).Proof: Suppose rst k =0and f 2S. We computef(x)==ZZe ix ^f() dfe ix ^f()(1 + jj 2 ) s=2 gf(1 + jj 2 ) ;s=2 g d:We apply the Cauchy-Schwarz inequality toestimate:jf(x)j 2 jfj 2 sZ(1 + jj 2 ) ;s d:Since 2s >m, (1+jj 2 ) ;s is integrable so jf(x)j Cjfj s . We take the supover x 2 R m to conclude jfj 10 Cjfj s for f 2 S . Elements of H s arethe limits in the j s norm of elements of S . The uniform limit of continuousfunctions is continuous so the elements of H s are continuous and the samenorm estimate extends to H s . If k>0, we use the estimate:jD x fj 10 CjD x fj s;jj Cjfj s for jj k and s ; k> m 2to conclude jfj 1k Cjfj s for f 2S. A similar argument shows that theelements of H s must be C k and that this estimate continues to hold.If s > t, we can estimate (1 + jj 2 ) s (1 + jj 2 ) t . This implies thatjfj s jfj t so the identity map on S extends to dene an injection ofH s ! H t which is norm non-increasing. The next lemma shows that thisinjection is compact if we restrict the supports involved.

8 1.1. Fourier Transform, Schwartz ClassLemma 1.1.5. Let ff n g2S be a sequence of functions with support ina xed compact set K. We suppose there is a constant C so jf n j s C forall n. Let s > t. There exists a subsequence f nk which converges in H t .(Rellich lemma).Proof: Choose g 2 C 0 (R m ) which is identically 1 on a neighborhood ofK. Then gf n = f n so by Lemma 1.1.2(c) ^f n = ^g ^f n . We let @ j = @@ jthen @ j (^g ^f n )=@ j ^g ^f n so that:j@ j ^fn ()j Zjf@ j^g( ; )g ^f n ()j d:We apply the Cauchy-Schwarz inequality toestimate:j@ j ^fn ()j jf n j s Zj@ j^g( ; )j 2 (1 + jj 2 ) ;s d 1=2 C h()where h is some continuous function of . A similar estimate holds forj ^f n ()j. This implies that the f ^f n g form a uniformly bounded equi-continuousfamily on compact subsets. We apply the Arzela-Ascoli theoremto extract a subsequence we again label by f n so that ^f n () convergesuniformly on compact subsets. We complete the proof by verifying that f nconverges in H t for s>t. We compute:jf j ; f k j 2 t =Zj ^f j ; ^f k j 2 (1 + jj 2 ) t d:We decompose this integral into two parts, jj r and jj r. On jj rwe estimate (1 + jj 2 ) t (1 + r 2 ) t;s (1 + jj 2 ) s so that:Zjjrj ^f j ; ^f k j 2 (1 + jj 2 ) t d (1 + r 2 ) t;s Z j ^f j ; ^f k j 2 (1 + jj 2 ) s d 2C(1 + r 2 ) t;s :If " > 0 is given, we choose r so that 2C(1 + r 2 ) t;s < ". The remainingpart of the integral is over jj r. The ^f j converge uniformly on compactsubsets so this integral can be bounded above by " if jk > j("). Thiscompletes the proof.The hypothesis that the supports are uniformly bounded is essential. Itis easy to construct a sequence ff n g with jf n j s = 1 for all n and such thatthe supports are pair-wise disjoint. In this case we can nd ">0 so thatjf j ; f k j t >"for all (jk) sothere is no convergent subsequence.

And Sobolev Spaces 9We x 2S and let " (x) =("x). We suppose (0) = 1 and x f 2S.We compute:D x (f ; " f)=(1; " )D x f + terms of the form " j D x ("x)D xf:As " ! 0, these other terms go to zero in L 2 . Since " ! 1 pointwise,(1 ; " )D x f goes to zero in L 2 . This implies " f ! f in H n for any n 0as " ! 0 and therefore " f ! f in H s for any s. If we take 2 C 1 0 (R m ),this implies C 1 0 (R m )isdense in H s for any s.Each H s space is a Hilbert space so it is isomorphic to its dual. Becausethere is no preferred norm for H s , it is useful to obtain an invariantalternative characterization of the dual space H s :Lemma 1.1.6. The L 2 pairing which maps SS ! C extends to a mapof H s H ;s ! C which is a perfect pairing and which identies H ;s withH s . That is:(a) j(fg)jjfj s jgj ;s for fg 2S,(b) given f 2S there exists g 2S so (fg)=jfj s jgj ;s and we can denejfj s =j(fg)jsup :g2S g6=0 jgj ;sProof: This follows from the fact that H s is L 2 with the weight function(1+jj 2 ) s and H ;s is L 2 with the weight function (1+jj 2 ) ;s . We compute:Z(fg) =(^f^g) = ^f()(1 + jj 2 ) s=2^g()(1 + jj 2 ) ;s=2 dand apply the Cauchy-Schwartz inequality toprove (a).j(fg)jTo prove part (b), we note jfj s .jgj ;sdened by:supg2S g6=0^g = ^f(1 + jj 2 ) s 2SWe take g to beand note that (fg) = ( ^f^g) = jfj 2 s and that jgj 2 ;s = jfj 2 s to see thatequality can occur in (a) which proves (b)If s>t>uthen we can estimate:(1 + jj) 2t "(1 + jj) 2s + C(")(1 + jj) 2ufor any ">0. This leads immediately to the useful estimate:

10 1.1. Fourier Transform, Schwartz ClassLemma 1.1.7. Let s>t>uand let ">0 be given. Thenjfj t "jfj s + C(")jfj u :If V is a nite dimensional vector space, let C 1 (V ) be the space ofsmooth complex valued maps of R m ! V . We choose a xed Hermitianinner product on V and dene S (V ) and H s (V ) as in the scalar case. Ifdim(V ) = k and if we choose a xed orthonormal basis for V , then S (V )and H s (V ) become isomorphic to the direct sum of k copies of S and ofH s . Lemmas 1.1.1 through 1.1.7 extend in the obvious fashion.We conclude this subsection with an extremely useful if elementary estimate:Lemma 1.1.8. (Peetre's Inequality). Let s be real and x y 2 R m .Then (1 + jx + yj) s (1 + jyj) s (1 + jxj) jsj .Proof: We suppose rst s>0. We raise the triangle inequality:1+jx + yj < 1+jxj + jyj (1 + jyj)(1 + jxj)to the s th power to deduce the desired inequality. We now suppose s

1.2. Pseudo-Dierential Operators on R m .A linear partial P dierential operator of order d is a polynomial expressionP = p(x D) = jjd a (x)Dx where the a (x) are smooth. The symbolP = p is dened by:XP = p(x ) = a (x) jjdand is a polynomial of order d in the dual variable . It is convenient toregard the pair (x ) as dening a point of the cotangent space T (R m )we will return to this point again when we discuss the eect of coordinatetransformations. The leading symbol X L P is the highest order part: L P (x )= a (x) jj=dand is a homogeneous polynomial of order d in .We can use the Fourier inversion formula to express:Pf(x) =Ze ix p(x ) ^f() d =Ze i(x;y) p(x )f(y) dy dfor f 2S. We note that since the second integral does not converge absolutely,we cannot interchange the dy and d orders of integration. We usethis formalism to dene the action of pseudo-dierential operators ( DO's)for a wider class of symbols p(x ) than polynomials. We make the followingDefinition. p(x ) is a symbol of order d and we write p 2 S d if(a) p(x ) is smooth in (x ) 2 R m R m with compact x support,(b) for all ( ) there are constants C such thatjD x D p(x )j C (1 + jj) d;jj :For such a symbol p, we dene the associated operator P (x D) by:P (x D)(f)(x)=Ze ix p(x ) ^f() d =Ze (x;y) p(x )f(y) dy das a linear operator mapping S !S.A dierential operator has as its order a positive integer. The order ofa pseudo-dierential operator is not necessarily an integer. For example, iff 2 C 1 0 (R m ), dene:p(x ) =f(x)(1 + jj 2 ) d=2 2 S d for any d 2 R.

12 1.2. Pseudo-Differential OperatorsThis will be a symbol of order d. If p 2 S d for all d, then we say thatp 2 S ;1 is innitely smoothing. We adopt the notational conventionof letting p, q, r denote symbols and P , Q, R denote the correspondingDO's.Because we shall be interested in problems on compact manifolds, wehave assumed the symbols have compact x support to avoid a number oftechnical complications. The reader should note that there is a well denedtheory which does not require compact x support.When we discuss the heat equation, we shall have to consider a widerclass of symbols which depend on a complex parameter. We postponediscussion of this class until later to avoid unnecessarily complicating thediscussion at this stage. We shall phrase the theorems and proofs of thissection in such a manner that they will generalize easily to the wider classof symbols.Our rst task is to extend the action of P from S to H s .Lemma 1.2.1. Let p 2 S d then jPfj s;d Cjfj s for f 2S.to a continuous map P : H s ! H s;d for all s.P extendsProof: We compute Pf(x) = R e ix p(x ) ^f() d so that the Fouriertransform is given by:cPf() =Ze ix(;) p(x ) ^f() d dx:This integral is absolutely convergent since p has compact x support so wemay interchange the order of integration. If we deneq()=Ze ;ix p(x ) dxas the Fourier transform in the x direction, thencPf() =By Lemma 1.1.6, jPfj s;d = supg2S(Pfg)=ZZq( ; ) ^f() d:j(Pfg)j. We compute:jgj d;sq( ; ) ^f() ^g() d d:Dene:then:(Pfg)=K()=q( ; )(1 + jj) ;s (1 + jj) s;dZK() ^f()(1 + jj) s ^g()(1 + jj) d;s d d:

Operators on R m 13We apply the Cauchy-Schwarz inequality toestimate:j(Pfg)jZjK()jj^f()j 2 (1 + jj) 2s d dZ 1=2jK()jj^g()j 2 (1 + jj) 2d;2s d d 1=2:We complete the proof by showingZjK()j d C andZjK()j d Csince then j(Pfg)jCjfj s jgj d;s .By hypothesis, p has suppport in a compact set K and wehave estimates:Therefore:j q()j =jD x p(x )j C (1 + jj) d : Z e ;ix D x p(x ) dx C (1 + jj) d vol(K):Therefore, for any integer k, jq()jC k (1 + jj) d (1 + jj) ;k vol(K) and:jK()jC k (1 + jj) d;s (1 + jj) s;d (1 + j ; j) ;k vol(K):We apply Lemma 1.1.8 with x + y = and y = to estimate:If we choose k> m 2jK()jC k (1 + j ; j) jd;sj;k vol(K):+ jd ; sj, then this will be integrable and complete theproof.Our next task is to show that the class of DO's forms an algebra underthe operations of composition and taking adjoint. Before doing that, westudy the situation with respect to dierential operators to motivate theformulas we shall derive. Let P = P p (x)D x and let Q = P q (x)D xbe two dierential operators. We assume p and q have compact x support.It is immediate that:P = X D x p and PQ = X p (x)D x q (x)D xare again dierential operators in our class. Furthermore, using Leibnitz'sruleXDx (fg)= Dx (f) Dx(g) !!! +=d (+ ) = ( + )!!

14 1.2. Pseudo-Differential Operatorsit is an easy combinatorial exercise to compute that:(P )= X d D x p =! and (PQ)= X d p D x q=! :The perhaps surprising fact is that these formulas remain true in somesense for DO's, only the sums will become innite rather than nite.We introduce an equivalence relation on the class of symbols by deningp q if p ; q 2 S ;1 . We note that if p 2 S ;1 then P : H s ! H t for alls and t by Lemma 1.2.1. Consequently by Lemma 1.1.4, P : H s ! C 1 0 forall s so that P is innitely smoothing in this case. Thus we mod out byinnitely smoothing operators.Given symbols p j 2 S d jwhere d j !;1, we writep 1Xj=1if for every d there is an integer k(d) such that k k(d) implies thatp ; P kj=1 p j 2 S d . We emphasize that this sum does not in fact need toconverge. The relation p P p j simply means that the dierence betweenP and the partial sums of the P j is as smoothing as we like. It will turn outthat this is the appropriate sense in which we will generalize the formulasfor (P ) and (PQ) from dierential to pseudo-dierential operators.Ultimately, we will be interested in operators which are dened on compactmanifolds. Consequently, it poses no diculties to restrict the domainand the range of our operators. Let U be a open subset of R m with compactclosure. Let p(x ) 2 S d have x support in U. We restrict the domainof the operator P to C 1 0 (U) so P : C 1 0 (U) ! C 1 0 (U). Let d(U) denotethe space of all such operators. For d d 0 , then d(U) d0(U). Wedene [ \ (U) = d(U) and ;1(U) = d(U)ddto be the set of all pseudo-dierential operators on U and the set of innitelysmoothing pseudo-dierential operators on U.More generally, let p(x ) be a matrix valued symbol we suppose thecomponents of p all belong to S d . The corresponding operator P is givenby a matrix of pseudo-dierential operators. P is a map from vector valuedfunctions with compact support in U to vector valued functions withcompact support in U. We shall not introduce separate notation for theshape of p and shall continue to denote the collection of all such operatorsby d(U). If p and q are matrix valued and of the proper shape, we denepq and also the operator P Q by matrix product and by composition. Wealso dene p and P to be the matrix adjoint and the operator adjoint sothat (P fg) =(fP g) where f and g are vector valued and of compactsupport. Before studying the algebra structure on (U), we must enlargethe class of symbols which we can admit:p j

Operators on R m 15Lemma 1.2.2. Let r(x y) be a matrix valued symbol which is smoothin (x y). We suppose r has compact x support inside U and that thereare estimates:jD x D D y rjC (1 + jj) d;jjfor all multi-indices ( ). If f is vector valued with compact supportin U, we dene:Rf(x) =Ze i(x;y) r(x y)f(y) dy d:Then this operator is in d(U) and the symbol is given by:R(x ) Xd D y r(x y)=! x=y:Proof: We note that any symbol in S d belongs to this class of operatorsif we dene r(x y) = p(x ). We restricted to vector valued functionswith compact support in U. By multiplying r by a cut-o function in ywith compact support which is 1 over U, we may assume without loss ofgenerality the y support of r is compact as well. Dene:q(x )=Ze ;iy r(x y) dyto be the Fourier transform of r in the y variable. Using Lemma 1.1.2 wesee d (rf)=^r ^f. This implies that:Ze ;iy r(x y)f(y) dy =Zq(x ; ) ^f() d:The argument given in the proof of Lemma 1.2.1 gives estimates of theform:jq(x )jC k (1 + jj) d (1 + jj) ;k and j ^f()j C k (1 + jj) ;kfor any k. Consequently:jq(x ; ) ^f()j C k (1 + jj) d (1 + j ; j) ;k (1 + jj) ;k :We apply Lemma 1.1.8 to estimate:jq(x ; ) ^f()j C k (1 + jj) jdj;k (1 + jj) jdj;k

16 1.2. Pseudo-Differential Operatorsso this is absolutely integrable. We change the order of integration andexpress:Rf(x) =Ze ix q(x ; ) ^f() d d:We dene:p(x ) =Ze ix(;) q(x ; ) dand compute:Rf(x) =Ze ix p(x ) ^f() dis a pseudo-dierential operator once it is veried that p(x ) is a symbolin the correct form.We change variables to express:and estimate:p(x ) =Ze ix q(x + ) djq(x + )j C k (1 + j + j) d (1 + jj) ;k C k (1 + jj) d (1 + jj) jdj;k :This is integrable so jp(x )j Ck 0 (1 + jj)d . Similar estimates on jDx Dq(x + )j which arise from the given estimates for r show that p 2 S dso that R is a pseudo-dierential operator.We use Taylor's theorem on the middle variable of q(x +) to expand:X d q(x ) q(x + )= + q k (x ):!jjkThe remainder q k decays to arbitrarily high order in () and after integrationgives rise to a symbol in S d;k which may therefore be ignored. Weintegrate to concludeX Zp(x )= e ix d q(x )d + remainder!jjk= Xjjkd D yr(x y)! +aremainderx=yusing Lemma 1.1.2. This completes the proof of the lemma.We use this technical lemma to show that the pseudo-dierential operatorsform an algebra:

Operators on R m 17Lemma 1.2.3. Let P 2 d(U) and let Q 2 e(U). Then:(a) If U 0 is any open set with compact closure containing U, then P 2d(U 0 ) and (P ) P d D x p =! .(b) Assume that P and Q have the proper shapes so PQand pq are dened.Then PQ 2 d+e(U) and (PQ) P d p D x q=! .Proof: The fact that P lies in a larger space is only a slight bit oftechnical bother this fact plays an important role in considering boundaryvalue problems of course. Let (fg) = f g be the pointwise Hermitianinner product. Fix 2 C 1 0 (U 0 ) to be identically 1 on U and compute:(Pfg)==ZZe i(x;y) p(x )(y)f(y) g(x) dy d dxf(y) e i(y;x) p (x )(y)g(x) dy d dxsince the inner product is Hermitian. By approximating p (x ) by functionswith compact support, we can justify the use of Fubini's theoremto replace dy d dx by dx d dy and express:(Pfg)=Zf(y) e i(y;x) p (x )(y)g(x) dx d dy=(fP g)where we dene:P g(y) =Ze i(y;x) p (x )(y)g(x) dx d:This is an operator of the form discussed in Lemma 1.2.2 so P 2 d(U 0 )and we compute: X (P ) d D x p =!since =1on the support of p. This completes the proof of (a). We notethat we can delete the factor of from the expression for P g since it wasonly needed to prove P was a DO.We use (a) to prove (b). Since:Q g(y) =the Fourier inversion formula implies:Zd (Q g)=Ze i(y;x) q (x )g(x) dx de ;ix q (x )g(x) dx:

18 1.2. Pseudo-Differential OperatorsIf ~q is the symbol of Q ,thenwe interchange the roles of Q and Q to see:Z(Qg)= d e ;ix ~q (x )g(x) dx:Therefore:PQg(x)==ZZe ix p(x ) d (Qg)() de i(x;y) p(x )~q (y )g(y) dy dwhich is an operator of the form discussed in Lemma 1.2.2 if r(x y) =p(x )~q (y ). This proves PQ is a pseudo-dierential operator of thecorrect order. We compute the symbol of PQ to be: X d D y (p(x )~q (y ))=! evaluated at x = y.We use Leibnitz's formula and expand this in the form: X d p(x )D y d D y ~q =!! :The sum over yields the symbol of Q = Q so we conclude nally(PQ) X d p(x )D xq(x )=!which completes the proof.Let K(x y) be a smooth matrix valued function with compact x supportin U. If f is vector valued with compact support in U, we dene:P (K)(f)(x)=ZK(x y)f(y) dy:Lemma 1.2.4. Let K(x y) be smooth with compact x support in U, thenP (K) 2 ;1(U).Proof: We let () 2 C 1 0(R m ) with R () d =1. Dene:r(x y)=e i(y;x) ()K(x y)then this is a symbol in S ;1 of the sort discussed in Lemma 1.2.2. Itdenes an innitely smoothing operator. It is immediate that:P (K)(f)(x)=Ze i(x;y) r(x y)f(y) dy d:Conversely, it can be shown that any innitely smoothing map has asmooth kernel. In general, of course, it is not possible to represent an arbitrarypseudo-dierential operator by akernel. If P is smoothing enough,however, we can prove:

Operators on R m 19Lemma 1.2.5. Let r satisfy the hypothesis of Lemma 1.2.2 where d

20 1.2. Pseudo-Differential OperatorsK(x y) is arbitrarily smooth in (x y) and hence is C 1 . This completesthe proof.We note that in general K(x y) will become singular at x = y owing tothe presence of the terms jx ; yj ;2k if we do not assume the support of xis disjoint from the support of y.A dierential operator P is local in the sense that if f = 0 on some opensubset of U, then Pf = 0 on that same subset since dierentiation is apurely local process. DO's are not local in general since they are denedby theFourier transform which smears out the support. Nevertheless, theydo have a somewhat weaker property, they do not smear out the singularsupport of a distribution f. More precisely, let f 2 H s . If 2 C 1 0 (U),we dene the map f 7! f. If we take r(x y)=(x) and apply Lemma1.2.2, then we see that this is a pseudo-dierential operator of order 0.Therefore f 2 H s as well. This gives a suitable notion of restriction. Wesay thatf is smooth on an open subset U 0 of U if and only if f 2 C 1 forevery such . An operator P is said to be pseudo-local if f is smooth onU 0 implies Pf is smooth on U 0 .Lemma 1.2.7. Pseudo-dierential operators are pseudo-local.Proof: Let P 2 d(U) and let f 2 H s . Fix x 2 U 0 and choose 2C 1 0 (U 0 ) to be identically 1 near x. Choose 2 C 1 0 (U 0 ) with supportcontained in the set where is identically 1. We must verify that Pf issmooth. We compute:Pf = Pf + P(1 ; )f:By hypothesis, f is smooth so Pf is smooth. The operator P(1 ; )is represented by a kernel of the form (x)p(x )(1 ; (y)) which hasdisjoint x and y support. Lemma 1.2.6 implies P(1 ; )f is smoothwhich completes the proof.In Lemmas 1.2.2 and 1.2.3 we expressed the symbol of an operator as ainnite asymptotic series. We show that the algebra of symbols is completein a certain sense:Lemma 1.2.8. Let p j 2 S d j(U) where d j ! ;1. Then there existsp P j p j which is a symbol in our class. p is a unique modulo S ;1 .Proof: We may assume without loss of generality that d 1 >d 2 > !;1. We will construct p 2 S d 1. The uniqueness is clear so we mustprove existence. The p j all have support inside U we will construct p withsupport inside U 0 where U 0 is any open set containing the closure of U.Fix a smooth function such that:0 1 () =0for jj 1, () =1for jj 2.

Operators on R m 21We use to cut away the support near =0. Let t j ! 0 and dene:p(x ) = X j(t j )p j (x ):For any xed , (t j )=0for all but a nite number of j so this sum iswell dened and smooth in (x ). For j > 1 we havejp j (x )jC j (1 + jj) d j= C j (1 + jj) d 1(1 + jj) d j;d 1:If jj is large enough, (1 + jj) d j;d 1is as small as we like and therefore bypassing to a subsequence of the t j we can assumej(t j )p j (x )j 2 ;j (1 ;jj) d 1for j > 1.This implies that jp(x )j (C 1 + 1)(1 + jj) d 1. We use a similar argumentwith the derivatives and use a diagonalization argument on theresulting subsequences to conclude p 2 S d . The supports of all the p j arecontained compactly in U so the support of p is contained in U which iscontained in U 0 .We now apply exactly the same argument to p d2+ to assume thatp d2 + 2 S d 2. We continue in this fashion and use a diagonalizationargument on the resulting subsequences to conclude in the end that1Xj=j 0(t j )p j (x ) 2 S k for k = d j0 :Since p j ; (t j )p j 2 S ;1 , this implies p ; P j 0j=1p j 2 S k and completesthe proof.If K(x y) is smooth with compact x y support in U, then P (K) 2;1(U) denes a continuous operator from H s ! H t for any s t. LetjP j st denote the operator norm so jPfj t jP j st jfj s for any f 2S. It willbe convenient to be able to estimate jKj 1k in terms of these norms:Lemma 1.2.9. Let K(x y) be a smooth kernel with compact x y supportin U. Let P = P (K) be the operator dened by K. If k is a non-negativeinteger, then jKj 1k C(k)jP j ;kkProof: By arguing separately on each entry in the matrix K, we mayreduce ourselves to the R scalar case. Suppose rst k = 0. Choose 2C 1 0 (R m ) positive with (x) dx = 1. Fix points (x 0 y 0 ) 2 U U anddene:f n (x) =n m (n(x ; x 0 )) and g n (y) =n m (n(y ; y 0 )):

22 1.2. Pseudo-Differential OperatorsThen if n is large, f n and g n have compact support in U. Then:K(x 0 y 0 )= limn!1by Lemma 1.1.1. We estimateZf n (x)K(x y)g n (y) dy dx = limn!1 (f nPg n )j(f n Pg n )jjP j 00 jf n j 0 jg n j 0 = jP j 00 jj 2 0to complete the proof in this case.If jj k, jj k then:Dx D y K(x y)= limn!1Zf n (x) D x D y K(x y) g n(y) dy dx= limn!1Z ;Dx f n )K(x y)(D y g n(y) dy dx= limn!1 (D x f nPD y g n):We use Lemma 1.1.6 to estimate this byjD x f nj ;k jPD y g nj k jf n j 0 jP j ;kk jD y g nj ;kto complete the proof.jf n j 0 jg n j 0 jP j ;kk = jP j ;kk jj 2 0

1.3. Ellipticity and Pseudo-DierentialOperators on Manifolds.The norms wehave given to dene the spaces H s depend upon the Fouriertransform. In order to get a more invariant denition which can be usedto extend these notions to manifolds, we must consider elliptic pseudodierentialoperators.Let p 2 S d (U) be a square matrix and let U 1 be an open set withU 1 U. We say that p is elliptic on U 1 if there exists an open subset U 2with U 1 U 2 U 2 U and if there exists q 2 S ;d such that pq;I 2 S ;1and qp;I 2 S ;1 over U 2 . (Tosay that r 2 S ;1 over U 2 simply means theestimates of section 1.2 hold over U 2 . Equivalently, we assume r 2 S ;1for every 2 C 1 0 (U 2 )). This constant technical fuss over domains will beeliminated very shortly when we pass to considering compact manifoldsthe role of U 2 is to ensure uniform estimates over U 1 .It is clear that p is elliptic over U 1 if and only if there exists constantsC 0 and C 1 such that p(x ) is invertible for jj C 0 andjp(x ) ;1 jC 1 (1 + jj) ;d for jj C 0 , x 2 U 2 .We dene q = ()p ;1 (x ) where () is a cut-o function identically 0near =0and identically 1 near = 1. We used similar cuto functionsin the proof of Lemma 1.2.8. Furthermore, if p 0 2 S d;1 , then p is ellipticif and only if p + p 0 is elliptic adding lower order terms does not alterthe ellipticity. If p is a polynomial and P is a dierential P operator, thenp is elliptic if and only if the leading symbol L (p) = jj=d p (x) isinvertible for 6= 0.There exist elliptic operators of all orders. Let (x) 2 C 1 0 and denethe symbol p(x ) =(x)(1 + jj 2 ) d=2 I, then this is an elliptic symbol oforder d whenever (x) 6= 0.Lemma 1.3.1. Let P 2 d (U) be elliptic over U 1 then:(a) There exists Q 2 ;d (U) such that PQ ; I 0 and QP ; I 0over U 1 (i.e., (PQ; I) and (QP ; I) are innitely smoothing for any 2 C 1 0 (U 2 )).(b) P is hypo-elliptic over U 1 , i.e., if f 2 H s and if Pf is smooth over U 1then f is smooth over U 1 .(c) There exists a constant C such that jfj d C(jfj 0 + jPfj 0 ) for f 2C 1 0 (U 1 ). (Garding's inequality).Proof: We will dene Q to have symbol q 0 + q 1 + where q j 2 S ;d;j .We try to solve the equation(PQ; I) X jd p D x q j=! ; I 0:

24 1.3. Ellipticity and Pseudo-DifferentialWhen we decompose this sum into elements of S ;k , we conclude we mustsolveXd p D x q j=! =I if k =00 if k 6= 0:jj+j=kWe dene q 0 = q and then solve the equation inductively to dene:q k = ;q Xjj+j=kj

Operators on Manifolds 25disjoint from the support of j , then P ij is an innitely smoothing operatorwith a smooth kernel K ij (x y) by Lemma 1.2.6. Therefore h P ij is alsogiven by a smooth kernel and is a pseudo-dierential operator by Lemma1.2.4. Consequently, we may restrict attention to pairs (i j) such that thesupports of i and j intersect. We assume henceforth P is dened by asymbol p(x y) where p has arbitrarily small support in (x y).We rst suppose h is linear to motivate the constructions of the generalcase. Let h(x) =hx where h is a constant matrix. We equate:hx = x 1 hy = y 1 h t 1 = and denep 1 (x 1 1 y 1 )=p(x y):(In the above, h t denotes the matrix transpose of h). If f 2 C 1 0compute:(h P )f(x 1 )==ZZe i(x;y) p(x y)f(hy) dy de ih;1 (x 1 ;y 1 ) p(h ;1 x 1 h ;1 y 1 )f(y 1 )(e U), wejdet(h)j ;1 dy 1 d:We now use the identities h ;1 (x 1 ;y 1 ) =(x 1 ;y 1 ) 1 and j det(h)j d 1 =d to write:(h P )f(x 1 )==ZZe i(x 1;y 1 ) 1p(h ;1 x 1 h t 1 h ;1 y 1 )f(y 1 ) dy 1 d 1e i(x 1;y 1 ) 1p 1 (x 1 1 y 1 )f(y 1 ) dy 1 d 1 :This proves that (h P ) is a pseudo-dierential operator on e U. Since wedon't need to localize in this case, we compute directly that(h P )(x 1 1 )=p(h ;1 x 1 h t 1 ):We regard (x ) as P giving coordinates for T M when we expand any covectorin the form i dx i . This is exactly the transformation for thecotangent space so we may regard P as being invariantly dened onT R m .If h is not linear, the situation is somewhat more complicated. Letx ; y = h ;1 (x 1 ) ; h ;1 (y 1 )==Z 10Z 10ddt fh;1 (tx 1 +(1; t)y 1 )g dtd(h ;1 )(tx 1 +(1; t)y 1 ) (x 1 ; y 1 ) dt = T (x 1 y 1 )(x 1 ; y 1 )

26 1.3. Ellipticity and Pseudo-Differentialwhere T (x 1 y 1 ) is a square matrix. If x 1 = y 1 , then T (x 1 y 1 ) = d(h ;1 )is invertible since h is a dieomorphism. We localize using a partition ofunity to suppose henceforth the supports are small enough so T (x 1 y 1 ) isinvertible for all points of interest.We set 1 = T (x 1 y 1 ) t and compute:(h P )(f)(x 1 )===ZZZe i(x;y) p(x y)f(hy) dy de iT (x 1y 1 )(x 1 ;y 1 ) p(h ;1 x 1 h ;1 y 1 )f(y 1 )Jdy 1 de i(x 1;y 1 ) 1p 1 (x 1 1 y 1 )f(y 1 ) Jj det T (x 1 y 1 )j ;1 dy 1 d 1where J = j det(dh ;1 )j = j det T (y 1 y 1 )j. By Lemma 1.2.2, this denes apseudo-dierential operator of order d such that (h P )=p 1 modulo S d;1which completes the proof of (a). Since jdhj is uniformly bounded on U 1 ,jfj 0 Cjh fj 0 and jh fj 0 Cjfj 0 . If P is elliptic, then h P is elliptic ofthe same order. For d>0, choose P elliptic of order d and compute:jh fj d C(jh fj 0 + jPh fj 0 ) C(jfj 0 + j(h P )fj 0 ) Cjfj dwhich completes the proof of (b) if d 0. The result for d 0 follows byduality using Lemma 1.1.6(b).We introduce the spaces S d =S d;1 and dene L (P ) to be the elementdened by (P ) in this quotient. Let P and Q be pseudo-dierential operatorsof order d 1 and d 2 Then PQ is a pseudo-dierential operator oforder d 1 + d 2 and L (PQ)= L (P ) L (Q) since the remaining terms in theasymptotic series are of lower order. Similarly L (P ) = L (P ) . If wedene (x ) as coordinates for T (R m )by representing a cotangent vectorat a point x in the form P i dx i , then Lemma 1.3.2 implies L (P )isinvariantlydened on T (R m ). If P = P p D x isadierential operator thereis a natural identication of P jj=d p with the image of p in S d =S d;1so this denition of the leading symbol agrees with that given earlier.We now extend the results of section 1.2 to manifolds. Let M be asmooth compact Riemannian manifold without boundary. Let m be thedimension of M and let dvol or sometimes simply dx denote the Riemanianmeasure on M. In Chapter 2, we will use the notation j dvol j to denotethis measure in order to distinguish between measures and m-forms, butwe shall not bother with this degree of formalism here. We restrict toscalars rst. Let C 1 (M) be the space of smooth functions on M and letP : C 1 (M) ! C 1 (M) be a linear operator. We say that P is a pseudodierentialoperator of order d and write P 2 d(M)ifforevery open chart

Operators on Manifolds 27U on M and for every 2 C 1 0 (U), the localized operator P 2 d(U).We say that P is elliptic if P is elliptic where (x) 6= 0. If Q 2 d(U),we let P = Q for 2 C 1 0 (U). Lemma 1.3.2 implies P is a pseudodierentialoperator on M so there exists operators of all orders on M. Wedene:(M) = [ dd(M) and ;1(M) = \ dd(M)to be the set of all pseudo-dierential operators on M and the set of in-nitely smoothing operators on M.In any coordinate system, we dene (P )tothe symbol of the operatorP where =1near the point in question this is unique modulo S ;1 .The leading symbol is invariantly dened on T M, but the total symbolchanges under the same complicated transformation that the total symbolof a dierential operator does under coordinate transformations. Since weshall not need this transformaton law, we omit the statement it is implicitin the computations performed in Lemma 1.3.2.We dene L 2 (M) using the L 2 inner product(fg)=ZMf(x)g(x) dxjfj 2 0 =(ff):We let L 2 (M) be the completion of C 1 (M) in this norm. Let P : C 1 (M) !C 1 (M). We let P be dened by (Pfg)=(fP g) if such a P exists.Lemmas 1.3.2 and 1.2.3 imply that:Lemma 1.3.3.(a) If P 2 d(M), then P 2 d(M) and L (P )= L (P ) . In any coordinatechart, (P ) has a asymptotic expansion given by Lemma 1.2.3(a).(b) If P 2 d(M) and Q 2 e(M), then PQ 2 d+e(M) and L (PQ)= L (P ) L (Q). In any coordinate chart (PQ) has an asymptotic expansiongiven in Lemma 1.2.3(b).We use a partition of unity to dene the Sobolev spaces H s (M). Cover Mby a nite number of coordinate charts U i with dieomorphisms h i : O i !U i where the O i are open subsets of R m with compact closure. If f 2C 1 0 (U i ), we dene:jfj (i)s = jh i fj swhere we shall use the superscript (i) to denote the localized norm. Letf i g be a partition of unity subordinate to this cover and dene:jfj s = X j i fj (i)s :

28 1.3. Ellipticity and Pseudo-DifferentialIf 2 C 1 (M), we note that j i fj (i)s Cj i fj (i)s since multiplication bydenes a DO of order 0. Suppose fUj 0O0 j h0 j 0 jg is another possiblechoice to dene j 0 s. We estimate:j 0 j fj0(j) s X ij 0 j ifj 0(j)s :Since 0 j if 2 C 1 0by j (i)s soj 0 jfj 0(j)s(U i \ Uj 0 ), we can use Lemma 1.3.2 (b) to estimate j0(j) s X iCj 0 j i fj (i)s Cj i fj (i)s Cjfj sso that jfj 0 s Cjfj s . Similarly jfj s Cjfj 0 s. This shows these two normsare equivalent so H s (M) is dened independent of the choices made.We note that:XiX fj i fj (i)s g 2 jfj 2 s Cifj i fj (i)s g 2so by using this equivalent norm we conclude the H s (M) are topologicallyHilbert spaces. If f i g are given subordinate to the cover U i with i 0and = P i i > 0, we let i = i = and compute:XiXij i fj (i)sj i fj (i)s= X i= X ij i fj (i)sj ;1ifj (i)s C X i C X ij i fj (i)sj i fj (i)s= Cjfj sP to see the norm dened by i j ifj (i)s is equivalent to the norm jfj s aswell.Lemma 1.1.5 implies that jfj t jfj s if t < s and that the inclusion ofH s (M) ! H t (M) is compact. Lemma 1.1.7 implies given s > t > u and">0 we can estimate:jfj t "jfj s + C(")jfj u :We assume the coordinate charts U i are chosen so the union U i [ U j isalso contained in a larger P coordinate chart for all (i j). We decomposep 2 d(M) as P = ij P ij for P ij = i P j . By Lemma 1.2.1 we canestimate:j i P j fj (i)s Cj j fj (j)s+dso jPfj s Cjfj s+d and P extends to a continuous map from H s+d (M) !H s (M) for all s.

Operators on Manifolds 29We dene:jfj 1k = X ij i fj (i)1kas a measure of the sup norm of the k th derivatives of f. This is independentof particular choices made. Lemma 1.1.4 generalizes to:jfj 1k Cjfj s for s> m 2 + k:Thus H s (M) isa subset of C k (M) inthis situation.We choose i 2 C 1 (U i ) with P i 2 i =1then:j(fg)j = j X i( i f i g)j X ij( i f i g)j C X ij i fj (i)s j i gj (i);s Cjfj sjgj ;s :Thus the L 2 inner product gives a continuous map H s (M)H ;s (M) ! C.Lemma 1.3.4.(a) The natural inclusion H s ! H t is compact for s > t. Furthermore, ifs>t>uand if ">0, then jfj t "jfj s + C(")jfj u .(b) If s>k+ m then H 2 s(M) is contained in C k (M) and we can estimatejfj 1k Cjfj s .(c) If P 2 d(M) then P : H s+d (M) ! H s (M) is continuous for all s.(d) The pairing H s (M) H ;s (M) ! C given by the L 2 inner product isa perfect pairing.Proof: We have proved every assertion except the fact (d) that the pairingis a perfect pairing. We postpone this proof briey until after we havediscussed elliptic DO's.The sum of two elliptic operators need not be elliptic.However, thesum of two elliptic operators with positive symbols is elliptic. Let P i havesymbol (1+j i j 2 ) d=2 on U i and let i be a partition of unity. P = P i iP i i this is an elliptic DO of order d for any d, so elliptic operators exist.We let P be an elliptic DO of order d and let i be identically 1 onthe support of i . We use these functions to construct Q 2 ;d(M) soPQ; I 2 ;1(M) andQP ; I 2 ;1(M). In each coordinate chart, letP i = i P i then P i ; P j 2 ;1 on the support of i j . We construct Q ias the formal inverse to P i on the support of i , then Q i ; Q j 2 ;1 onthe support of i j since the formal inverse is unique. Modulo ;1 wehave P = P i iP P i iP i . We dene Q = P j Q j j and note Q has thedesired properties.It is worth noting we could also construct the formal inverse using aNeumann series. By hypothesis, there exists q so qp;I 2 S ;1 and pq ;I 2

30 1.3. Ellipticity and Pseudo-Differential OperatorsS ;1 , where p = L (P ). We construct Q 1 using a partition of unity so L (Q 1 )=q. We let Q r and Q l be dened by the formal series:(;1) k (PQ 1 ; I) kQ l = Q 1 XkQ r = Xk(;1) k (Q 1 P ; I) k Q 1to construct formal left and right inverses so Q = Q r = Q l modulo ;1.Let P be elliptic of order d>0. We estimate:jfj d j(QP ; I)fj d + jQP fj d Cjfj 0 + CjPfj 0 Cjfj dso we could dene H d using the norm jfj d = jfj 0 + jPfj 0 . We specializeto the following case. Let Q be elliptic of order d=2 andlet P = Q Q +1.Then Q Q is self-adjoint and non-negative sowe can estimate jfj 0 jPfj 0and we can dene j

1.4. Fredholm Operators and theIndex of a Fredholm Operator.Elliptic DO's are invertible modulo ;1. Lemma 1.3.4 will imply thatelliptic DO's are invertible modulo compact operators and that such operatorsare Fredholm. We briey review the facts we shall need concerningFredholm and compact operators.Let H be a Hilbert space and let END(H) denote the space of all boundedlinear maps T : H ! H. There is a natural norm on END(H) dened by:jTxjjT j = supx2H jxjwhere the sup ranges over x 6= 0. END(H) becomes a Banach space underthis norm. The operations of addition, composition, and taking adjoint arecontinuous. We let GL(H) be the subset of END(H) consisting of maps Twhich are 1-1 and onto. The inverse boundedness theorem shows that ifT 2 END(H) is 1-1 and onto, then there exists ">0 such that jTxj"jxjso T ;1 is bounded as well. The Neuman series:(1 ; z) ;1 =converges for jzj < 1. If jI ; T j < 1, we may express T = I ; (I ; T ). Ifwe deneS =1Xk=01Xk=0z k(I ; T ) kthen this converges in END(H) to dene an element S 2 END(H) soST = TS = I. Furthermore, this shows jT j ;1 (1 ;jI ; T j) ;1 so GL(H)contains an open neighborhood of I and the map T ! T ;1 is continuousthere. Using the group operation on GL(H), we see that GL(H) isaopensubset of END(H) and is a topological group.We say that T 2 END(H) is compact if T maps bounded sets to precompactsets|i.e., if jx n jC is a bounded sequence, then there exists asubsequence x nk so Tx nk ! y for some y 2 H. We let COM(H) denotethe set of all compact maps.Lemma 1.4.1. COM(H) is a closed 2-sided -ideal of END(H).Proof: It is clear the sum of two compact operators is compact. LetT 2 END(H) and let C 2 COM(H). Let fx n g be a bounded sequence inH then fTx n g is also a bounded sequence. By passing to a subsequence,we may assume Cx n ! y and CTx n ! z. Since TCx n ! Ty, this impliesCT and TC are compact so COM(H) is a ideal. Next let C n ! C inEND(H), and let x n be a bounded sequence in H. Choose a subsequence

32 1.4. Fredholm Operatorsx 1 n so C 1 x 1 n ! y 1 . We choose a subsequence of the x 1 n so C 2 x 2 n ! y 2 .By continuing in this way and then using the diagonal subsequence, wecan nd a subsequence we denote by x n n so C k (x n n) ! y k for all k. Wenote jCx n n ; C k x n nj jC ; C k jc. Since jC ; C k j ! 0 this shows thesequence Cx n n is Cauchy so C is compact and COM(H) is closed. Finallylet C 2 COM(H) and suppose C =2 COM(H). We choose jx n j 1 sojC x n ; C x m j " > 0 for all n, m. We let y n = C x n be a boundedsequence, then (Cy n ; Cy m x n ; x m )=jC x n ; C x m j 2 " 2 . Therefore" 2 jCy n ; Cy m jjx n ; x m j 2jCy n ; Cy m j so Cy n has no convergentsubsequence. This contradicts the assumption C 2 COM(H) and provesC 2 COM(H).We shall assume henceforth that H is a separable innite dimensionalspace. Although any two such Hilbert spaces are isomorphic, it is convenientto separate the domain and range. If E and F are Hilbert spaces, wedene HOM(EF) to be the Banach space of bounded linear maps fromE to F with the operator norm. We let ISO(EF) be the set of invertiblemaps in HOM(EF) and let COM(EF) be the closed subspace ofHOM(EF) of compact maps. If we choose a xed isomorphism of E withF , we may identify HOM(EF) = END(E), ISO(EF) = GL(E), andCOM(EF) =COM(E). ISO(EF) is a open subset of HOM(EF) andthe operation of taking the inverse is a continuous map from ISO(EF)toISO(F E). If T 2 HOM(EF), we dene:N(T )=f e 2 E : T (E) =0gR(T )=f f 2 F : f = T (e) for some e 2 E g(the null space)(the range).N(T )isalways closed, but R(T ) need not be. If ? denotes the operation oftaking orthogonal complement, then R(T ) ? =N(T ). We let FRED(EF)be the subset of HOM(EF) consisting of operators invertible modulo compactoperators:FRED(EF)=f T 2 HOM(EF):9S 1 S 2 2 HOM(F E) soS 1 T ; I 2 COM(E) and TS 2 ; I 2 COM(F ) g:We note this condition implies S 1 ;S 2 2 COM(F E)sowe can assume S 1 =S 2 if we like. The following lemma provides another useful characterizationof FRED(EF):Lemma 1.4.2. The following are equivalent:(a) T 2 FRED(EF)(b) T 2 END(EF) has dim N(T ) < 1, dim N(T ) < 1, R(T ) is closed,and R(T ) is closed.Proof: Let T 2 FRED(EF) and let x n 2 N(T ) with jx n j = 1. Thenx n = (I ; S 1 T )x n = Cx n . Since C is compact, there exists a convegent

Index of a Fredholm Operator 33subsequence. This implies the unit sphere in N(T ) is compact so N(T )is nite dimensional. Next let y n = Tx n and y n ! y. We may assumewithout loss of generality that x n 2 N(T ) ? . Suppose there exists a constantC so jx n j C. We have x n = S 1 y n +(I ; S 1 T )x n . Since S 1 y n ! S 1 yand since (I ; S 1 T ) is compact, we can nd a convergent subsequence sox n ! x and hence y = lim n y n = lim n Tx n = Tx is in the range of T soR(T ) will be closed. Suppose instead jx n j!1. If x 0 n = x n =jx n j we haveTx 0 n = y n =jx n j ! 0. We apply the same argument to nd a subsequencex 0 n ! x with Tx = 0, jxj = 1, and x 2 N(T ) ? . Since this is impossible,we conclude R(T ) is closed (by passing to a subsequence, one of these twopossibilities must hold). Since T S 1 ; I = C 1 and S 2 T ; I = C 2 weconclude T 2 FRED(F E) so N(T ) is nite dimensional and R(T ) isclosed. This proves (a) implies (b).Conversely, suppose N(T ) and N(T ) are nite dimensional and thatR(T )isclosed. We decompose:E =N(T ) N(T ) ? F =N(T ) R(T )where T :N(T ) ? ! R(T ) is 1-1 and onto. Consequently, we can nd abounded linear operator S so ST = I on N(T ) ? and TS = I on R(T ). Weextend S to be zero on N(T ) and computeST ; I = N(T ) and TS ; I = N(T )where denotes orthogonal projection on the indicated subspace. Sincethese two projections have nite dimensional range, they are compact whichproves T 2 FRED(EF).If T 2 FRED(EF) we shall say that T is Fredholm. There is a naturallaw of composition:Lemma 1.4.3.(a) If T 2 FRED(EF) then T 2 FRED(F E).(b) If T 1 2 FRED(EF) and T 2 2 FRED(F G) then T 2 T 1 2 FRED(EG).Proof: (a) follows from Lemma 1.4.2. If S 1 T 1 ;I 2 C(E) and S 2 T 2 ;I 2C(F ) then S 1 S 2 T 2 T 1 ; I = S 1 (S 2 T 2 ; I)T 1 +(S 1 T 1 ; I) 2 C(E). SimilarlyT 2 T 1 S 1 S 2 ; I 2 C(E).If T 2 FRED(EF), then we dene:index(T )=dimN(T ) ; dim N(T ):We note that ISO(EF)iscontained in FRED(EF) and that index(T )=0if T 2 ISO(EF).

34 1.4. Fredholm OperatorsLemma 1.4.4.(a) index(T )=; index(T ).(b) If T 2 FRED(EF) and S 2 FRED(F G) thenindex(ST) = index(T ) + index(S):(c) FRED(EF) is a open subset of HOM(EF).(d) index: FRED(EF) ! Z is continuous and locally constant.Proof: (a) is immediate from the denition. We computeso thatN(ST)=N(T ) T ;1 (R(T ) \ N(S))N(T S )=N(S ) (S ) ;1 (R(S ) \ N(T ))=N(S ) (S ) ;1 (R(T ) ? \ N(S) ? )index(ST) = dim N(T ) + dim(R(T ) \ N(S)) ; dim N(S ); dim(R(T ) ? \ N(S) ? )= dim N(T ) + dim(R(T ) \ N(S)) + dim(R(T ) ? \ N(S)); dim N(S ) ; dim(R(T ) ? \ N(S) ? ); dim(R(T ) ? \ N(S))= dim N(T )+dimN(S) ; dim N(S ) ; dim(R(T ) ? )= dim N(T )+dimN(S) ; dim N(S ) ; dim N(T )= index(S) + index(T )We prove (c) and (d) as follows. Fix T 2 FRED(EF). We decompose:E =N(T ) N(T ) ? and F =N(T ) R(T )where T :N(T ) ? ! R(T ) is 1-1 onto. We let 1 : E ! N(T ) be orthogonalprojection. We dene E 1 = N(T ) E and F 1 = N(T ) F tobe Hilbert spaces by requiring the decompositon to be orthogonal. LetS 2 HOM(EF), we dene S 1 2 HOM(E 1 F 1 ) by:S 1 (f 0 e) = 1 (e) (f 0 + S 1 (e)):It is clear jS 1 ; S 0 1j = jS ; S 0 j so the map S ! S 1 denes a continuousmap from HOM(EF) ! HOM(E 1 F 1 ). Let i 1 : E ! E 1 be the naturalinclusion and let 2 : F 1 ! F be the natural projection. Since N(T ) andN(T ) are nite dimensional, these are Fredholm maps. It is immediatefrom the denition that if S 2 HOM(EF) thenS = 2 S 1 i 1 :

Index of a Fredholm Operator 35If we let T = S and decompose e = e 0 e 1 and f = f 0 f 1 for e 0 2 N(T )and f 0 2 N(T ), then:T 1 (f 0 e 0 e 1 )=e 0 f 0 Te 1so that T 1 2 ISO(E 1 F 1 ). Since ISO(E 1 F 1 ) is an open subset, there exists">0sojT ;Sj

36 1.4. Fredholm OperatorsLemma 1.4.5. Let P : C 1 (V ) ! C 1 (W ) be an elliptic DO of order dover a compact manifold without boundary. Then:(a) N(P ) is a nite dimensional subset of C 1 (V ).(b) P : H s (V ) ! H s;d (W ) is Fredholm. P has closed range. index(P ) doesnot depend on the particular s chosen.(c) index(P ) only depends on the homotopy type of the leading symbol ofP within the class of elliptic DO's of order d.

1.5. Elliptic Complexes,The Hodge Decomposition Theorem,And Poincare Duality.Let V be a graded vector bundle. V is a collection of vector bundlesfV j g j2Z such that V j 6= f0g for only a nite number of indices j. We letP be a graded DO of order d. P is a collection of d th order pseudodierentialoperators P j : C 1 (V j ) ! C 1 (V j+1 ). We say that (P V ) is acomplex if P j+1 P j = 0 and L P j+1 L P j = 0 (the condition on the symbolfollows from P 2 = 0 for dierential operators). Wesay that (P V ) is ellipticif:N( L P j )(x )=R( L P j;1 )(x ) for 6= 0or equivalently if the complex is exact on the symbol level.We dene the cohomology of this complex by:H j (VP)=N(P j )= R(P j;1 ):We shall show later in this section that H j (VP) is nite dimensional if(P V ) is an elliptic complex. We then deneindex(P )= X j(;1) j dim H j (VP)as the Euler characteristic of this elliptic complex.Choose a xed Hermitian inner product on the bers of V . We usethatinner product together with the Riemannian metric on M to dene L 2 (V ).We dene adjoints with respect to this structure. If (P V ) is an ellipticcomplex, we construct the associated self-adjoint Laplacian: j =(P P ) j = P j P j + P j;1 P j;1: C 1 (V j ) ! C 1 (V j ):If p j = L (P j ), then L ( j )=p j p j + p j;1 p j;1. We can also express thecondition of ellipticity interms of j :Lemma 1.5.1. Let (P V ) be a d th order partial dierential complex.Then (P V ) is elliptic if and only if j is an elliptic operator of order 2dfor all j.Proof: We suppose that (P V ) is elliptic we must check L ( j ) is nonsingularfor 6= 0. Suppose (p j p j + p j;1 p j;1)(x )v = 0. If we dot thisequation with v, weseep j v p j v + p j;1 v p j;1 v = 0 so that p jv = p j;1 v =0. Thus v 2 N(p j ) so v 2 R(p j;1 ) so we can write v = p j;1 w. Sincep j;1 p j;1w = 0, we dot this equation with w to see p j;1 w p j;1 w =0sov = p j;1 w =0which proves j is elliptic. Conversely, let L ( j )benonsingularfor 6= 0. Since (P V ) is a complex, R(p j;1 ) is a subset of N(p j ).

38 1.5. Elliptic Complexes, Hodge DecompositionConversely, let v 2 N(p j ). Since L ( j ) is non-singular, we can expressv = (p j p j + p j;1 p j;1 )w. We apply p j to conclude p j p j p jw = 0. We dotthis equation with p j w to conclude p j p j p jw p j w = p j p jw p j p jw =0sop j p jw = 0. This implies v = p j;1 p j;1 w 2 R(p j;1) which completes theproof.We can now prove the following:Theorem 1.5.2 (Hodge decomposition theorem). Let (P V ) bea d th order DO elliptic complex. Then(a) We can decompose L 2 (V j ) = N( j )R(P j;1 )R(Pj ) as an orthogonaldirect sum.(b) N( j ) is a nite dimensional vector space and there is a natural isomorphismof H j (P V ) ' N( j ). The elements of N( j ) are smooth sectionsto V j .Proof: We regard j : H 2d (V j ) ! L 2 (V j ). Since this is elliptic, it is Fredholm.This proves N( j ) is nite dimensional. Since j is hypoelliptic,N( j ) consists of smooth sections to V . Since j is self adjoint and Fredholm,R( j )isclosedsowemay decompose L 2 (V j ) = N( j )R( j ). It isclear R( j ) is contained in the span of R(P j;1 ) and R(Pj ). We computethe L 2 inner product:(P j fP j;1g) =(fP j P j;1 g)=0since P j P j;1 = 0. This implies R(P j;1 ) and R(Pj ) are orthogonal. Letf 2 N( j ), we take the L 2 inner product with f to conclude0=( j ff) =(P j fP j f)+(P j;1fP j;1f)so N( j ) = N(P j ) \ N(P j;1 ). This implies R( j) contains the span ofR(P j;1 ) and R(Pj ). Since these two subspaces are orthogonal and R( j)is closed, R(P j;1 ) and R(Pj ) are both closed and we have an orthogonaldirect sum:L 2 (V j ) = N( j ) R(P j;1 ) R(Pj ):This proves (a).The natural inclusion of N( j ) into N(P j ) denes a natural map ofN( j ) ! H j (P V ) = N(P j )= R(P j;1 ). Since R(P j;1 ) is orthogonal toN( j ), this map is injective. If f 2 C 1 (V j ) and P j f =0,we can decomposef = f 0 j f 1 for f 0 2 N( j ). Since f and f 0 are smooth, j f 1 issmooth so f 1 2 C 1 (V j ). P j f = P j f 0 + P j j f 1 =0implies P j j f 1 =0.We dot this equation with P j f 1 to conclude:0=(P j f 1 P j P j P jf 1 + P j P j;1 P j;1 f 1)=(P j P jf 1 P j P jf 1 )

And Poincare Duality 39so Pj P jf 1 =0so j f 1 = P j;1 P j;1 f 1 2 R(P j;1 ). This implies f and f 0represent the same element of H j (P V ) so the map N( j ) ! H j (P V ) issurjective. This completes the proof.To illustrate these concepts, we discuss briey the de Rham complex.Let T M be the cotangent space of M. The exterior algebra (T M) isthe universal algebra generated by T M subject to the relation ^ = 0for 2 T M. If fe 1 ...e m g is a basis for T M and if I = f1 i 1

40 1.5. Elliptic Complexes, Hodge DecompositionA Riemannian metric on M denes ber metrics on p (T M). If fe i g isan orthonormal local frame for TM we take the dual frame fe i g for T M.For notational simplicity, we will simply denote this frame again by fe i g.The corresponding fe I g dene an orthonormal local frame for (T M).We deneinterior multiplication int(): p (T M) ! p;1 (T M)tobethedual of exterior multiplication. Then:int(e 1 )e I =ej for J = fi 2 ...i p g id i 1 =10 if i 1 > 1soint(e 1 ) ext(e 1 ) + ext(e 1 )int(e 1 )=I:If jj 2 denotes the length of the covector , then more generally:(i ext() ; i int()) 2 = jj 2 I:If : C 1 ( p (T M)) ! C 1 ( p;1 (T M)) is the adjoint of d, then L =i int(). We let d + d = (d + ) 2 = . L = jj 2 is elliptic. Thede Rham theorem gives a natural isomorphism from the cohomology of Mto H (d):H p (M C) =N(d p )= R(d p;1 )where we take closed modulo exact forms. The Hodge decomposition theoremimplies these groups are naturally isomorphic to the harmonic p-forms N( p ) which are nite dimensional. The Euler-Poincare characteristic(M) is given by:(M) = X (;1) p dim H p (M C) = X (;1) p dim N( p ) = index(d)is therefore the index of an elliptic complex.If M is oriented, we let dvol be the oriented volume element. The Hodge operator : p (T M) ! m;p (T M)isdened by the identity:! ^! =(! !)dvolwhere \" denotes the inner product dened by the metric. If fe i g is anoriented local frame for T M, then dvol = e 1 ^^e m and(e 1 ^^e p )=e p+1 ^^e m :The following identities are immediate consequences of Stoke's theorem: =(;1) p(m;p) and =(;1) mp+m+1 d :

And Poincare Duality 41Since=(d+) 2 = d+d we compute = so :N( p ) ! N( m;p )is an isomorphism. We may regard as an isomorphism : H p (M C) !H m;p (M C) in this description it is Poincare duality.The exterior algebra is not very suited to computations owing to thelarge number of signs which enter in the discussion of . When we discussthe signature and spin complexes in the third chapter, we will introduceCliord algebras whichmake the discussion of Poincaredualitymuch easier.It is possible to \roll up" the de Rham complex and dene:where e (T M)= M 2k(d + ) e : C 1 ( e (T M)) ! C 1 ( o (T M))M 2k (T M) and o (T M)= 2k+1 (T M)2k+1denote the dierential forms of even and odd degrees. (d + ) is an ellipticoperator since (d + ) e(d + ) e = is elliptic since dim e = dim o . (Inthis representation (d + ) e is not self-adjoint since the range and domainare distinct, (d + ) e =(d + ) o ). It is clear index(d + ) e = dim N( e )=dim N( o ) = (M). We can always \roll up" any elliptic complex toform an elliptic complex of the same index with two terms. Of course,the original elliptic complex does not depend upon the choice of a bermetric to dene adjoints so there is some advantage in working with thefull complex occasionally as we shall see later.We note nally that if m is even, we can always nd a manifold with(M) arbitrary so there exist lots of elliptic operators with non-zero index.We shall see that index(P )=0ifm is odd (and one must consider pseudodierentialoperators to get a non-zero index in that case).We summarize these computations for the de Rham complex in the following:Lemma 1.5.3. Let (M) =(T M) be the complete exterior algebra.(a) d: C 1 ( p (T M)) ! C 1 ( p+1 (T M)) is an elliptic complex. The symbolis L (d)(x )=i ext().(b) If is the adjoint, then L ()(x )=;i int().(c) If p = (d + d) p is the associated Laplacian, then N( p ) is nitedimensional and there are natural identications:N( p ) ' N(d p )= R(d p;1 ) ' H p (M C):(d) index(d) =(M) is the Euler-Poincare characteristic of M.(e) If M is oriented, we let be the Hodge operator. Then =(;1) p(m;p) and =(;1) mp+m+1 d :Furthermore, :N( p ) ' N( m;p ) gives Poincare duality.In the next chapter, we will discuss the Gauss-Bonnet theorem whichgives a formula for index(d) =(M) in terms of curvature.

1.6. The Heat Equation.Before proceeding with our discussion of the index of an elliptic operator,we must discuss spectral theory. We restrict ourselves to the context of acompact self-adjoint operator to avoid unnecessary technical details. LetT 2 COM(H) be a self-adjoint compact operator on the Hilbert space H.Letspec(T )=f 2 C :(T ; ) =2 GL(H) g:Since GL(H) is open, spec(T ) is a closed subset of C. If jj > jT j, theseriesg() =1Xn=0T n = n+1converges to dene an element of END(H). As (T ;)g() =g()(T ;) =;I, =2 spec(T ). This shows spec(T ) is bounded.Since T is self-adjoint, N(T ; )=f0g if =2 R. This implies R(T ; )is dense in H. Since Tx x is always real, j(T ; )x xj im()jjxj 2 sothat j(T ; )xj im()jjxj. This implies R(T ; ) is closed so T ; issurjective. Since T ; is injective, =2 spec(T )if 2 C ; R so spec(T )isa subset of R. If 2 [;jT j jT j], we dene E() =f x 2 H : Tx = x g.Lemma 1.6.1. Let T 2 COM(H) be self-adjoint. ThendimfE(;jT j) E(jT j)g > 0:Proof: If jT j = 0 then T = 0 and the result is clear. Otherwise choosejx n j =1so that jTx n j!jT j. We choose a subsequence so Tx n ! y. Let = jT j, we compute:jT 2 x n ; 2 x n j 2 = jT 2 x n j 2 + j 2 x n j 2 ; 2 2 T 2 x n x n 2 4 ; 2 2 jTx n j 2 ! 0:Since Tx n ! y, T 2 x n ! Ty. Thus 2 x n ! Ty. Since 2 6= 0, thisimplies x n ! x for x = Ty= 2 . Furthermore jT 2 x ; 2 xj = 0. Since(T 2 ; 2 ) = (T ; )(T + ), either (T + )x = 0 so x 2 E(;) 6= f0g or(T ; )(y 1 )=0for y 1 =(T + )x 6= 0so E() 6= f0g. This completes theproof.If 6= 0,the equation Tx = x implies the unit disk in E() iscompactand hence E() is nite dimensional. E() is T -invariant. Since T is selfadjoint,the orthogonal complement E() is also T -invariant. We take anorthogonal decomposition H = E(jT j) E(;jT j) H 1 . T respects thisdecomposition we let T 1 be the restriction of T to H 1 . Clearly jT 1 jjT j.If we have equality, then Lemma 1.6.1 implies there exists x 6= 0 in H 1

The Heat Equation 43so T 1 x = jT jx, which would be false. Thus jT 1 j < jT j. We proceedinductively to decompose:H = E( 1 ) E(; 1 ) E( n ) E(; n ) H nwhere 1 > 2 > > n and where jT n j < n . We also assume E( n ) E(; n ) 6= f0g. We suppose that the n do not converge to zero butconverge to " positive. For each n, we choose x n so jx n j = 1 and Tx n = n x n . Since T is self-adjoint, this is an orthogonal decomposition jx j ;x k j = p 2. Since T is compact, we can choose a convergent subsequence n x n ! y. Since n ! " positive, this implies T x n ! x. This is impossibleso therefore the n ! 0. We dene H 0 = n H n as a closed subset of H.Since jT j < n for all n on H 0 , jT j = 0 so T = 0 and H 0 = E(0). Thisdenes a direct sum decomposition of the form:H = M kE( k ) E(0)where the k 2 R are the non-zero subspace of the E( n ) and E(; n ).We construct a complete orthonormal system f n g 1 n=1 for H with either n 2 E(0) or n 2 E( k ) for some k. Then T n = n n so n is aneigenvector of T . This proves the spectral decomposition for self-adjointcompact operators:Lemma 1.6.2. Let T 2 COM(H) be self-adjoint. We can nd a completeorthonormal system for H consisting of eigenvectors of T .We remark that this need not be true if T is self-adjoint but not compactor if T is compact but not self-adjoint.We can use this lemma to prove the following:Lemma 1.6.3. Let P : C 1 (V ) ! C 1 (V ) be an elliptic self-adjoint DOof order d>0.(a) We can nd a complete orthonormal basis f n g 1 n=1 for L 2 (V ) of eigenvectorsof P . P n = n n .(b) The eigenvectors n are smooth and lim n!1 j n j = 1.(c) If we order the eigenvalues j 1 jj 2 jthen there exists a constantC > 0 and an exponent > 0 such that j n jCn if n>n 0 is large.Remark: The estimate (c) can be improved to show j n j n d=m but theweaker estimate will suce and is easier to prove.Proof: P : H d (V ) ! L 2 (V )isFredholm. P :N(P ) ? \ H d (M) ! N(P ) ? \L 2 (V ) is 1-1 and onto by Garding's inequality P 2 L 2 implies 2 H d .We let S denote the inverse of this map and extend S to be zero on thenite dimensional space N(P ). Since the inclusion of H d (M) into L 2 (V )is compact, S is a compact self-adjoint operator. S is often referred to

44 1.6. The Heat Equationas the Greens operator. We nd a complete orthonormal system f n g ofeigenvectors of S. If S n = 0 then P n = 0 since N(S) = N(P ). IfS n = n n 6=0then P n = n n for n = ;1 n . Since the n ! 0, thej n j ! 1. If k is an integer such that dk > 1, then P k ; k n is elliptic.Since (P k ; k n) n = 0, this implies n 2 C 1 (V ). This completes theproof of (a) and (b).By replacing P with P k we replace n by k n. Since kd > m if k is large,2we may assume without loss of generality that d > m in the proof of (c).2We dene:jfj 10 = sup jf(x)j for f 2 C 1 (V ):x2MWe estimate:jfj 10 Cjfj d C(jPfj 0 + jfj 0 ):Let F (a) be the space spanned by the j where j j j a and let n =dim F (a). We estimate n = n(a) asfollows. We havejfj 10 C(1 + a)jfj 0on F (a):Suppose rst V = M C is the trivial line bundle. Let c j be complexconstants, then this estimate shows:nX nXc j j (x) C(1 + a) j=1j=1If we take c j = j (x) then this yields the estimate:nXj=1 j (x) j (x) C(1 + a)jc j j 2 1=2: nX 1=2 j (x) j (x)j=1i.e.,nXj=1 j (x) j (x) C 2 (1 + a) 2 :We integrate this estimate over M to concluden C 2 (1 + a) 2 vol(M)or equivalentlyC 1 (n ; C 2 ) 1=2 a = j n jfrom which the desired estimate follows.

The Heat Equation 45If dim V = k, we choose a local orthonormal frame for V to decompose j into components u j for 1 u k. We estimate:nX nXc u j u j (x) C(1 + a) j=1j=1jc u j j 2 1=2If we let c u j = u j (x), then summing over u yields:nXnX jj (x) kC(1 + a) j=1j=1for u =1 ...k.1=2 jj (x)since each term on the left hand side of the previous inequality can beestimated seperately by the right hand side of this inequality. This meansthat the constants which arise in the estimate of (c) depend on k, butthiscauses no diculty. This completes the proof of (c).The argument given above for (c) was shown to us by Prof. B. Allard(Duke University) and it is a clever argument to avoid the use of theSchwarz kernel theorem.Let P : C 1 (V ) ! C 1 (V ) be an elliptic DO of order d > 0 which isself-adjoint. We say P has positive denite leading symbol if there existsp(x ): T M ! END(V ) such that p(x ) isa positive denite Hermitianmatrix for 6= 0and such that P ; p 2 S d;1 in any coordinate system.The spectrum of such aP is not necessarily non-negative, but it is boundedfrom below as we shall show. We construct Q 0 with leading symbol p pand let Q = Q 0 Q 0. Then by hypothesis P ; Q 2 S d;1 . We compute:(Pff)=((P ; Q)ff)+(Qf f)j((P ; Q)ff)j Cjfj d=2 j(P ; Q)fj d=2 Cjfj d=2 jfj d=2;1(Qf f) =(Q 0 fQ 0 f) and jfj 2 d=2

46 1.6. The Heat EquationLemma 1.6.4. Let P : C 1 (V ) ! C 1 (V ) be an elliptic DO of orderd > 0 which is self-adjoint with positive denite leading symbol. Thenspec(P ) is contained in [;C 1) for some constant C.We xsuch a P henceforth. The heat equation is the partial dierentialequation:ddt + P f(x t) =0for t 0 with f(x 0) = f(x):At least formally, this has the solution f(x t) =e ;tP f(x). We decomposef(x) = P c n n (x) for c n = (f n ) in a generalized Fourier series. Thesolution of the heat equation is given by:f(x t) = X ne ;t nc n n (x):Proceeding formally, we dene:K(t x y)= X ne ;t n n (x) n (y): V y ! V xso that:e ;tP f(x)=ZM= X nK(t x y)f(y) dvol(y)e ;t n n (x)ZMf(y) n (y) dvol(y):We regard K(t x y) as an endomorphism from the ber of V over y to theber of V over x.We justify this purely formal procedure using Lemma 1.3.4. We estimatethat:j n j 1k C(j n j 0 + jP j j 0 )=C(1 + j n j j ) where jd>k+ m 2 .Only a nite number of eigenvalues of P are negative by Lemma 1.6.4.These will not aect convergence questions, so we may assume > 0.Estimate:e ;t j t ;j C(j)e ;t=2to compute:jK(t x y)j 1k t ;j(k) C(k) X ne ;t n=2 :Since jj Cn for > 0 and n large, the series can be bounded byXn>0e ;tn=2

The Heat Equation 47which converges. This shows K(t x y) is an innitely smooth functionof (t x y) for t > 0 and justies all the formal procedures involved. WecomputeTr L 2 e ;tP = X e ;t n=ZMTr Vx K(t x x) dvol(x):We can use this formula to compute the index of Q:Lemma 1.6.5. Let Q: C 1 (V ) ! C 1 (W ) be an elliptic DO of orderd>0. Then for t>0, e ;tQQ and e ;tQQ are in ;1 with smooth kernelfunctions andindex(Q) =Tr e ;tQQ ; Tr e ;tQQfor any t>0.Proof: Since e ;tQQ and e ;tQQ have smooth kernel functions, they arein ;1 so wemust only prove the identityonindex(Q). We dene E 0 () =f 2 L 2 (V ) : Q Q = g and E 1 () = f 2 L 2 (W ) : QQ = g.These are nite dimensional subspaces of smooth sections to V and W .Because Q(Q Q)=(QQ )Q and Q (QQ )=(Q Q)Q , Q and Q denemaps:Q: E 0 () ! E 1 () and Q : E 1 () ! E 0 ():If 6= 0, = Q Q: E 0 () ! E 1 () ! E 0 () is an isomorphism sodim E 0 () =dimE 1 (). We compute:Tr e ;tQQ ; Tr e ;tQQ = X e ;t fdim E 0 () ; dim E 1 ()g= e ;t0 fdim E 0 (0) ; dim E 1 (0)g= index(Q)which completes the proof.If (Q V ) is a elliptic complex, we use the same reasoning to concludee ;t iis in ;1 with a smooth kernel function. We dene E i () =f 2L 2 (V i ): i = g. Then Q i : E i () ! E P i+1 () denes an acyclic complex(i.e., N(Q i )=R(Q i;1 )) if 6= 0so that (;1) i dim E i () =0for 6= 0.ThereforeX index(Q) = (;1) i Tr(e ;t i)iand Lemma 1.6.5 generalizes to the case of elliptic complexes which are notjust two term complexes.Let P be an elliptic DO of order d>0 which is self-adjoint with positivedenite leading symbol. Then spec(P ) is contained in [;C 1) for someconstant C. For =2 spec(P ), (P ; ) ;1 2 END(L 2 (V )) satises:j(P ; ) ;1 j = dist( spec(P )) ;1 =infn j ; nj ;1 :

48 1.6. The Heat EquationThe function j(P ; ) ;1 j is a continuous function of 2 C ; spec(P ). Weuse the Riemann integral with values in the Banach space END(L 2 (V )) todene:e ;tP = 1 Ze ;t (P ; ) ;1 d2iwhere is the path about [;C 1) given by the union of the appropriatepieces of the straight lines Re(+C +1) = Im() pictured below. We letR be the closed region of C consisting of together with that componentof C ; which does not contain [;C 1).Ry-axis;;;;;;;@@@@@@@R'?spectrum of Px-axisWe wish to extend (P ; ) ;1 to H s . We note thatj ; j ;1 C for 2R, 2 spec(P )so that j(P ; ) ;1 fj 0 0, choose k so kd s>kd; d and estimate:j(P ; ) ;1 fj s Cj(P ; ) ;1 fj kd C(1 + jj) k;1 jfj kd;d C(1 + jj) k;1 jfj s :

Similarly, ifs0 and ~ 2 R + . Therefore j( k n ; )j ;1 "j k nj ;1 " ;1 n ;kby Lemma 1.6.3. The convergence of the sum dening K k then followsusing the same arguments as given in the proof that e ;tP is a smoothingoperator.This technical lemma will be used in the next subsection to estimatevarious error terms which occur in the construction of a parametrix.

1.7. Local Formula forThe Index of an Elliptic Operator.In this subsection, let P : C 1 (V ) ! C 1 (V ) be a self-adjoint ellipticpartial dierential operator of order d>0. Decompose the symbol (P )=p d + + p 0 into homogeneous polymonials p j of order j in 2 T M. Weassume p d is a positive denite Hermitian matrix for 6= 0. Let the curve and the region R be as dened previously.The operator (P ; ) ;1 for 2Ris not a pseudo-dierential operator.We will approximate (P ; ) ;1 by a pseudo-dierential operator R() andthen use that approximation to obtain properties of exp(;tP ). Let U bean open subset of R m with compact closure. Fix d 2 Z. We make thefollowing denition to generalize that given in section 1.2:Definition. q(x ) 2 S k ()(U) is a symbol of order k depending onthe complex parameter 2Rif(a) q(x ) is smooth in (x ) 2 R m R m R, has compact x-supportin U and is holomorphic in .(b) For all ( ) there exist constants C such that:jD x D D q(x )jC (1 + jj + jj 1=d ) k;jj;djj :We say that q(x ) ishomogeneous of order k in () ifq(x t t d )=t k q(x ) for t 1:We think of the parameter as being of order d. It is clear if q is homogeneousin (), then it satises the decay conditions (b). Since thep j are polynomials in , they are regular at = 0 and dene elements ofS j (). If the p j were only pseudo-dierential, we would have to smoothout the singularity at =0whichwould destroy the homogeneity in()and they would not belong to S j () equivalently, d p j will not exhibitdecay in for jj > j if p j is not a polynomial. Thus the restriction todierential operators is an essential one, although it is possible to discusssome results in the pseudo-dierential case by taking more care with theestimates involved.We also note that (p d ; ) ;1 2 S ;d () and that the spaces S () forma symbol class closed under dierentiation and multiplication. They aresuitable generalizations of ordinary pseudo-dierential operators discussedearlier.We let k()(U) be the set of all operators Q(): C 1 0 (U) ! C 1 0 (U) withsymbols q(x ) inS k () having x-support in U we let q = Q. For anyxed , Q() 2 k(U) is an ordinary pseudo-dierential operator of orderk. The new features arise from the dependence on the parameter . Weextend the denition of given earlier to this wider class by:q X jq j

Local Formula for the Index 51if for every k > 0 there exists n(k) so n n(k) implies q ; P jn q j 2S ;k (). Lemmas 1.2.1, 1.2.3(b), and 1.3.2 generalize easily to yield thefollowing Lemma. We omit the proofs in the interests of brevity as they areidentical to those previously given with only minor technical modications.Lemma 1.7.1.(a) Let Q i 2 k i()(U) with symbol q i . Then Q 1 Q 2 2 k 1 +k 2()(U) hassymbol q where:q X d q 1D x q 2=! :(b) Given n>0 there exists k(n) > 0 such thatQ 2 ;k(n) ()(U) impliesQ() denes a continuous map from H ;n ! H n and we can estimate theoperator norm by:jQ()j ;nn C(1 + jj) ;n :(c) If h: U ! e U is a dieomorphism, then h : k()(U) ! k()(e U) and(h P ) ; p(h ;1 x 1 (dh ;1 (x 1 )) t 1 ) 2 S k;1 ()(e U):As before, we let ()(U) = S n n()(U) be the set of all pseudodierentialoperators depending on a complex parameter 2 R denedover U. This class depends on the order d chosen and also on the region R,but we supress this dependence in the interests of notational clarity. Thereis no analogue of the completeness of Lemma 1.2.8 for such symbols sincewe require analyticity in. Thus in constructing an approximation to theparametrix, we will always restrict to a nite sum rather than an innitesum.Using (c), we extend the class () to compact manifolds using a partitionof unity argument. (a) and (b) generalize suitably. We now turn tothe question of ellipticity. We wish to solve the equation:(R()(P ; )) ; I 0:Inductively we dene R() with symbol r 0 + r 1 + where r j 2 S ;d;j ().We dene p 0 j (x ) P = p j(x ) for j < d and p 0 d (x ) = p d(x ) ; .dThen (P ; ) =j=0 p0 j . We note that p0 j 2 S j () and that p 0 d;1 2S ;d () sothat (P ; ) is elliptic in a suitable sense. The essential featureof this construction is that the parameter is absorbed in the leadingsymbol and not treated as a lower order perturbation.The equation (R()(P ; )) I yields:Xjkd r j Dx p0 k =! I:

52 1.7. Local Formula for the IndexWe decompose this series into terms homogeneous of order ;n to write:XnXjj+j+d;k=nd r j D x p0 k =! Iwhere jk 0 and k d. There are no terms with n < 0, i.e., positiveorder. If we investigate the term with n = 0, we arrive at the conditionr 0 p 0 d = I so r 0 =(p d ; ) ;1 and inductively:r n = ;r 0Xjj+j+d;k=nj 0 so D x p0 k = D x p k in this sum. Therefore we mayreplace p 0 k by p k and writer n = ;r 0Xjj+j+d;k=nj0 be given. We can choose n 0 = n 0 (k) so thatjf(P ; ) ;1 ; R()gfj k C k (1 + jj) ;k jfj ;k for 2R, f 2 C 1 (V ).Thus (P ; ) ;1 is approximated arbitrarily well by the parametrix R()in the operator norms as !1.Proof: We compute:jf(P ; ) ;1 ; R()gfj k = j(P ; ) ;1 fI ; (P ; )R()gfj k C k (1 + jj )j(I ; (P ; )R())fj kby Lemma 1.6.6. Since I ; (P ; )R() 2 S ;n 0, we use Lemma 1.7.1 tocomplete the proof.RWe dene E(t) =2i1 e;t R() d. We will show shortly that this hasa smooth kernel K 0 (t x y). Let K(t x y) be the smooth kernel of e ;tP . We

Of an Elliptic Operator 53will use Lemma 1.2.9 to estimate the dierence between these two kernels.We compute:E(t) ; e ;tP = 12iZe ;t (R() ; (P ; ) ;1 ) d:We assume 0 < t < 1 and make a change of variables to replace t by .We use Cauchy's theorem to shift the resulting path t inside R back tothe original path where we have uniform estimates. This expresses:E(t) ; e ;tP = 12iWe estimate therefore:ZjE(t) ; e ;tP j ;kk C kZe ; (R(t ;1 ) ; (P ; t ;1 ))t ;1 d C k t kprovided n 0 is large enough. Lemma 1.2.9 impliesje ; (1 + t ;1 jj) ;k djLemma 1.7.3. Let k be given. If n 0 is large enough, we can estimate:jK(t x y) ; K 0 (t x y)j 1k C k t kfor 0

54 1.7. Local Formula for the IndexWe compute:K n (t x x)= 1 ZZe ;t r n (x ) d d:2i We make a change of variables to replace by t ;1 and by t ;1=d tocompute:K n (t x x)=t ; m d ;1 1 ZZe ; r n (x t ;1d t ;1 ) d d2i = t ; m d;1+n+dd= t n;md e n (x)12iZZe ; r n (x ) d dwhere we let this integral dene e n (x).Since p d (x ) is assumed to be positive denite, d must be even since p dis a polynomial in . Inductively we express r n as a sum of terms of theform:r j 10 q 1r j 20 q 2 ...r j k0where the q k are polynomials in (x ). The sum of the degrees of the q k isodd if n is odd and therefore r n (x ;) = ;r n (x ). If we replace by ; in the integral dening e n (x), we conclude e n (x) =0if n is odd.Lemma 1.7.4. Let P be a self-adjoint elliptic partial dierential operatorof order d > 0 such that the leading symbol of P is positive denite for 6= 0. Then:(a) If wechoose a coordinate system for M near a point x 2 M and choose alocal frame for V ,we can dene e n (x) using the complicated combinatorialrecipe given above. e n (x) depends functorially on a nite number of jetsof the symbol p(x ).(b) If K(t x y) is the kernel of e ;tP thenK(t x x) 1Xn=0t n;md e n (x) as t ! 0 +i.e., given any integer k there exists n(k) such that: XK(t x x) ;nn(k)t n;md e n (x)1k

(d) e n (x) =0if n is odd.Of an Elliptic Operator 55Proof: (a) is immediate. We computed thatK 0 (t x x)=Xn 0n=0t n;md e n (x)so (b) follows from Lemma 1.7.3. Since K(t x x) does not depend on thecoordinate system chosen, (c) follows from (b). We computed (d) earlierto complete the proof.We remark that this asymptotic representation of K(t x x) exists fora much wider class of operators P . We refer to the literature for furtherdetails. We shall give explicit formulas for e 0 , e 2 , and e 4 in section 4.8 forcertain examples arising in geometry.The invariants e n (x) =e n (x P ) are sections to the bundle of endomorphisms,END(V ). They have anumber of functorial properties. We sumarizesome of these below.Lemma 1.7.5.(a) Let P i : C 1 (V i ) ! C 1 (V i ) be elliptic self-adjoint partial dierentialoperators of order d > 0 with positive denite leading symbol. We formP = P 1 P 2 : C 1 (V 1 V 2 ) ! C 1 (V 1 V 2 ). Then P is an elliptic selfadjointpartial dierential operator of order d > 0 with positive deniteleading symbol and e n (x P 1 P 2 )=e n (x P 1 ) e n (x P 2 ).(b) Let P i : C 1 (V i ) ! C 1 (V i ) be elliptic self-adjoint partial dierentialoperators of order d>0 with positive denite leading symbol dened overdierent manifolds M i . We letP = P 1 1+1 P 2 : C 1 (V 1 V 2 ) ! C 1 (V 1 V 2 )over M = M 1 M 2 . Then P is an elliptic self-adjoint partial dierentialoperator of order d>0 with positive denite leading symbol over M ande n (x P )=Xp+q=ne p (x 1 P 1 ) e q (x 2 P 2 ):(c) Let P : C 1 (V ) ! C 1 (V ) be an elliptic self-adjoint partial dierentialoperator of order d>0 with positive denite leading symbol. We decomposethe total symbol of P in the form:Xp(x ) = p (x) :jjdFix a local frame for V and a system of local coordinates on M. We letindices a, b index the local frame and let p = p ab give the components

56 1.7. Local Formula for the Indexof the matrix p . We introduce formal variables p ab= = Dx p ab forthe jets of the symbol of P . Dene ord(p ab= ) = jj + d ;jj. Thene n (x P ) can be expressed as a sum of monomials in the fp ab= g variableswhich are homogeneous of order n in the jets of the symbol as discussedabove and with coecients which depend smoothly on the leading symbolfp ab g jj=d .(d) If the leading symbol of P is scalar, then the invariance theory can besimplied. We donotneedto introduce the components of the symbol explicitlyand can compute e n (x P ) as a non-commutative polynomial whichis homogeneous of order n in the fp = g variables with coecients whichdepend smoothly upon the leading symbol.Remark: The statement of this lemma is somewhat technical. It will beconvenient, however, to have these functorial properties precisely stated forlater reference. This result suces to study the index theorem. We shallgive one more result which generalizes this one at the end of this sectionuseful in studying the eta invariant. The objects we are studying have abigrading one grading comes from counting the number of derivatives inthe jets of the symbol, while the other measures the degree of homogeneityin the () variables.Proof: (a) and (b) follow from the identities:so the kernels satisfy the identities:e ;t(P 1P 2 ) = e ;tP 1 e ;tP 2e ;t(P 11+1P 2 ) = e ;tP 1 e ;tP 2K(t x x P 1 P 2 )=K(t x x P 1 ) K(t x x P 2 )K(t x x P 1 1+1 P 2 )=K(t x 1 x 1 P 1 ) K(t x 2 x 2 P 2 ):We equate equal powers of t in the asymptotic series:Xtn;md e n (x P 1 P 2 )X X t n;md e n (x P 1 ) t n;md e n (x P 2 )X n;mt d e n (x P 1 1+1 P 2 ) Xtp;m 1d e p (x 1 P 1 ) X q;m 2t d e q (x 2 P 2 )to complete the proof of (a) and (b). We note that the multiplicative property(b) is a direct consequence of the identity e ;t(a+b) = e ;ta e ;tb it wasfor this reason we worked with the heat equation. Had we worked insteadwith the zeta function to study Tr(P ;s ) the corresponding multiplicativeproperty would have been much more dicult to derive.

Of an Elliptic Operator 57We prove (c) as follows: expand p(x ) = P j p j(x ) into homogeneouspolynomials where p j (x ) = P jj=j p (x) . Suppose for the momentthat p d (x ) =p d (x )I V is scalar so it commutes with every matrix. Theapproximate resolvant is given by:r 0 =(p d (x ) ; ) ;1r n = ;r 0Xjj+j+d;k=nj

58 1.7. Local Formula for the Indexthe symbol. The remainder of the argument is the same performing thed d integral yields a smooth function of the leading symbol as a coecientof such a term, but this function is not in general rational of course. Thescalar case is much simpler as we don't need to introduce the componentsof the matrices explicitly (although the frame dependence is still there ofcourse) since r 0 can be commuted. This is atechnical point, but one oftenuseful in making specic calculations.We dene the scalar invariantsa n (x P )=Tr e n (x P )where the trace is the ber trace in V over the point x. These scalarinvariants a n (x P ) inherit suitable functorial properties from the functorialproperties of the invariants e n (x P ). It is immediate that:Tr L 2 e ;tP =ZM1Xn=01Xn=0Tr Vx K(t x x)dvol(x)t m;ndZMt m;nd a n (P )a n (x P )dvol(x):R where a n (P ) =M a n(x P )dvol(x) is the integrated invariant. This isa spectral invariant of P which can be computed from local informationabout the symbol of P .Let (P V ) be an elliptic complex of dierential operators and let i bethe associated Laplacians. We dene:a n (x P )= X i(;1) i Tr e n (x i )then Lemma 1.6.5 and the remark which follows this lemma implyindex(P )= X i(;1) i Tr e ;t i1Xn=0t n;mdZMa n (x P )dvol(x):Since the left hand side does not depend on the parameter t, we conclude:Theorem 1.7.6. Let (P V ) be a elliptic complex of dierential operators.(a) a n (x P ) can be computed in any coordinate system and relative toanylocal frames as a complicated combinatorial expression in the jets of P andof P up to some nite order. a n =0if n is odd.

(b)ZMOf an Elliptic Operator 59a n (x P )dvol(x) =index(P ) if n = m0 if n 6= m:We note as an immediate consequence that index(P ) = 0 if m is odd.The local nature of the invariants a n will play avery important role in ourdiscussion of the index theorem. We will develop some of their functorialproperties at that point. We give one simple example to illustrate this fact.Let : M 1 ! M 2 be a nite covering projection with ber F . We canchoose the metric on M 1 to be the pull-back of a metric on M 2 so that is an isometry. Let d be the operator of the de Rham complex. a n (x d) =a n (x d) since a n is locally dened. This implies:(M 1 )=ZM 1a m (x d)dvol(x) =jF jZM 2a m (x d)dvol(x) =jF j(M 2 )so the Euler-Poincare characteristic is multiplicative under nite coverings.A similar argument shows the signature is multiplicative under orientationpreserving coverings and that the arithmetic genus is multiplicative underholomorphic coverings. (We will discuss the signature and arithmetic genusin more detail in Chapter 3).We conclude this section with a minor generalization of Lemma 1.7.5which will be useful in discussing the eta invariant in section 1.10:Lemma 1.7.7. Let P : C 1 (V ) ! C 1 (V ) be an elliptic self-adjoint partialdierential operator of order d > 0 with positive denite leading symboland let Q be an auxilary partial dierential operator on C 1 (V ) of ordera 0. Qe ;tP is an innitely smoothing operator with kernel QK(t x y).There is an asymptotic expansion on the diagonal:fQK(t x y)gj x=y 1Xn=0t (n;m;a)=d e n (x Q P ):The e n are smooth local invariants of the jets of the symbols of P and Q ande n =0if n + a is odd. If we let Q = P q D x ,we dene ord(q )=a ;jj.If the leading symbol of P is scalar, we can compute e n (x Q P ) as anon-commutative polynomial in the variables fq p = g which is homogeneousof order n in the jets with coecients which depend smoothly onthe fp g jj=d variables. This expression is linear in the fq g variables anddoes not involve the higher jets of these variables. If the leading symbolof P is not scalar, there is a similar expression for the matrix componentsof e n in the matrix components of these variables. e n (x Q P ) is additiveand multiplicative inthe sense of Lemma 1.7.5(a) and (b) with respect todirect sums of operators and tensor products over product manifolds.Remark: In fact it is not necessary to assume P is self-adjoint to dene e ;tPand to dene the asymptotic series. It is easy to generalize the techniques

60 1.7. Local Formula for the Indexwe have developed to prove this lemma continues to hold true if we onlyasume that det(p d (x ) ; ) 6= 0 for 6= 0 and Im() 0. This impliesthat the spectrum of P is pure point and contained in a cone about thepositive real axis. We omit the details.Proof: Qe ;tP has smooth kernel QK(t x y) where Q acts as a dierentialoperator on the x variables. One representative piece is:q (x)D x K n(t x y)=q (x)D x= q (x)ZX+=e i(x;y) e n (t x ) d!! !Ze i(x;y) D xe n (t x ) d:We dene q (x) = q (x) ! where + = , then a representative! !term of this kernel has the form:q (x)Ze i(x;y) D x e n(t x ) dwhere q (x) is homogeneous of order a;jj;jj in the jets of the symbolof Q.We suppose the leading symbol is scalar to simplify the computationsthe general case is handled similarly using Cramer's rule. We expresse n = 12iZe ;t r n (x ) dwhere r n is a sum of terms of the form r nj (x) r j 0. When we dierentiatesuch a term with respect to the x variables, wechange the order in the jets ofthe symbol, but do not change the () degree of homogeneity. ThereforeQK(t x y) isa sum of terms of the form:where:q (x)Z Ze ;t r j 0 (x ) d e i(x;y) ~r nj (x) dr nj is of order n + jj in the jets of the symbol,; d ; n = ;jd+ jj:We evaluate on the diagonal to set x = y. This expression is homogeneousof order ;d ; n + jj in (), so when we make the appropriate change ofvariables we calculate this term becomes:t (n;m;jj)=d q (x)~r nj (x) Z ne ; r j 0 (x ) d o d:

Of an Elliptic Operator 61This is of order n+jj+a;jj;jj = n+a;jj = in the jets of the symbol.The exponentoft is (n;m;jj)=d =(;a+jj;m;jj)=d =(;a;m)=d.The asymptotic formula has the proper form if we index by the order inthe jets of the symbol . It is linear in q and vanishes if jj + jj is odd.As d is even, we compute: + a = n + a ;jj + a = jd; d ;jj +2a ;jj jj + jj mod 2so that this term vanishes if + a is odd. This completes the proof.

1.8. Lefschetz Fixed Point Theorems.Let T : M ! M be a continuous map. T denes a map on the cohomologyof M and the Lefschetz number of T is dened by:L(T )= X (;1) p Tr(T on H p (M C)):This is always an integer. We present the following example to illustratethe concepts involved. Let M = S 1 S 1 be the two dimensional toruswhich we realize as R 2 modulo the integer lattice Z 2 . We dene T (x y) =(n 1 x + n 2 y n 3 x + n 4 y) where the n i are integers. Since this preserves theinteger lattice, this denes a map from M to itself. We compute:T (1) = 1T (dx) =n 1 dx + n 2 dyT (dx ^ dy) =(n 1 n 4 ; n 2 n 3 )dx ^ dyL(T )=1+(n 1 n 4 ; n 2 n 3 ) ; (n 1 + n 4 ):T (dy) =n 3 dx + n 4 dyOf course, there are many other interesting examples.We computed L(T ) in the above example using the de Rham isomorphism.We let T = p (dT ): p (T M) ! p (T M) to be the pull-backoperation. It is a map from the ber over T (x) to the ber over x. SincedT = T d, T induces a map on H p (MC) =ker(d p )= image(d p;1 ). If Tis the identity map, then L(T )=(M). The perhaps somewhat surprisingfact is that Lemma 1.6.5 can be generalized to compute L(T ) in terms ofthe heat equation.Let p = (d + d) p : C 1 ( p T M) ! C 1 ( p T M) be the associatedLaplacian. We decompose L 2 ( p T M) = L E p() into the eigenspacesof p . We let (p ) denote orthogonal projection on these subspaces, andwe dene T (p ) =(p )T : E p () ! E p (). It is immediate that:Tr(T e ;t p)= X e ;t Tr(T (p )):Since dT = T d and d = d, for 6= 0we get a chain map between theexact sequences:E p;1 ()d;! E p ()d;! E p+1 () ?yT ? yT?yT L 2 ( p;1 )?yd;! L 2 ( p )?yd;! L 2 ( p+1 ) ?yE p;1 ()d;! E p ()d;! E p+1 ()

Lefschetz Fixed Point Theorems 63Since this diagram commutes and since the two rows are long exact sequencesof nite dimensional vector spaces, a standard result in homologicalalgebra implies:Xp(;1) p Tr(T (p )) = 0 for 6= 0:It is easy to see the corresponding sum for = 0 yields L(T ), so Lemma1.6.5 generalizes to give a heat equation formula for L(T ).This computation was purely formal and did not depend on the fact thatwe were dealing with the de Rham complex.Lemma 1.8.1. Let (P V ) be an elliptic complex over M and let T : M !M be smooth. We assume given linear maps V i (T ): V i (Tx) ! V i (x) so thatP i (V i (T )) = V i (T )P i . Then T induces a map on H p (P V ). We deneL(T ) P = X p(;1) p Tr(T on H p (P V )):Then we can compute L(T ) P = P p Tr(V i(T )e ;t p) where p = (P P +PP ) p is the associated Laplacian.The V i (T ) are a smooth linear action of T on the bundles V i . They willbe given by the representations involved for the de Rham, signature, spin,and Dolbeault complexes as we shall discuss in Chapter 4. We usuallydenote V i (T ) by T unless it is necessary to specify the action involved.This Lemma implies Lemma 1.6.5 if we take T = I and V i (T ) = I.To generalize Lemma 1.7.4 and thereby get a local formula for L(T ) P , wemust place some restrictions on the map T . We assume the xed pointset of T consists of the nite disjoint union of smooth submanifolds N i .Let (N i )=T (M)=T (N i ) be the normal bundle over the submanifold N i .Since dT preserves T (N i ), it induces a map dT on the bundle (N i ). Wesuppose det(I ; dT ) 6= 0 as a non-degeneracy condition there are nonormal directions left xed innitesimally by T .If T is an isometry, this condition is automatic. We can construct anon-example by dening T (z) =z=(z + 1): S 2 ! S 2 . The only xed pointis at z =0and dT (0) = I, so this xed point is degenerate.If K is the kernel of e ;tP ,we pull back the kernel to dene T (K)(t x y)= T (x)K(t T x y). It is immediate T K is the kernel of T e ;tP .Lemma 1.8.2. Let P be an elliptic partial dierential operator of orderd>0 which is self-adjoint and which has a positive denite leading symbolfor 6= 0. Let T : M ! M be a smooth non-degenerate map and letT : V T (x) ! V x R be a smooth linear action. If K is the kernel of e ;tP thenTr(T e ;tP )=M Tr(T K)(t x x) dvol(x). Furthermore:(a) If T has no xed points, j Tr(T e ;tP )jC n t ;n as t ! 0 + for any n.

64 1.8. Lefschetz Fixed Point Theorems(b) If the xed point set of T consists of submanifolds N i of dimension m i ,we will construct scalar invariants a n (x) which depend functorially upon anite number of jets of the symbol and of T . The a n (x) are dened overN i andTr(T e ;tP ) X i1Xn=0t n;m idZN ia n (x) dvol i (x):dvol i (x) denotes the Riemannian measure on the submanifold.It follows that if T has no xed points, then L(T ) P =0.Proof: Let fe n r n K n g be as dened in section 1.7. The estimates ofLemma 1.7.3 show that jT K ; P nn 0T K n j 1k C(k)t ;k as t ! 0 +for any k if n 0 = n 0 (k). We may therefore replace K by K n in proving (a)and (b). We recall that:e n (t x )= 12iWe use the homogeneity ofr n to express:Ze n (t x )= 12i = t n 1d2iZe ;t r n (x ) d:e ; r n (x t ;1 )t ;1 dZ= t n d e n (x t 1 d )e ; r n (x t 1 d ) dwhere we dene e n (x ) =e n (1x) 2 S ;1 . Then:K n (t x y)=Z= t n;mde i(x;y) e n (t x ) dZe i(x;y)t; 1 d e n (x ) d:This shows that:T K n (t x x) =t n;mdZT (x)e i(Tx;x)t;1=d e n (Tx) d:We must study terms which have the form:Ze i(Tx;x)t;1=d Tr(T (x)e n (Tx)) d dxwhere the integral is over the cotangent space T (M). We use the methodof stationary phase on this highly oscillatory integral. We rst bound

Lefschez Fixed Point Theorems 65jTx ; xj " > 0. Using the argument developed in Lemma 1.2.6, weintegrate by parts to bound this integral by C(n k ")t k as t ! 0 for anyk. If T has no xed points, this proves (a). There is a slight amount ofnotational sloppiness here since we really should introduce partitions ofunity and coordinate charts to dene Tx; x, but we supress these detailsin the interests of clarity.We can localize the integral to an arbitrarily small neighborhood of thexed point set in proving (b). We shall assume for notational simplicity thatthe xed point set of T consists of a single submanifold N of dimension m 1 .The map dT on the normal bundle has no eigenvalue 1. We identify withthe span of the generalized eigenvectors of dT on T (M)j N which correspondto eigenvalues not equal to 1. This gives a direct sum decomposition overN:T (M) =T (N) and dT = I dT :We choose a Riemannian metric for M so this splitting is orthogonal. Weemphasize that these are bundles over N and not over the whole manifoldM.We describe the geometry near the xed manifold N using the normalbundle . Let y = (y 1 ...y m1 ) be local coordinates on N and letf~s 1 ...~s m;m1 g be a local orthonormal frame for . We use this localorthonormal frame to introduce ber coordinates z =(z 1 ...z m;m1 ) for by decomposing any ~s 2 in the form:~s = X jz j ~s j (y):We let x =(y z) be local coordinates for . The geodesic ow identies aneighborhood of the zero section of the bundle with a neighborhood ofN in M so we can also regard x =(y z) as local coordinates on M.We decomposeT (x) =(T 1 (x)T 2 (x))into tangential and ber coordinates. Because the Jacobian matrix has theform:dT (y 0) = I 00 dT !we conclude that T 1 (x) ; y must vanish to second order in z along N.We integrate Tr(T K n )(t x x) along a small neighborhood of the zerosection of . We shall integrate along the bers rst to reduce this toan integral along N. We decompose = ( 1 2 ) corresponding to thedecomposition of x =(y z). LetD() =f (y z) :jzj 1 g

66 1.8. Lefschetz Fixed Point Theoremsbe the unit disk bundle of the normal bundle. Let U = T (D()) be thecotangent bundle of the unit disk bundle of the normal bundle. We assumethe metric chosen so the geodesic ow embeds D() inM. We parametrizeU by f (y z 1 2 ):jzj 1 g. Let s = t 1 d . Modulo terms which vanish toinnite order in s, we compute:I def=ZMTr((T K n )(t x x)) dxZ= s n;m e i(T 1 (yz);y) 1 s ;1 e i(T 2 (yz);z) 2 s ;1U Tr(T (x)e n (Tx 1 2 )) d 1 d 2 dz dy:The non-degeneracy assumption on T means the phase function w =T 2 (y z);z denes a non-degenerate change of variables if we replace (y z)by (y w). This transforms the integral into the form:ZI = s n;m e i(T 1 (yw);y) 1 s ;1 e iw 2s ;1Ujdet(I ; dT 2 )j ;1 Tr(T (y w)e n (T (y w) 1 2 ) d 1 d 2 dwdywhere U is the image of U under this change. We nowmake another changeof coordinates to let w = s ;1 w. As s ! 0, s ;1 U will converge to T (v).Since dw = s m;m 1dw this transforms the integral I to the form:I = s n;m 1Zs ;1 Ue i(T 1(ysw);y) 1 s ;1 e iw 2j det(I ; dT 2 ) ;1 j(y sw) Tr(T (y sw)e n (T (y sw) 1 2 )) d 1 d 2 dw dy:Since the phase function T 1 (y w) ; y vanishes to second order at w =0,the function (T 1 (y sw) ; y)s ;1 is regular at s =0. Dene:e 0 n(y w 1 2 s)=e i(T 1(ysw);y) 1 s ;1 j det(I ; dT 2 )(y sw)j Tr(T (y sw)e n (T (y sw) 1 2 )):This vanishes to innite order in 1 , 2 at 1 and is regular at s = 0. Tocomplete the proof of Lemma 1.8.2, we must evaluate:s (n;m 1)Zs ;1 Ue 0 n(y w 1 2 s)e iw 2d 1 d 2 dw dy:We expand e 0 n in a Taylor series in s centered at s =0. If we dierentiatee 0 n with respect to s a total of k times and evaluate at s = 0, then theexponential term disappears and we are left with an expression which is

Lefschez Fixed Point Theorems 67polynomial in w and of degree at worst 2k. It still vanishes to innite orderin ( 1 2 ). We decompose:e 0 n = X jj 0s jXjj2jwhere " is the remainder term.We rst study a term of the form:s n+j;m 1Zc j (y 1 2 )w + s j 0"(y w 1 2 s)jwj

68 1.8. Lefschetz Fixed Point TheoremsLemma 1.8.3. Let P be a second order elliptic partial dierential operatorwith leading symbol jj 2 I. Let T : M ! M be smooth with nondegenerateisolated xed points. Then:Tr T e ;tP = X iTr(T ) j det(I ; dT )j ;1 (x i )summed over the xed point set.We combine Lemmas 1.8.1 and 1.8.2 to constuct a local formula forL(T ) P to generalize the local formula for index(P )given by Theorem 1.7.6we will discuss this further in the fourth chapter.We can use Lemma 1.8.3 to prove the classical Lefschetz xed pointformula for the de Rham complex. Let T : V ! V be a linear map, then itis easily computed that:We compute:Xp(;1) p Tr p (T )=det(I ; T ):L(T )= X p(;1) p Tr(T e ;t p)= X pi(;1) p Tr( p dT ) j det(I ; dT )j ;1 (x i )= X i= X idet(I ; dT ) j det(I ; dT )j ;1 (x i )sign det(I ; dT )(x i )summed over the xed point set of T . This proves:Theorem 1.8.4 (Classical Lefschetz Fixed Point Formula).Let T : M ! M be smooth with isolated non-degenerate xed points. Then:L(T )= X p= X i(;1) p Tr(T on H p (M C))sign det(I ; dT )(x i )summed over the xed point set.Remark: We can generalize Lemma 1.8.2 to study Tr(T Qe ;tP ) where Qis an auxilary dierential operator of order a. Just as in lemma 1.7.7 wemay obtain an asymptotic series:Tr(T Qe ;tP ) X i1Xn=0t (n;m i;a)=d a n (x Q P )dvol i (x):

Lefschez Fixed Point Theorems 69We shall omit the details as the additional terms created by the operatorQ are exactly the same as those given in the proof of Lemma 1.7.7. Eachterm a n is homogeneous of order n in the jets of the symbols of (Q P ) andof the map T in a suitable sense.

1.9. Elliptic Boundary Value Problems.In section 1.7 we derived a local formula for the index of an ellipticpartial-dierential complex using heat equation methods. This formula willlead to a heat equation proof of the Atiyah-Singer index theorem whichwe shall discuss later. Unfortunately, it is not known at present how togive a heat equation proof of the Atiyah-Bott index theorem for manifoldswith boundary in full generality. We must adopt a much stronger notionof ellipticity to deal with the analytic problems involved. This will yielda heat equation proof of the Gauss-Bonnet theorem for manifolds withboundary which we shall discuss in the fourth chapter.Let M be a smooth compact manifold with smooth boundary dM andlet P : C 1 (V ) ! C 1 (V ) be a partial dierential operator of order d > 0.We let p = L (P )betheleading symbol of P . We assume henceforth thatp is self-adjoint and elliptic|i.e., det p(x ) 6= 0for 6= 0. Let R denotethe non-zero positive/negative real numbers. It is immediate thatdetfp(x ) ; g 6=0for () 6= (0 0) 2 T (M) fC ; R + ; R ; gsince p is self-adjoint and elliptic.We x a ber metric on V and a volume element on M to dene theglobal inner product ( ) on L 2 (V ). We assume that P is formally selfadjoint:(Pfg)=(fPg)for f and g smooth section with supports disjoint from the boundary dM.We must impose boundary conditions to make P self-adjoint. For example,if P = ;@ 2 =@x 2 on the line segment [0A], then P is formally self-adjointand elliptic, but we must impose Neumann or Dirichlet boundary conditionsto ensure that P is self-adjoint with discrete spectrum.Near dM we let x =(y r) wherey =(y 1 ...y m;1 ) is a system of localcoordinates on dM and where r is the normal distance to the boundary. Weassume dM = fx : r(x) =0g and that @=@r is the inward unit normal. Wefurther normalize the choice of coordinate by requiring the curves x(r) =(y 0 r) for r 2 [0) are unit speed geodesics for any y 0 2 dM. The inwardgeodesic ow identies a neighborhood of dM in M with the collar dM [0) for some > 0. The collaring gives a splitting of T (M) =T (dM) T (R) and a dual splitting T (M) = T (dM) T (R). We let = (z)for 2 T (dM) and z 2 T (R) reect this splitting.It is convenient to discuss boundary conditions in the context of gradedvector bundles. A graded bundle U over M is a vector bundle U togetherwith a xed decompositionU = U 0 U d;1into sub-bundles U j where U j = f0g is permitted in this decomposition.We let W = V 1 d = V V restricted to dM be the bundle of

Elliptic Boundary 71Cauchy data. If W j = V j dM is the (j +1) st factor in this decomposition,then this denes a natural grading on W we identify W j with the bundleof j th normal derivatives. The restriction map:dened by:: C 1 (V ) ! C 1 (W )(f) =(f 0 ...f d;1 )where f j = D j r fj dM =(;i) j @j f@r j dMassigns to any smooth section its Cauchy data.Let W 0 be an auxiliary graded vector bundle over dM. We assume thatdim W = d dim V is even and that 2dimW 0 =dimW . Let B: C 1 (W ) !C 1 (W 0 ) be a tangential dierential operator over dM. Decompose B =B ij forB ij : C 1 (W i ) ! C 1 (W 0 j)and assume thatord(B ij ) j ; i:It is natural to regard a section to C 1 (W i ) as being of order i and to denethe graded leading symbol of B by:L (B g (B) ij (y )= ij )(y ) if ord(B ij )=j ; i0 if ord(B ij )

72 1.9. Elliptic BoundaryWe say that (P B) is elliptic with respect to C;R + if det(p(x );) 6= 0on the interior for all () 6= (0 0) 2 T (M) C ; R + and if on theboundary there always exists a unique solution to this ordinary dierentialequation such that g (B)(y )f = f 0 for any prescribed f 0 2 W 0 . In asimilar fashion, we dene ellipticity with respect to C ; R + ; R ; if theseconditions hold for 2 C ; R + ; R ; .Again, we illustrate these notions for the operator P = ;@ 2 =@x 2 and aboundary condition at x =0. Since dim M = 1, there is no dependence on and we must simply study the ordinary dierential equation:;f 00 = f with lim f(r) =0 ( 6= 0):r!1If 2 C ;R + , then we can express = 2 for Im() > 0. Solutions to theequation ;f 00 = f are of the form f(r) =ae ir + be ;ir . The decay at1implies b = 0 so f(r) =ae ir . Such a function is uniquely determined byeither its Dirichlet or Neumann data at r =0and hence P is elliptic withrespect to either Neumann or Dirichlet boundary conditions.We assume henceforth that P B is self-adjoint and that (P B) is ellipticwith respect to either the cone C ; R + or the cone C ; R + ; R ; . It isbeyond the scope of this book to develop the analysis required to discusselliptic boundary value problems we shall simply quote the required resultsand refer to the appropriate papers of Seeley and Greiner for further details.Lemmas 1.6.3 and 1.6.4 generalize to yield:Lemma 1.9.1. Let P : C 1 (V ) ! C 1 (V ) be an elliptic partial dierentialoperator of order d>0. Let B be a boundary condition. We assume (P B)is self-adjoint and elliptic with respect to C ; R + ; R ; .(a) We can nd a complete orthonormal system f n g 1 n=1 for L 2 (V ) withP n = n n .(b) n 2 C 1 (V ) and satisfy the boundary condition B n =0.(c) n 2 R and lim n!1 j n j = 1. If we order the n so j 1 j j 2 jthen there exists n 0 and > 0 so that j n j >n for n>n 0 .(d) If (P B) is elliptic with respect to the cone C ; R + , then the n arebounded from below and spec(P B ) is contained in [;C 1) for some C.on [0] with Dirichlet boundary condi-For example, if P = ; @2@x 2tions, then the spectral resolution becomesnq2 sin(nx) o 1n=1and the correspondingeigenvalues are n 2 .If (P B) is elliptic with respect to the cone C ; R + , then e ;tP Bsmoothing operator with smooth kernel K(t x x 0 ) dened by:is aK(t x x 0 )= X ne ;t n n (x) n (x 0 ):Lemma 1.7.4 generalizes to yield:

Value Problems 73Lemma 1.9.2. Let (P B) be elliptic with respect to the cone C ; R + ,and be of order d>0.(a) e ;tP Bis an innitely smoothing operator with smooth kernel K(t x x 0 ).(b) On the interior we dene a n (x P )=Tr e n (x P ) be as in Lemma 1.7.4.On the boundary we dene a n (y P B) using a complicated combinatorialrecipe which depends functorially on a nite number of jets of the symbolsof P and of B.(c) As t ! 0 + we have an asymptotic expansion:Tr e ;tP B= X ne ;t n=ZM1Xn=0+Tr K(t x x)dvol(x)t n;md1Xn=0ZMt n;m+1da n (x P )dvol(x)ZdMa n (y P B)dvol(y):(d) a n (x P ) and a n (y P B) are invariantly dened scalar valued functionswhich do not depend on the coordinate system chosen nor on the local framechosen. a n (x P )=0if n is odd, but a n (y P B) is in general non-zero forall values of n.The interior term a n (x P ) arises from the calculus described previously.We rst construct a parametrix on the interior. Since this parametrix willnot have the proper boundary values, it is necessary to add a boundarycorrection term which gives rise to the additional boundary integrandsa n (y P B).To illustrate this asymptotic series, we let P = ; @2; e(x) on the interval[0A] where e(x) is a real potential. Let B be the modied Neumann@x2 boundary conditions: f 0 (0) + s(0)f(0) = f 0 (A)+s(A)f(A)=0. This is ellipticand self-adjoint with respect to the cone C;R + . It can be computedthat:Tr e ;tP B (4t) ;1=2 Zf1+t e(x)+t 2 (e 00 (x)+3e(x) 2 )=6+g dx 1=2; s(0) ; s(A)+ 1 2 + t+ t 4 ; e(0) + e(A)+2s 2 (0) + 2s 2 (A) + The terms arising from the interior increase by integer powers of t whilethe terms from the boundary increase by powers of t 1=2 . In this integral,dx is ordinary unnormalized Lebesgue measure.

74 1.9. Elliptic BoundaryIn practice, we shall be interested in rst order operators which areelliptic with respect to the cone C;R + ;R ; and which have both positiveand negative spectrum. An interesting measure of the spectral asymmetryof such an operator is obtained by studying Tr(P B e ;tP B 2 ) which will bediscussed in more detail in the next section. To study the index of anelliptic operator, however, it suces to study e ;tP B 2 it is necessary towork with PB 2 of course to ensure that the spectrum is positive so thisconverges to dene a smoothing operator. There are two approaches whichare available. The rst is to work with the function e ;t2 and integrateover a path of the form:XXXXXXXXXXXXXX:9XXXXXXXXXXXXXXto dene e ;tP B 2 directly using the functional calculus. The second approachis to dene e ;tP B 2 using the operator P 2 with boundary conditions Bf =BPf = 0. These two approaches both yield the same operator and givean appropriate asymptotic expansion which generalizes Lemma 1.9.2.We use the heat equation to construct a local formula for the indexof certain elliptic complexes. Let Q: C 1 (V 1 ) ! C 1 (V 2 ) be an ellipticdierential operator of order d > 0. Let Q : C 1 (V 2 ) ! C 1 (V 1 ) be theformal adjoint. Let B 1 : C 1 (W 1 ) ! C 1 (W1) 0 be a boundary condition forthe operator Q. We assume there exists a boundary condition B 2 for Q of the same form so that(Q B1 ) =(Q ) B2 :We form:P = Q Q and B = B 1 B 2so P : C 1 (V 1 V 2 ) ! C 1 (V 1 V 2 ). Then P B will be self-adjoint. Weassume that (P B) is elliptic with respect to the cone C ; R + ; R ; .This is a very strong assmption which rules out many interesting cases,but is necessary to treat the index theorem using heat equation methods.We emphasize that the Atiyah-Bott theorem in its full generality doesnotfollow from these methods.

Value Problems 75Since P 2 = Q Q + QQ , it is clear that PB 2 decomposes as the sumof two operators which preserve C 1 (V 1 ) and C 1 (V 2 ). We let S be theendomorphism +1 on V 1 and ;1 on V 2 to take care of the signs it is clearPB 2 S = S P B 2 . The same cancellation lemma we have used previously yields:index(Q 1 B 1 ) = dim N(Q B1 ) ; dim N(Q B 2)=Tr(S e ;tP 2 B ):(The fact that this denition of the index agrees with the denition givenin the Atiyah-Bott paper follows from the fact that B: C 1 (W ) ! C 1 (W 1 )is surjective). Consequently, the application of Lemma 1.9.2 yields:Theorem 1.9.3. Let Q: C 1 (V 1 ) ! C 1 (V 2 ) be an elliptic dierentialoperator of order d>0. Let Q : C 1 (V 2 ) ! C 1 (V 1 ) be the formal adjointand dene P = QQ : C 1 (V 1 V 2 ) ! C 1 (V 1 V 2 ). Let B = B 1 B 2 bea boundary condition such that (P B) is elliptic with respect to the coneC ; R + ; R ; and so that P B is self-adjoint. Dene S =+1on V 1 and ;1on V 2 then:(a) index(Q B1 )=Tr(S e ;tP 2 B ) for all t.(b) There exist local invariants a n (x Q) and a n (y Q B 1 ) such that:Tr(S e ;tP 2 B ) 1Xn=0t n;m2d+ZM1Xn=0a n (x Q)dvol(x)t n;m+12dZdMa n (y Q B 1 )dvol(y):(c)Za n (x Q)dvol(x)+MZdMa n;1 (y Q B 1 )dvol(y)=index(QB1 ) if n = m0 if n 6= m:This gives a local formula for the index there is an analogue of Lemma1.7.5 giving various functorial properties of these invariants we will discussin the fourth chapter where we shall discuss the de Rham complex and theGauss-Bonnet theorem for manifolds with boundary.We now specialize henceforth to rst order operators. We decomposep(x ) = X je j (x) jand near the boundary express:p(y 0z)=e 0 (y) z +m;1Xj=1e j (y) j = e 0 (y) z + p(y 00):

76 1.9. Elliptic BoundaryWe study the ordinary dierential equation:or equivalently:fip 0 @=@r + p(y 00)gf(r)=f(r);ip 0 f@=@r + ip ;10With this equation in mind, we dene:p(y 00) ; ip;1 0 )gf(r) =0:(y )=ip ;10 (p(y 00) ; ):Lemma 1.9.4. Let p(x ) be self-adjoint and elliptic. Then (y ) hasno purely imaginary eigenvalues for () 6= (0 0) 2 T (dM) (C ; R + ;R ; ).Proof: We suppose the contrary and set (y )v = izv where z is realand v 6= 0. This implies that:(p(y 00) ; )v = p 0 zvor equivalently that:p(y 0;z)v = v:Since p is self-adjoint, this implies = 0 so 6= 0 which contradicts theellipticity of p.We dene bundles ()over T (dM)fC;R + ;R ; g;(0 0) to be thespan of the generalized eigenvectors of which correspond to eigenvalueswith positive/negative real part + ; = V . The dierential equationhas the form:f@=@r + gf =0so the condition lim r!1 f(r) =0implies that f(0) 2 + (). Thus + ()is the bundle of Cauchy data corresponding to solutions to this ODE. Sinced =1,the boundary condition B is just an endomorphism:and we conclude:B: V jdM ! W 0Lemma 1.9.5. Let P be a rst order formally self-adjoint elliptic dierentialoperator. Let B be a 0 th order boundary condition. Dene(y )=ip ;10 (p(y 00) ; ):Then has no purely imaginary eigenvalues and we dene () to bebundles of generalized eigenvectors corresponding to eigenvalues with positivereal and negative real parts. (P B) is elliptic with respect to the cone

Value Problems 77C ; R + ; R ; if and only if B: + () ! W 0 is an isomorphism for all() 6= (0 0) 2 T (dM) fC ; R + ; R ; g. P B is self-adjoint if and onlyif p 0 N(B) is perpendicular to N(B).Proof: We have checked everything except the condition that P B be selfadjoint.Since P is formally self-adjoint, it is immediate that:(Pfg) ; (fPg)=ZdM(;ip 0 fg):We know that on the boundary, both f and g have values in N(B) sincethey satisfy the boundary condition. Thus this vanishes identically if andonly if (p 0 fg) =0for all fg 2 N(B) which completes the proof.We emphasize that these boundary conditions are much stronger thanthose required in the Atiyah-Bott theorem to dene the index. They aremuch more rigid and avoid some of the pathologies which can occur otherwise.Such boundary conditions do not necessarily exist in general as weshall see later.We shall discuss the general case in more detail in Chapter 4. We completethis section by discussing the one-dimensional case case to illustratethe ideas involved. We consider V =[0 1] C 2 and let P be the operator:P = ;i @ @r0 11 0!:At the point 0, we have:!() =;i 0 1 :1 0; aThus if Im() > 0 we have + () = a while if Im() < 0 we haven; + () = ao;a. We let B ; be the boundary projection which projects0on the rst factor|N(B) = a we take the same boundary conditionat x = 1 to dene an elliptic self-adjoint operator P B . Let P 2 = ; @2@x 2with boundary conditions: Dirichlet boundary conditions on the rst factorand Neumann boundary conditions on the second factor. The index of theproblem is ;1.

1.10. Eta and Zeta Functions.We have chosen to work with the heat equation for various technicalreasons. However, much of the development of the subject has centeredon the zeta function so we shall briey indicate the relationship betweenthese two in this section. We shall also dene the eta invariant which playsan important role in the Atiyah-Singer index theorem for manifolds withboundary.Recall that ; is dened by:;(s) =Z 10t s;1 e ;t dtfor Re(s) > 0. We use the functional equation s;(s) = ;(s +1) to extend; to a meromorphic function on C with isolated simple poles ats = 0 ;1 ;2 ... . Let P : C 1 (V ) ! C 1 (V ) be an elliptic self-adjointpartial dierential operator of order d > 0 with positive denite leadingsymbol. We showed in section 1.6 that this implies spec(P ) is containedin [;C 1) for some constant C. We now assume that P itself is positivedenite|i.e., spec(P ) is contained in [" 1) for some ">0.Proceeding formally, we dene P s by:P s = 12iZ s (P ; ) ;1 dwhere is a suitable path in the half-plane Re() > 0. The estimates ofsection 1.6 imply this is smoothing operator if Re(s) 0 with a kernelfunction given by:L(s x y)= X n s n n(x) n (y)this converges to dene a C k kernel if Re(s)

Eta and Zeta Functions 79P away from zero. Using the growth estimates of section 1.6 it is immediatethat:Z 1t s;1 Tr e ;tP dt = r(s P )1denes an entire function of s. If 0 < t < 1 we use the results of section1.7 to expandXTr e ;tP = d a n (P )+Ot n 0 ;mda n (P )=nn 0t n;mZMa n (x P )dvol(x)where a n (x P ) is a local scalar invariant of the jets of the total symbol ofP given by Lemma 1.7.4.If we integrate the error term which isOt n 0 ;md from 0 to 1, we denea holomorphic function of s for Re(s)+ n 0;mdbetween 0 and 1 to conclude:X;(s)Tr(P ;s )=;(s)(s P )=> 0. We integrate t s;1 t (n;m)dn ; (n 0;m)d. This proves ;(s)(s P )extends to a meromorphic function on C with isolated simple poles. Furthermore,the residue at these poles is given by a local formula. Since ;has isolated simple poles at s =0 ;1 ;2 ... we conclude that:Lemma 1.10.1. Let P be a self-adjoint, positive, elliptic partial dierentialoperator of order d>0 with positive denite symbol. We dene:Z(s P )=Tr(P ;s 1)=;(s) ;1 t s;1 Tr e ;tP dt:This is well dened and holomorphic for Re(s) 0. It has a meromorphicextension to C with isolated simple poles at s = (m ; n)=d for n =0 1 2 .... The residue of at these local poles is a n (P );((m ; n)=d) ;1 :If s = 0 ;1 ... is a non-positive integer, then (s P ) is regular at thisvalue of s and its value is given by a n (P )Res s=(m;n)=d ;(s). a n (P ) isthe invariant R given in the asymptotic expansion of the heat equation.a n (P ) =M a n(x P )dvol(x) where a n (x P ) is a local invariant of thejets of the total symbol of P . a n vanishes if n is odd.Remark: If A is an auxilary dierential operator of order a, we can deneZ(s A P )=Tr(AP ;s 1)=;(s) ;1 t s;1 Tr(Ae ;tP ) dt:00

80 1.10. Eta and Zeta FunctionsWe apply Lemma 1.7.7 to see this has a meromorphic extension to C withisolated simple poles at s = (m + a ; n)=d. The residue at these poles isgiven by the generalized invariants of the heat equationa n (A P );f(m + a ; n)=dg :There is a similar theorem if dM 6= and if B is an elliptic boundarycondition as discussed in section 1.9.If P is only positive semi-denite so it has a nite dimensional spacecorresponding to the eigenvalue 0, we dene:X(s P )= n >0and Lemma 1.10 extends to this case as well with suitable modications.For example, if M = S 1 is the unit circle and if P = ;@ 2 =@ 2 is theLaplacian, then:X (s P )=2 n ;2sn>0 ;snis essentially just the Riemann zeta function. This has a simple isolatedpole at s =1=2.If Q: C 1 (V 1 ) ! C 1 (V 2 ), then the cancellation lemma of section 1.6implies that:" ;s index(Q) =(s Q Q + ") ; (s QQ + ")for any " > 0 and for any s. is regular at s = 0 and its value is givenby a local formula. This gives a local formula for index(Q). Using Lemma1.10.1 and some functorial properties of the heat equation, it is not dicultto show this local formula is equivalent to the local formula given by theheat equation so no new information has resulted. The asymptotics of theheat equation are related to the values and residues of the zeta function byLemma 1.10.1.If we do not assume that P is positive denite, it is possible to denea more subtle invariant which measures the dierence between positiveand negative spectrum. Let P be a self-adjoint elliptic partial dierentialoperator of order d>0 which is not necessarily positive denite. We denethe eta invariant by:X X X(s P )= ; (; n ) ;s = n >0 ;sn n

Eta and Zeta Functions 81Again, this is absolutely convergent and denes a holomorphic function ifRe(s) 0.We can also discuss the eta invariant using the heat equation. Theidentity:implies that:Z 10t (s;1)=2 e ;2t dt =;((s +1)=2) sign()jj ;sZ 1(s P ) = ;((s +1)=2) ;1 t (s;1)=2 TrfPe ;tP 2 g dt:0Again, the asymptotic behavior at t = 0 is all that counts in producing polessince this decays exponentially at 1 assuming P has no zero eigenvectorif dim N(P ) > 0 a seperate argument must be made to take care of thiseigenspace. This can be done by replacing P by P + " and letting " ! 0.Lemma 1.7.7 shows that there is an asymptotic series of the form:forTr(Pe ;tP 2 ) a n (P P 2 )=ZM1Xn=0t n;m;d2d a n (P P 2 )a n (x P P 2 )dvol(x):This is a local invariant of the jets of the total symbol of P .We substitute this asymptotic expansion into the expression for to see:X(s P );((s +1)=2) ;1 2d=ds + n ; m a n(P P 2 )+r n0nn 0where r n0 is holomorphic on a suitable half-plane. This proves has asuitable meromorphic extension to C with locally computable residues.Unfortunately, while it was clear from the analysis that was regular ats =0,it is not immediate that is regular at s =0since ; does not havea pole at 1 to cancel the pole which may be introduced when n = m. in2fact, if one works with the local invariants involved, a n (x P P 2 ) 6= 0 ingeneral so the local poles are in fact present at s =0. However, it is a factwhich we shall discuss later that is regular at s =0and we will dene~(p) = 1 f(0p) + dim N(p)g2 modZ(we reduce modulo Z since has jumps in 2Z as eigenvalues cross theorigin).

82 1.10. Eta and Zeta FunctionsWe compute a specic example to illustrate the role of in measuringspectral asymmetry. Let P = ;i@=@ on C 1 (S 1 ), then the eigenvalues ofp are the integers so (s p) = 0 since the spectrum is symmetric about theorigin. Let a 2 R and dene:P a = P ; a(s P a )= X n2Zsignfn ; agjn ; aj ;s :We dierentiate this with respect to the parameter a to conclude:d Xda (s P (a)) = s((n ; a) 2 ) ;(s+1)=2 :n2ZIf we compare this with the Riemann zeta function, thenXn2Z((n ; a) 2 ) ;(s+1)=2has a simple pole at s =0with residue 2 and therefore:dda (s P (a)) s=0=2:Since vanishes when a =0,weintegrate this with respect to a to conclude:(0P(a)) = 2a +1 mod 2Z and ~(P (a)) = a + 1 2is non-trivial. (We must work modulo Z since spec P (a) is periodic withperiod 1 in a).We used the identity:dda (s P a)=;s Tr( _ P a (P 2 a ) ;(s+1)=2 )in the previous computation it in fact holds true in general:lemma 1.10.2. Let P (a) be a smooth 1-parameter family of elliptic selfadjointpartial dierential operators of order d > 0. Assume P (a) has nozero spectrum for a in the parameter range. Then if \" denotes dierentiationwith respect to the parameter a,_(s P (a)) = ;s Tr( _ P (a)(P (a) 2 ) ;(s+1)=2 ):If P (a) has zero spectrum, we regard (s P (a)) 2 C=Z.Proof: If we replaceP by P k for an odd positive integer k, then (s P k )= (ks P). This shows that it suces to prove Lemma 1.10.2 for d 0.

Eta and Zeta Functions 83By Lemma 1.6.6, (P ; ) ;1 is smoothing and hence of trace class for dlarge. We can take trace inside the integral to compute:(s P (a)) = 12iZ 1 ;s Tr((P (a) ; ) ;1 ) d;Z 2(;) ;s Tr((P (a) ; ) ;1 dwhere 1 and 2 are paths about the positive/negative real axis whichare oriented suitably. We dierentiate with respect to the parameter a toexpress:dda ((P (a) ; );1 = ;(P (a) ; ) ;1 _ P (a)(P (a) ; ) ;1 :Since the operators involved are of Trace class,Tr dda (P (a) ; );1 = ; Tr( _ P (a)(P (a) ; ) ;2 ):We use this identity and bring trace back outside the integral to compute:(s P (a)) = 12i Tr( _ P (a))We now use the identity: Z 1; ;s (P (a) ; ) ;2 d+Z 2(;) ;s (P (a) ; ) ;2 ddd ((P (a) ; );1 )=(P (a) ; ) ;2to integrate by parts in in this expression. This leads immediately to thedesired formula.We could also calculate using the heat equation. We proceed formally:dZ 1 dda (s P (a)) = ;f(s +1)=2g;1 t (s;1)=2 Tr0da (P (a)e;tP (a)2 ) dtZ 1 dP=;f(s +1)=2g ;1 t (s;1)=2 Tr0da (1 ; 2tP 2 )e ;tP 2 dtZ 1 =;f(s +1)=2g ;1 t (s;1)=2 1+2t d dPTr0dt da e;tP 2 dt:We now integrate by parts to compute::

84 1.10. Eta and Zeta FunctionsZ 1=;f(s +1)=2g ;1 ;st (s;1)=2 Tr;s;f(s +1)=2g ;1 X n;s;f(s +1)=2g ;1 X nIn particular, this shows that d~da0Z 1 dPda e;tP 2 dtt (s;1)=2 t (n;m;d)=2d a n dPda P2 dt0 dP2=(s +(n ; m)=d)a nda P2 :is regular at s =0and the value; 1 ;1 dP;;2 amda P2is given by a local formula.The interchange of order involved in using global trace is an essentialpart of this argument. Tr( P _ (a)(P 2 ) ;(s+1)=2 ) has a meromorphic extensionto C with isolated simple poles. The pole at s = ;1=2 can be present, butis cancelled o by the factor of ;s which multiplies the expression. Thusdda Res s=0 (s P (a)) = 0. This shows the global residue is a homotopyinvariant this fact will be used in Chapter 4 to show that in fact isregular at the origin. This argument does not go through locally, and infact it is possible to construct operators in any dimension m 2 so thatthe local eta function is not regular at s =0.We have assumed that P (a) has no zero spectrum. If we supress anite number of eigenvalues which may cross the origin, this makes nocontribution to the residue at s = 0. Furthermore, the value at s = 0changes by jumps of an even integer as eigenvalues cross the origin. Thisshows ~ is in fact smooth in the parameter a and proves:Lemma 1.10.3.(a) Let P be a self-adjoint elliptic partial dierential operator of positiveorder d. Dene:Z 1(s P )=Tr(P (P 2 ) ;(s+1)=2 )=;f(s+1)=2g ;1 t (s;1)=2 Tr(Pe ;tP 2 ) dt:This admits a meromorphic extension to C with isolated simple poles ats =(m ; n)=d. The residue of at such a pole is computed by:ZRes s=(m;n)=d =2;f(m + d ; n)=2dg ;1 a n (x P P 2 )dvol(x)where a n (x P P 2 ) is dened in Lemma 1.7.7 it is a local invariant in thejets of the symbol of P .0M

Eta and Zeta Functions 85(b) Let P (a) be a smooth 1-parameter family of such operators.assume eigenvalues do not cross the origin, then:dda (s P (a)) = ;s Tr dPda (P (a)2 ) ;(s+1)=2 = ;2s;f(s +1)=2g ;1 Z 10If we dPt (s;1)=2 Trda e;tP 2 dt:Regardless of whether or not eigenvalues cross the origin, ~(P (a)) is smoothin R mod Z in the parameter a andddPda ~(P (a)) = ;;( 1 2 );1 a mda P2= ;;( 1 2 );1 Z M a m x dPda P2 dvol(x) dPda P2is the local invariant in the jets of the symbols ofLemma 1.7.7.given byIn the example on the circle, the operator P (a) is locally isomorphic toP . Thus the value of at the origin is not given by a local formula. Thisis a global invariant of the operator only the derivative is locally given.It is not necessary to assume that P is a dierential operator to denethe eta invariant. If P is an elliptic self-adjoint pseudo-dierential operatorof order d>0, then the sum dening (s P ) converges absolutely for s 0to dene a holomorphic function. This admits a meromorphic extensionto the half-plane Re(s) > ; for some > 0 and the results of Lemma1.10.3 continue to apply. This requires much more delicate estimates thanwe have developed and we shall omit details. The reader is referred to thepapers of Seeley for the proofs.We also remark that it is not necessary to assume that P is self-adjointit suces to assume that det(p(x );it) 6= 0 for (t) 6= (0 0) 2 T M R.Under this ellipticity hypothesis, the spectrum of P is discrete and only anite number of Xeigenvalues lie on Xor near the imaginary axis. We dene:(s P )= ( i ) ;s ; (; i ) ;s~(P )= 1 2Re( i )>0Re( i )

86 1.10. Eta and Zeta FunctionsIn section 4.3, we will discuss the eta invariant in further detail and use itto dene an index with coecients in a locally at bundle using secondarycharacteristic classes.If the leading symbol of P is positive denite, the asymptotics of Tr(e ;tP )as t ! 0 + are given by local formulas integrated over the manifold. LetA(x) =a 0 x a + +a a and B(x) =b 0 x b + +b b be polynomials where b 0 >0. The operator A(P )e ;tB(P ) is innitely smoothing. The asymptotics ofTr(A(P )e ;tB(P ) ) are linear combinations of the invariants a N (P ) givingthe asymptotics of Tr(e ;tP ). Thus there is no new information whichresults by considering more general operators of heat equation type. If theleading symbol of P is indenite, one must consider both the zeta and etafunction or equivalently:Tr(e ;tP 2 ) and Tr(Pe ;tP 2 ):Then if b 0 > 0iseven, we obtain enough invariants to calculate the asymptoticsof Tr(A(P )e ;tB(P ) ). We refer to Fegan-Gilkey for further details.

CHAPTER 2CHARACTERISTIC CLASSESIntroductionIn the second chapter, we develop the theory of characteristic classes. Insection 2.1, we discuss the Chern classes of a complex vector bundle andin section 2.2 we discuss the Pontrjagin and Euler classes of a real vectorbundle. We shall dene the Todd class, the Hirzebruch L-polynomial, andthe A-roof genus which will play a central role in our discussion of the indextheorem. We also discuss the total Chern and Pontrjagin classes as well asthe Chern character.In section 2.3, we apply these ideas to the tangent space of a real manifoldand to the holomorphic tangent space of a holomorphic manifold. Wecompute several examples dened by Cliord matrices and compute theChern classes of complex projective space. We show that suitable productsof projective spaces form a dual basis to the space of characteristic classes.Such products will be used in chapter three to nd the normalizing constantswhich appear in the formula for the index theorem.In section 2.4 and in the rst part of section 2.5, we give a heat equationproof of the Gauss-Bonnet theorem. This is based on rst giving anabstract characterization of the Euler form in terms of invariance theory.This permis us to identify the integrand of the heat equation with the Eulerintegrand. This gives a more direct proof of Theorem 2.4.8 which was rstproved by Patodi using a complicated cancellation lemma. (The theoremfor dimension m =2is due to McKean and Singer).In the remainder of section 2.5, we develop a similar characterization ofthe Pontrjagin forms of the real tangent space. Weshallwait until the thirdchapter to apply these results to obtain the Hirzebruch signature theorem.There are two dierent approaches to this result. We have presented bothour original approach and also one modeled on an approach due to Atiyah,Patodi and Bott. This approach uses H. Weyl's theorem on the invariantsof the orthogonal group and is not self-contained as it also uses facts aboutgeodesic normal coordinates we shall not develop. The other approach ismore combinatorial in avor but is self-contained. It also generalizes todeal with the holomorphic case for a Kaehler metric.The signature complex with coecients in a bundle V gives rise to invariantswhich depend upon both the metric on the tangent space of Mand on the connection 1-form of V . In section 2.6, we extend the results ofsection 2.5 to cover more general invariants of this type. We also constructdual bases for these more general invariants similar to those constructedin section 2.3 using products of projective spaces and suitable bundles overthese spaces.

88 Chapter 2The material of sections 2.1 through 2.3 is standard and reviews thetheory of characteristic classes in the context we shall need. Sections 2.4through 2.6 deal with less standard material. The chapter is entirely selfcontainedwith the exception of some material in section 2.5 as noted above.We have postponed a discussion of similar material for the holomorphicKaehler case until sections 3.6 and 3.7 of chapter three since this materialis not needed to discuss the signature and spin complexes.

2.1. Characteristic ClassesOf a Complex Vector Bundle.The characteristic classes are topological invariants of a vector bundlewhich are represented by dierential forms. They are dened in terms ofthe curvature of a connection.Let M be a smooth compact manifold and let V be a smooth complexvector bundle of dimension k over M. A connection r on V is a rst orderpartial dierential operator r: C 1 (V ) ! C 1 (T M V ) such that:r(fs)=df s + frs for f 2 C 1 (M) and s 2 C 1 (V ):Let (s 1 ...s k ) be a local frame for V . We can decompose any sections 2 C 1 (V ) locally in the form s(x) = f i (x)s i (x) for f i 2 C 1 (M). Weadopt the convention of summing over repeated indices unless otherwisespecied in this subsection. We compute:rs = df i s i + f i rs i = df i s i + f i ! ij s j where rs i = ! ij s j :The connection 1-form ! dened by! = ! ijis a matrix of 1-forms. The connection r is uniquely determined by thederivation property and by the connection 1-form. If we specify! arbitrarilylocally, we can dene r locally. The convex combination of connectionsis again a connection, so using a partition of unity we can always constructconnections on a bundle V .If we choose another frame for V ,we can express s 0 i = h ijs j . If h ;1ij s0 j =s i represents the inverse matrix, then we compute:rs 0 i = ! 0 ij s 0 j = r(h ik s k )=dh ik s k + h ik ! kl s l=(dh ik h ;1kj + h ik! kl h ;1lj ) s0 j :This shows ! 0 obeys the transformation law:! 0 ij = dh ik h ;1kj + h ik! kl h ;1lj i.e., ! 0 = dh h ;1 + h!h ;1in matrix notation. This is, of course, the manner in which the 0 th ordersymbol of a rst order operator transforms.We extend r to be a derivation mappingC 1 ( p T M V ) ! C 1 ( p+1 T M V )so that:r( p s) =d p s +(;1) p p ^rs:

90 2.1. Characteristic ClassesWe compute that:r 2 (fs)=r( df s + frs) =d 2 f s ; df ^rs + df ^rs + fr 2 s = fr 2 sso instead of being a second order operator, r 2 is a 0 th order operator. Wemay therefore expressr 2 (s)(x 0 )=(x 0 )s(x 0 )where the curvature isa section to the bundle 2 (T M) END(V ) is a2-form valued endomorphism of V .In local coordinates, we compute:so that:(s i )= ij s j = r(! ij s j )= d! ij s j ; ! ij ^ ! jk s k ij = d! ij ; ! ik ^ ! kj i.e., = d! ; ! ^ !.Since r 2 is a 0 th order operator, must transform like atensor: 0 ij = h ik kl h ;1lj i.e., 0 = hh ;1 .This can also be veried directly from the transition law for! 0 . The readershould note that in some references, the curvature is dened by = d! +! ^ !. This sign convention results from writing V T M instead ofT M V and corresponds to studying left invariant rather than rightinvariant vector elds on GL(k C).It is often convenient to normalize the choice of frame.Lemma 2.1.1. Let r be a connection on a vector bundle V . We canalways choose a frame s so that at a given point x 0 we have!(x 0 )=0 and d(x 0 )=0:Proof: We nd a matrix h(x) dened near x 0 so that h(x 0 ) = I anddh(x 0 ) = ;!(x 0 ). If s 0 i = h ijs j , then it is immediate that ! 0 (x 0 ) = 0.Similarly we compute d 0 (x 0 )=d(d! 0 ; ! 0 ^ ! 0 )(x 0 )=! 0 (x 0 ) ^ d! 0 (x 0 ) ;d! 0 (x 0 ) ^ ! 0 (x 0 )=0.We note that as the curvature is invariantly dened, we cannot in generalnd a parallel frame s so ! vanishes in a neighborhood of x 0 since this wouldimply r 2 =0near x 0 which need not be true.Let A ij denote the components of a matrix in END(C k )andletP (A) =P (A ij ) be a polynomial mapping END(C k ) ! C. We assume that P isinvariant|i.e., P (hAh ;1 )=P (A) for any h 2 GL(k C). We dene P ()as an even dierential form on M by substitution. Since P is invariant

Of a Complex Vector Bundle 91and since the curvature transforms tensorially, P () def= P (r) is denedindependently of the particular local frame which is chosen.There are two examples which are of particular interest and which willplay an important role in the statement of the Atiyah-Singer index theorem.We dene:c(A) = detI + i2 A =1+c 1 (A)++ X c k (A)ch(A) =Tr e iA=2 iA= Tr2j(the total Chern form) j j! (the total Chern character)The c j (A) represent the portion of c(A) which is homogeneous of order j.c j (r) 2 2j (M). In a similar fashion, we decompose ch(A) = P j ch j(A)for ch j (A) =Tr(iA=2) j =j!Strictly speaking, ch(A) is not a polynomial. However, when we substitutethe components of the curvature tensor, this becomes a nite sumsince Tr( j ) = 0 if 2j > dim M. More generally, we can dene P () ifP (A) is an invariant formal power series by truncating the power seriesappropriately.As a dierential form, P (r) depends on the connection chosen. Weshow P (r) isalways closed. As an element of de Rham cohomology, P (r)is independent of the connection and denes a cohomology class we shalldenote by P (V ).Lemma 2.1.2. Let P be an invariant polynomial.(a) dP (r) =0so P (r) is a closed dierential form.(b) Given two connections r 0 and r 1 , we can dene a dierential formTP(r 0 r 1 ) so that P (r 1 ) ; P (r 0 )=dfTP(r 1 r 0 )g.Proof: By decomposing P as a sum of homogeneous polynomials, wemayassume without loss of generality that P is homogeneous of order k. LetP (A 1 ...A k ) denote the complete polarization of P . We expand P (t 1 A 1 ++t k A k ) as a polynomial in the variables ft j g. 1=k! times the coecientof t 1 ...t k is the polarization of P . P is a multi-linear symmetric functionof its arguments. For example, if P (A) =Tr(A 3 ), then the polarization isgiven by 1 2 Tr(A 1A 2 A 3 + A 2 A 1 A 3 ) and P (A) =P (A A A).Fix a point x 0 of M and choose a frame so !(x 0 )= d(x 0 )=0. ThendP ()(x 0 )= dP ( ... )(x 0 )=kP(d ... )(x 0 )=0:Since x 0 is arbitrary and since dP () is independent of the frame chosenthis proves dP () = 0 which proves (a).

92 2.1. Characteristic ClassesLet r t = tr 1 +(1; t)r 0 have connection 1-form ! t = ! 0 + t where = ! 1 ; ! 0 . The transformation law for! implies relative to a new frame: 0 = ! 0 1 ; ! 0 0 =(dh h ;1 + h! 1 h ;1 ) ; (dh h ;1 + h! 0 h ;1 )= h(! 1 ; ! 0 )h ;1 = hh ;1so transforms like a tensor. This is of course nothing but the fact that thedierence between two rst order operators with the same leading symbolis a 0 th order operator.Let t be the curvature of the connection r t . This is a matrix valued2-form. Since is a 1-form, it commutes with 2-forms and we can deneP ( t ... t ) 2 2k;1 (T M)by substitution. Since P is invariant, its complete polarization is also invariantsoP (h ;1 h h ;1 t h ...h ;1 t h)=P ( t ... t )isinvariantlydened independent of the choice of frame.We compute that:Z 1dZ 1P (r 1 ) ; P (r 0 )=0 dt P ( t ... t ) dt = k P ( 0 t t ... t ) dt:0We dene:Z 1TP(r 1 r 0 )=k P ( t ... t ) dt:0To complete the proof of the Lemma, it suces to checkdP ( t ... t )=P ( 0 t t ... t ):Since both sides of the equation are invariantly dened, we can choose asuitable local frame to simplify the computation. Let x 0 2 M and x t 0 .We choose a frame so ! t (x 0 t 0 )=0and d t (x 0 t 0 )=0. We compute:and 0 t = fd! 0 + td ; ! t ^ ! t g 0 0 t(x 0 t 0 )= d= d ; ! 0 t ^ ! t ; ! t ^ ! 0 tdP ( t ... t )(x 0 t 0 )=P (d t ... t )(x 0 t 0 )which completes the proof.= P ( 0 t t ... t )(x 0 t 0 )TP is called the transgression of P and will play an important role in ourdiscussion of secondary characteristic classes in Chapter 4 when we discussthe eta invariant with coecients in a locally at bundle.

Of a Complex Vector Bundle 93We suppose that the matrix A = diag( ; 1 ... k ) is diagonal.imodulo suitable normalizing constantsthe j th elementary symmetric function of the i 's sinceThen2 j, it is immediate that cj (A) isdet(I + A) = Y (1 + j )=1+s 1 ()++ s k ():If P () is any invariant polynomial, then P (A) is a symmetric function ofthe i 's. The elementary symmetric functions form an algebra basis forthe symmetric polynomials so there is a unique polynomial Q so thatP (A) =Q(c 1 ...c k )(A)i.e., we can decompose P as a polynomial in the c i 's for diagonal matrices.Since P is invariant, this is true for diagonalizable matrices. Since thediagonalizable matrices are dense and P is continuous, this is true for allA. This proves:Lemma 2.1.3. Let P (A) be a invariant polynomial. There exists a uniquepolynomial Q so that P = Q(c 1 (A) ...c k (A)).It is clear that any polynomial in the c k 's is invariant, so this completelycharacterizes the ring of invariant polynomials it is a free algebra in kvariables fc 1 ...c k g.If P (A) is homogeneous of degree k and is dened on k k matrices, weshall see that if P (A) 6= 0 as a polynomial, then there exists a holomorphicmanifold M so that if T c (M) is the holomorphic tangent space, thenRM P (T c(M)) 6= 0where dim M =2k. This fact can be used to show thatin general if P (A) 6= 0 as a polynomial, then there exists a manifold Mand a bundle V over M so P (V ) 6= 0in cohomology.We can apply functorial constructions on connections. Dene:r 1 r 2 on C 1 (V 1 V 2 )r 1 r 2 on C 1 (V 1 V 2 )r 1 on C 1 (V 1 )by:(r 1 r 2 )(s 1 s 2 )=(r 1 s 1 ) (r 2 s 2 )with ! = ! 1 ! 2 and = 1 2(r 1 r 2 )(s 1 s 2 )=(r 1 s 1 ) (r 2 s 2 )with ! = ! 1 1+1 ! 2 and = 1 1+1 2(r 1 s 1 s 1)+(s 1 r 1 s 1)=d(s 1 s 1)relative to the dual frame ! = ;! t 1 and =; t 1.

94 2.1. Characteristic ClassesIn a similar fashion we can dene an induced connection on p (V ) (thebundle of p-forms) and S p (V ) (the bundle of symmetric p-tensors). If Vhas a given Hermitian ber metric, the connection r is said to be unitaryor Riemannian if (rs 1 s 2 )+(s 1 rs 2 ) = d(s 1 s 2 ). If we identify V withV using the metric, this simply means r = r . This is equivalent tothe condition that ! is a skew adjoint matrix of 1-forms relative toalocalorthonormal frame. We can always construct Riemannian connections locallyby taking ! = 0 relative to a local orthonormal frame and then usinga partition of unity to construct a global Riemannian connection.If we restrict to Riemannian connections, then it is natural to considerpolynomials P (A) which are dened for skew-Hermitian matrices A+A =0 and which are invariant under the action of the unitary group. Exactlythe same argument as that given in the proof of Lemma 2.1.3 shows thatwe can express such a P in the form P (A) =Q(c 1 (A) ...c k (A)) so thatthe GL( C) and the U() characteristic classes coincide. This is not truein the real category as we shall see the Euler form is a characteristic formof the special orthogonal group which can not be dened as a characteristicform using the general linear group GL( R).The Chern form and the Chern character satisfy certain identities withrespect to functorial constructions.Lemma 2.1.4.(a)c(V 1 V 2 )=c(V 1 )c(V 2 )c(V )=1; c 1 (V )+c 2 (V ) ; +(;1) k c k (V ):(b)ch(V 1 V 2 )=ch(V 1 )+ch(V 2 )ch(V 1 ) V 2 )=ch(V 1 )ch(V 2 ):Proof: All the identities except the one involving c(V ) are immediatefrom the denition if we use the direct sum and product connections. If wex a Hermitian structure on V , the identication of V with V is conjugatelinear. The curvature of r is ; t so we compute:c(V )=detwhich gives the desired identity.I ; i2 t = detI ; i2 If we choose a Riemannian connection on V , then + t = 0. Thisimmediately yields the identities:ch(V )=ch(V ) and c(V )=c(V )

Of a Complex Vector Bundle 95so these are real cohomology classes. In fact the normalizing constantswere chosen so c k (V ) is an integral cohomology R class|i.e., if N 2k is anyoriented submanifold of dimension 2k, thenN 2 k c k(V ) 2 Z. The ch k (V )are not integral for k>1, but they are rational cohomology classes. As weshall not need this fact, we omit the proof.The characteristic classes give cohomological invariants of avector bundle.We illustrate this by constructing certain examples over even dimensionalspheres these examples will play an important role in our laterdevelopment of the Atiyah-Singer index theorem.Definition. A set of matrices fe 0 ...e m g are Cliord matrices if thee i are self-adjoint and satisfy the commutation relations e i e j + e j e i =2 ijwhere ij is the Kronecker symbol.For example, if m =2we can take:e 0 = 1 00 ;1! e 1 = 0 11 0! e 2 =!0 i;i 0to be the Dirac matrices. More generally, we can take the e i 's to be givenby the spin representations. If x 2 R m+1 , we dene:e(x) = X ix i e i so e(x) 2 = jxj 2 I:Conversely let e(x) be a linear map from R m to the set of self-adjointmatrices with e(x) = jxj 2 I. If fv 0 ...v m g is any orthonormal basis forR m , fe(v 0 ) ...e(v m )g forms a set of Cliord matrices.If x 2 S m , we let (x) be the range of 1 2 (1 + e(x)) = (x). Thisis the span of the 1 eigenvectors of e(x). If e(x) is a 2k 2k matrix,then dim (x) = k. We have a decomposition S m C 2k = + ; .We project the at connection on S m C 2k to the two sub-bundles todene connections r on . If e 0 is a local frame for (x 0 ), we denee (x) = e 0 as a frame in a neighborhood of x 0 . We computer e = d e 0 e = d d e 0 :Since e 0 = e (x 0 ), this yields the identity: (x 0 )= d d (x 0 ):Since is tensorial, this holds for all x.

96 2.1. Characteristic ClassesLet m = 2j be even. We wish to compute ch j . Suppose rst x 0 =(1 0 ... 0) is the north pole of the sphere. Then: + (x 0 )= 1 2 (1 + e 0)d + (x 0 )= 1 2Xi1dx i e i + (x 0 )= 1 2 (1 + e 0) + (x 0 ) j = 1 2 (1 + e 0) 12Xi1 1 X2i1dx i e i 2dx i e i 2j=2 ;m;1 m!(1 + e 0 )(e 1 ...e m )(dx 1 ^^dx m ):The volume form at x 0 is dx 1 ^ ^dx m . Since e 1 anti-commutes withthe matrix e 1 ...e m , this matrix has trace 0 so we compute: i jch j ( + )(x 0 )= 2 ;m;1 m!Tr(e 0 ...e m )dvol(x 0 )=j! :2A similar computation shows this is true at any point x 0 of S m so that:Z i jch j ( + )= 2 ;m;1 m!Tr(e 0 ...e m )vol(s m )=j! :S m 2Since the volume of S m is j!2 m+1 j =m! we conclude:Lemma 2.1.5. Let e(x) be a linear map from R m+1 to the set of selfadjointmatrices. We suppose e(x) 2 = jxj 2 I and dene bundles (x) overS m corresponding to the 1 eigenvalues of e. Let m =2j be even, then:Zch j ( + )=i j 2 ;j Tr(e 0 ...e m ):S mIn particular, this is always an integer as we shall see later when weconsider the spin complex. Ife 0 = 1 00 ;1! e 1 = 0 11 0! e 2 =!0 i;i 0then Tr(e 0 e 1 e 2 ) = ;2i so R S 2 ch 1 ( + ) = 1 which shows in particular +is a non-trivial line bundle over S 2 .

Of a Complex Vector Bundle 97There are other characteristic classes arising in the index theorem. Theseare most conveniently discussed using generating functions. Letx j =i2 jbe the normalized eigenvalues of the matrix A. We dene:c(A) =kYj=1(1 + x j ) and ch(A) =If P (x) is a symmetric polynomial, we express P (x) =Q(c 1 (x) ...c k (x))to show P (A) is a polynomial in the components of A. More generally ifP is analytic, we rst express P in a formal power series and then collecthomogeneous terms to dene P (A). We dene:Todd class: Td(A) =kYj=1x j1 ; e ;x jkXj=1e x j:=1+ 1 2 c 1(A)+ 1 12 (c2 1 + c 2 )(A)+ 124 c 1(A)c 2 (A)+The Todd class appears in the Riemann-Roch theorem. It is clear that it ismultiplicative with respect to direct sum|Td(V 1 V 2 )=Td(V 1 ) Td(V 2 ).The Hirzebruch L-polynomial and the ^A polynomial will be discussed inthe next subsection. These are real characteristic classes which also aredened by generating functions.If V is a bundle of dimension k, then V 1 will be a bundle of dimensionk+1. It is clear c(V 1) = c(V )andTd(V 1) = Td(V ) so these are stablecharacteristic classes. ch(V ) on the other hand is not a stable characteristicclass since ch 0 (V )=dim V depends explicitly on the dimension of V andchanges if we alter the dimension of V by adding a trivial bundle.

2.2. Characteristic Classes of a Real Vector Bundle.Pontrjagin and Euler Classes.Let V be a real vector bundle of dimension k and let V c = V C denotethe complexication of V . We place a real ber metric on V to reducethe structure group from GL(k R) to the orthogonal group O(k). We restricthenceforth to Riemannian connections on V , and to local orthonormalframes. Under these assumptions, the curvature is a skew-symmetricmatrix of 2-forms:+ t =0:Since V c arises from a real vector bundle, the natural isomorphism of Vwith V dened by the metric induces a complex linear isomorphism of V cwith Vc so c j (V c )=0for j odd by Lemma 2.1.4. Expressed locally,detI + i2 A = detI + i2 At = detI ; i2 A if A + A t =0soc(A) is an even polynomial in A. To avoid factors of i wedene the Pontrjagin form:p(A) =detI + 12 A =1+p 1 (A)+p 2 (A)+where p j (A) is homogeneous of order 2j in the components of A the correspondingcharacteristic class p j (V ) 2 H 4j (M R). It is immediate that:p j (V )=(;1) j c 2j (V c )where the factor of (;1) j comes form the missing factors of i.The set of skew-symmetric matrices is the Lie algebra of O(k). Let P (A)be an invariant polynomial under the action of O(k). We dene P () forr a Riemannian connection exactly as in the previous subsections. Thenthe analogue of Lemma 2.1.3 is:Lemma 2.2.1. Let P (A) be a polynomial in the components of a matrixA. Suppose P (A) = P (gAg ;1 ) for every skew-symmetric A and forevery g 2 O(k). Then there exists a polynomial Q(p 1 ...) so P (A) =Q(p 1 (A) ...) for every skew-symmetric A.Proof: It is important to note that we are not asserting that we haveP (A) =Q(p 1 (A) ...) for every matrix A, but only for skew-symmetric A.For example, P (A) = Tr(A) vanishes for skew symmetric A but does notvanish in general.

Classes of a Real Bundle 99It is not possible in general to diagonalize a skew-symmetric real matrix.We can, however, put it in block diagonal form:A =0B@0 ; 1 0 0 ...... 1 0 0 0 ......0 0 0 ; 2 ......0 0 2 0 ......::::::::::::::::::::::::::::::::::::::::::::::::::::If k is odd, then the last block will be a 1 1 block with a zero in it.We letx j = ; j =2 the sign convention is chosen to make the Euler formhave the right sign. Then:1CAp(A) = Y j(1 + x 2 j ) kwhere the product ranges from 1 through 2 .If P (A) is any invariant polynomial, then P is a symmetric function inthe fx j g. By conjugating A by an element of O(k), we can replace any x jby ;x j so P is a symmetric function of the fx 2 j g. The remainder of theproof of Lemma 2.2.1 is the same we simply express P (A) in terms of theelementary symmetric functions of the fx 2 j g.Just as in the complex case, it is convenient to dene additional characteristicclasses using generating functions. The functions:z= tanh z and zf2sinh(z=2)gare both even functions of the parameter z. We dene:The Hirzebruch L-polynomial:L(x)= Y jx jtanh x j=1+ 1 3 p 1 + 1 45 (7p 2 ; p 2 1) The ^A (A-roof ) genus:A(x)= Y jx j2 sinh(x j =2)=1; 1 24 p 1 + 15760 (7p2 1 ; 4p 2 )+ :These characteristic classes appear in the formula for the signature andspin complexes.

100 2.2. Classes of a Real BundleFor both the real and complex case, the characteristic ring is a purepolynomial algebra without relations. Increasing the dimension k just addsgenerators to the ring. In the complex case, the generators are the Chernclasses f1c 1 ...c k g. In the real case, the generators are the Pontrjaginclasses f1p 1 ...p [k=2] g where [k=2] is the greatest integer in k=2. Thereis one nal structure group which will be of interest to us.Let V be avector bundle of real dimension k. V is orientable if we canchoose a ber orientation consistently. This is equivalent to assuming thatthe real line bundle k (V ) is trivial. We choose a ber metric for V andan orientation for V and restrict attention to oriented orthonormal frames.This restricts the structure group from O(k) to the special orthogonal groupSO(k).If k is odd, no new characteristic classes result from the restriction of theber group. We use the nal 1 1 block of0 in the representation of A toreplace any i by ; i by conjugation by anelement of SO(k). If n is even,however, we cannot do this as we would be conjugating by a orientationreversing matrix.Let P (A) be a polymonial in the components of A. Suppose P (A) =P (gAg ;1 ) for every skew-symmetric A and every g 2 SO(k). Let k = 2kbe even and let P (A) =P (x 1 ...x k ). Fix g 0 2 O(k) ; SO(k) and dene:P 0 (A) = 1 2 (P (A)+P (g 0Ag ;10 )) and P 1(A) = 1 2 (P (A) ; P (g 0Ag ;10 )):Both P 0 and P 1 are SO(k) invariant. P 0 is O(k) invariant whileP 1 changessign under the action of an element of O(k) ; SO(k).We can replace x 1 by ;x 1 by conjugating by a suitable orientation reversingmap. This shows:P 1 (x 1 x 2 ...)=;P (;x 1 x 2 ...)so that x 1 must divide every monomial of P 0 . By symmetry, x j dividesevery monomial of P 1 for 1 j k so we can express:P 1 (A) =x 1 ...x k P 0 1(A)where P 0 1(A) is now invariant under the action of O(k). Since P 0 1 is polynomialin the fx 2 j g, we conclude that both P 0 and P 0 1 can be representedas polynomials in the Pontrjagin classes. We dene:e(A) =x 1 ...x k so e(A) 2 = det(A) =p k (A) = Y jx 2 jand decompose:P = P 0 + e(A)P 0 1 :

Pontrjagin and Euler Classes 101e(A) is a square root of the determinant of A.It is not, of course, immediate that e(A) is a polynomial in the componentsof A. Let fv i g be an oriented orthonormal basis for R k . We letAv i = P j A ijv j and dene!(A) = 12Xi

102 2.2. Classes of a Real BundleThe Euclidean connection e r is easily computed to be:er e1 e 1 = (sin v) ;1 (; cos u ; sin u 0) = ;(cos v= sin v)e 2 ; e 3er e1 e 2 = (cos v= sin v)(; sin u cos u 0) = (cos v= sin v)e 1er e2 e 1 =0er e2 e 2 =(; cos u sin v ; sin u sin v ; cos v) =;e 3 :Covariant dierentiation r on the sphere is given by projecting back toT (S 2 ) so that:r e1 e 1 =(; cos v= sin v)e 2 r e1 e 2 = (cos v= sin v)e 1r e2 e 1 =0 r e2 e 2 =0and the connection 1-form is given by:re 1 =(; cos v= sin v)e 1 e 2 so ! 11 =0and ! 12 = ;(cos v= sin v)e 1re 2 = (cos v= sin v)e 1 e 1 so ! 22 =0and ! 21 = (cos v= sin v)e 1 .As du = e 1 = sin v we compute ! 12 = ; cos vdu so 12 = sin vdv^ du =;e 1 ^ e 2 and 21 = e 1 ^ e 2 . From this we calculate that:e() = ; 14 ( 12 ; 21 )= e1 ^ e 22and consequently R S 2 E 2 =vol(S 2 )=2 =2=(S 2 ).There is a natural relation between the Euler form and c k . Let V bea complex vector space of dimension k and let V r be the underlying realvector space of dimention 2 k = k. If V has a Hermitian inner product,then V r inherits a natural real inner product. V r also inherits a naturalorientation from the complex structure on V . If fe j g is a unitary basis forV ,thenfe 1 ie 1 e 2 ie 2 ...g is an oriented orthonormal basis for V r . Let Abe a skew-Hermitian matrix on V . The restriction of A to V r denes a skewsymmetricmatrix A r on V r . We choose a basis fe j g for V so Ae j = i j e j .Then:x j = ; j =2 and c k (A) =x 1 ...x kdenes the k th Chern class. If v 1 = e 1 , v 2 = ie 1 , v 3 = e 2 , v 4 = ie 2 ...then:A r v 1 = 1 v 2 A r v 2 = ; 1 v 1 A r v 3 = 2 v 4 A r v 4 = ; 2 v 3 ...so thate(A r )=c k (A):We summarize these results as follows:

Pontrjagin and Euler Classes 103Lemma 2.2.2. Let P (A) be a polynomial with P (A) = P (gAg ;1 ) forevery skew symmetric A and every g 2 SO(k).(a) If k is odd, then P (A) is invariant under O(k) and is expressible interms of Pontrjagin classes.(b) If k =2 k is even, then we can decompose P (A) =P 0 (A)+e(A)P 1 (A)where P i (A) are O(k) invariant and are expressible in terms of Pontrjaginclasses.(c) e(A) is dened by:e(A) =(;4) ; k X sign()A (1)(2) ...A (k;1)(k) =k!This satises the identity e(A) 2 = p k (A). (We dene e(A) =0if k is odd).This completely describes the characteristic ring. We emphasize theconclusions are only applicable to skew-symmetric real matrices A. Wehave proved that e(A) has the following functorial properties:Lemma 2.2.3.(a) If we take the direct sum connection and metric, then e(V 1 V 2 ) =e(V 1 )e(V 2 ).(b) If we take the metric and connection of V r obtained by forgetting thecomplex structure on a complex bundle V then e(V r )=c k (V ).This lemma establishes that the top dimensional Chern class c k does notreally depend on having a complex structure but only depends on havingan orientation on the underlying real vector bundle. The choice of the signin computing det(A) 1=2 is, of course, motivated by this normalization.

2.3. Characteristic Classes of Complex Projective Space.So far, we have only discussed covariant dierentiation from the pointof view of the total covariant derivative. At this stage, it is convenientto introduce covariant dierentiation along a direction. Let X 2 T (M)be a tangent vector and let s 2 C 1 (V ) be a smooth section. We dener X s 2 C 1 (V ) by:r X s = X rswhere \" denotes the natural pairing from T (M) T (M) V ! V . Let[X Y ]=XY ; YX denote the Lie-bracket of vector elds. We dene:and compute that:(X Y )=r X r Y ;r Y r X ;r [XY ](fXY)s =(X fY )s =(X Y )fs = f(X Y )sforf 2 C 1 (M) s 2 C 1 (V ) and X Y 2 C 1 (TM):If fe i g is a local frame for T (M), let fe i g be the dual frame for T (M).It is immediate that: X rs = e i r ei (s)iso we can recover the total covariant derivative from the directional derivatives.We can also recover the curvature:s = X i

They are related by the formula:Characteristic Classes 105; ijk = X l; l ij g lk ; k ij = X l; ijl g lkwhere g lk =(dx l dx k )isthe inverse of the matrix g ij . It is not dicult tocompute that:; ijk = 1 2 fg jk=i + g ik=j ; g ij=k gwhere we use the notation \=" to denote (multiple) partial dierentiation.The complete curvature tensor is dened by:(@=@x i @=@x j )@=@x k = X lR ijk l @=@x lor equivalently if we lower indices:((@=@x i @=@x j )@=@x k @=@x l )=R ijkl :The expression of the curvature tensor in terms of the derivatives of themetric is very complicated in general. By making a linear change of coordinateswe can always normalize the metric so g ij (x 0 )= ij is the Kroneckersymbol. Similarly, by making a quadratic change of coordinates, we canfurther normalize the metric so g ij=k (x 0 )=0. Relative to such a choice ofcoordinates:; k ij(x 0 )=; ijk (x 0 )=0andR ijkl (x 0 )=R ijk l (x 0 )= 1 2 fg jl=ik + g ik=lj ; g jk=il ; g il=jk g(x 0 ):At any other point, of course, the curvature tensor is not as simply expressed.In general it is linear in the second derivatives of the metric,quadratic in the rst derivatives of the metric with coecients which dependsmoothly on the g ij 's.We choose a local orthonormal frame fe i g for T (M). Let m = 2m beeven and let orn = 1=dvol be the oriented volume. Lete = E m (g) ornbe the Euler form. If we change the choice of the local orientation, then eand orn both change signs so E m (g) is scalar invariant of the metric. Interms of the curvature, if 2m = m, then:c(m) =(;1) m (8) ; m 1(m)!XE m = c(m) sign() sign()R (1)(2)(1)(2) ...R (m;1)(m)(m;1)(m)

106 2.3. Characteristic Classeswhere the sum ranges over all permutations of the integers 1 throughm. For example:E 2 = ;(2) ;1 R 1212E 4 =(2) ;2XijklfR ijij R klkl + R ijkl R ijkl ; 4R ijik R ljlk g=8:Let dvol be the Riemannian measure on M. If M is oriented, thenZME m dvol =is independent of the orientation of M and of R the metric. If M is notorientable, we pass to the double cover to seeM E m dvol is a topologicalinvariant of the manifold M. We shall prove later this integral is the Eulercharacteristic (M) but for the moment simply note it is not dependentupon a choice of orientation of M and is in fact dened even if M is notorientable.It is worth computing an example. We let S 2 be the unit sphere in R 3 .Since this is homogeneous, E 2 is constant on S 2 . We compute E 2 at thenorth pole (0 0 1) and parametrize S 2 by (u v (1 ; u 2 ; v 2 ) 1=2 ). Then@=@u =(1 0 ;u=(1 ; u 2 ; v 2 ) 1=2 ) @=@v =(0 1 ;v=(1 ; u 2 ; v 2 ) 1=2 )g 11 =1+u 2 =(1 ; u 2 ; v 2 )g 22 =1+v 2 =(1 ; u 2 ; v 2 )g 12 = uv=(1 ; u 2 ; v 2 ):It is clear g ij (0) = ij and g ij=k (0) = 0. Therefore at u = v =0,so thatE 2 = ;(2) ;1 R 1212 = ;(2) ;1 fg 11=22 + g 22=11 ; 2g 12=12 g=2= ;(2) ;1 f0+0; 2g=2 =(2) ;1ZS 2 E 2 dvol = (4)(2) ;1 =2=(S 2 ):More generally, we can let M = S 2 S 2 where we have m factorsand 2m = m. Since e(V 1 V 2 ) = e(V 1 )e(V 2 ), when we take the productmetric we have E m =(E 2 ) m =(2) ; m for this example. Therefore:ZS 2 S 2 E m dvol = 2 m = (S 2 S 2 ):We will use this example later in this chapter to prove R M E m = (M) ingeneral. The natural examples for studying the Euler class are productsZMe

Of Complex Projective Space 107of two dimensional spheres. Unfortunately, the Pontrjagin classes vanishidentically on products of spheres so we must nd other examples.It is convenient at this point to discuss the holomorphic category. Amanifold M of real dimension m =2m is said to be holomorphic if we havelocal coordinate charts z =(z 1 ...z m ): M ! C m such thattwo charts arerelated by holomorphic changes of coordinates. We expand z j = x j + iy jand dene:@=@z j = 1 2 (@=@x j ; i@=@y j )dz j = dx j + idy j @=@z j = 1 2 (@=@x j + i@=@y j )dz j = dx j + idy j :We complexify T (M) and T (M) and dene:T c (M) = spanf@=@z j g 10 (M) = spanfdz j g 01 (M) = spanfdz j g:Then 1 (M)= 10 (M) 01 (M)T c (M) = 10 (M):As complex bundles, T c (M) ' 01 (M). Dene:@(f) = X j@(f) = X j@f=@z j dz j : C 1 (M) ! C 1 ( 10 (M))@f=@z j dz j : C 1 (M) ! C 1 ( 01 (M))then the Cauchy-Riemann equations show f is holomorphic if and only if@f =0.This decomposes d = @ + @ on functions. More generally, we dene: pq = spanfdz i1 ^ ^dz ip ^ dz j1 ^ ^dz jq g n =Mp+q=n pq :This spanning set of pq is closed. We decompose d = @ + @ where@: C 1 ( pq ) ! C 1 ( p+1q ) and @: C 1 ( pq ) ! C 1 ( pq+1 )so that @@ = @ @ = @@ + @ @ = 0. These operators and bundles are allinvariantly dened independent of the particular holomorphic coordinatesystem chosen.

108 2.3. Characteristic ClassesA bundle V over M is said to be holomorphic if we can cover M byholomorphic charts U and nd frames s over each U so that on U \U the transition functions s = f s dene holomorphic maps to GL(n C).For such a bundle, we say a local section s is holomorphic if s = P a s for holomorphic functions a . For example, T c (M) and p0 (M) are holomorphicbundles over M 01 is anti-holomorphic.If V is holomorphic, we use a partition of unity to construct a Hermitianber metric h on V . Let h = (s s ) dene the metric locally thisis a positive denite symmetric matrix which satises the transition ruleh = f h f . We dene a connection 1-form locally by:and compute the transition rule:! = @h h ;1! = @ff h f gf ;1 h;1 f ;1 :Since f is holomorphic, @f =0. This implies df = @f and @ f =0 so that:! = @f f ;1 + f @h h ;1 f ;1 = df f ;1 + f ! f ;1 :Since this is the transition rule for a connection, the f! g patch togetherto dene a connection r h .It is immediate from the denition that:(r h s s )+(s r h s )=! h + h ! = @h + @h = dh so r h is a unitary connection on V . Since r h s 2 C 1 ( 10 V ) weconclude r h vanishes on holomorphic sections when dierentiated in antiholomorphicdirections (i.e., r h is a holomorphic connection). It is easilyveried that these two properties determine r h .We shall be particularly interested in holomorphic line bundles. If L is aline bundle, then h is a positive function on U with h = jf j 2 h . Thecurvature in this case is a 2-form dened by: = d! ; ! ^ ! = d(@h h ;1 )= @@log(h )=;@ @ log(h ):Therefore:c 1 (L) = 12i @ @ log(h)is independent of the holomorphic frame chosen for evaluation. c 1 (L) 2C 1 ( 11 (M)) and dc 1 = @c 1 = @c 1 = 0 so c 1 (L) is closed in all possiblesenses.

Of Complex Projective Space 109Let CP n be the set of all lines through 0 in C n+1 . Let C = C ; 0 acton C n+1 ; 0 by complex multiplication, then CP n =(C n+1 ; 0)=C andwe give CP n the quotient topology. Let L be the tautological line bundleover CP n :L = f (x z) 2 CP n C n+1 : z 2 x g:Let L be the dual bundle this is called the hyperplane bundle.Let (z 0 ...z n ) be the usual coordinates on C n+1 . Let U j = f z : z j 6=0 g. Since U j is C invariant, it projects to dene an open set on CP n weagain shall denote by U j . We denez j k = z k=z jover U j . Since these functions are invariant under the action of C , theyextend to continuous functions on U j . We let z j = (z j 0 ... c z j j ...zj n)where we have deleted z j j =1. This gives local coordinates on U j in CP n .The transition relations are:z j =(z k j ) ;1 z k which are holomorphic so CP n is a holomorphic manifold.We let s j =(z j 0 ...zj n) be a section to L over U j . Then s j =(zj k);1 s ktransform holomorphically so L is a holomorphic line bundle over CP n . Thecoordinates z j on C n+1 give linear functions on L and represent globalholomorphic sections to the dual bundle L . There is a natural innerproduct on the trivial bundle CP n C n+1 which denes abermetric onL. We dene:x = ;c 1 (L) =i2 @ @ log(1 + jz j j 2 )then this is a closed 2-form over CP n .U(n + 1) acts on C n+1 this action induces a natural action on CP n andon L. Since the metric on L arises from the invariant metric on C n+1 ,the2-form x is invariant under the action of U(n +1).Lemma 2.3.1. Let x = ;c 1 (L) over CP n . Then:(a) R CP nx n =1.(b) H (CP n C) is a polynomial ring with additive generators f1x...x n g.(c) If i: CP n;1 ! CP n is the natural inclusion map, then i (x) =x.Proof: (c) is immediate from the naturality of the constructions involved.Standard methods of algebraic topology give the additive structure ofH (CP n C) = C 0 C 0 C. Since x 2 H 2 (CP n C) satisesx n 6= 0, (a) will complete the proof of (b). We x a coordinate chart U n .Since CP n ; U n = CP n;1 , it has measure zero. It suces to check that:ZU nx n =1:

110 2.3. Characteristic ClassesWe identify z 2 U n = C n with (z 1) 2 C n+1 . We imbed U(n) inU(n +1)as the isotropy group of the vector (0 ... 0 1). Then U(n) actson (z 1) in C n+1 exactly the same way as U(n) acts on z in C n . Letdvol = dx 1 ^ dy 1 ^ ^dx n ^ dy n be ordinary Lebesgue measure on C nwithout any normalizing constants of 2. We parametrize C n in the form(r) for 0 r1Consequently at this point,x n = ;2i 1 nn!(1 + r 2 ) ;n;1 dz 1 ^ dz 1 ^ ^dz n ^ dz n= ;n n!(1 + r 2 ) ;n;1 dx 1 ^ dy 1 ^ ^dx n ^ dy n= ;n n!(1 + r 2 ) ;n;1 dvol :We use the spherical symmetry to conclude this identitiy holds for all z.We integrate over U n = C n and use spherical coordinates:Zx n = n! ;n Z (1 + r 2 ) ;n;1 dvol = n! ;n Z (1 + r 2 ) ;n;1 r 2n;1 dr d= n! ;n vol(S 2n;1 )= 1 2 n! ;n vol(S 2n;1 )Z 10Z 10(1 + r 2 ) ;n;1 r 2n;1 dr(1 + t) ;n;1 t n;1 dt:We compute the volume of S 2n;1 using the identity p R 1 =;1 e;t2 dt.Thus:ZZ n 1= e ;r2 dvol = vol(S 2n;1 ) r 2n;1 e ;r2 dr= 1 2 vol(S 2n;1 )Z 100t n;1 e ;t dt =(n ; 1)!2:vol(S 2n;1 ):

Of Complex Projective Space 111We solve this for vol(S 2n;1 ) and substitute to compute:Zx n = n=Z 10Z 10(1 + t) ;n;1 t n;1 dt =(n ; 1)(1 + t) ;2 dt =1which completes the proof.Z 10(1 + t) ;n t n;2 dt = x can also be used to dene a U(n +1)invariant metric on CP n calledthe Fubini-Study metric. We shall discuss this in more detail in Chapter 3as this gives a Kaehler metric for CP n .There is a relation between L and 10 (CP n ) which it will be convenientto exploit:Lemma 2.3.2. Let M = CP n .(a) There is a short exact sequence of holomorphic bundles:0 ! 10 (M) ! L 1 n+1 ! 1 ! 0:(b) There is a natural isomorphism of complex bundles:| {z }n+1 timesT c (M) 1 ' L 1 n+1 = L L :Proof: Rather than attempting to give a geometric proof of this fact, wegive a combinatorial argument. Over U k we have functions zik which givecoordinates 0 i n for i 6= k. Furthermore, we have a section s k toL. We let fs k i gn i=0 give a frame for L 1n+1 = L L. We deneF : 10 (M) ! L 1 n+1 on U k by:F (dz k i )=s k i ; z k i s k k:We note that z k k =1and F (dzk k )=0sothis is well dened on U k. On theoverlap, we have the relations z k i =(zj k );1 z j i and:s k i =(z j k );1 s j i and dz k i =(z j k );1 dz j i ; (zj k );2 z j i dzj k :Thus if we compute in the coordinate system U j we have:F (dz k i )=(z j k );1 (s j i ; zj i sj j ) ; (zj k );2 z j i (sj k ; zj k sj j )= s k i ; z j i sk j ; z k i sk k + z j i sk j = s k i ; z k i sk kwhich agrees with the denition of F on the coordinate system U k . ThusF is invariantly dened. It is clear F is holomorphic and injective. We let

112 2.3. Characteristic Classes = L 1 n+1 = image(F ), and let be the natural projection. It is clearthat s k k is never in the image of F so sk k 6=0. SinceF (dz k j )=s k j ; z k j s k k = z k j (s j j ; sk k)we conclude that s j j = sk k so s = sk k is a globally dened non-zerosection to . This completes the proof of (a). We dualize to get a shortexact sequence:0 ! 1 ! L 1 n+1 ! T c (M) ! 0:These three bundles have natural ber metrics. Any short exact sequenceof vector bundles splits (although the splitting is not holomorphic) and thisproves (b).From assertion (b) it follows that the Chern class of T c (CP n ) is givenby:c(T c )=c(T c 1) = c(L L )=c(L ) n+1 =(1+x) n+1 :For example, if m = 1, then c(T c ) = (1 + x) 2 = 1+2x. In this case,c 1 (T c ) = e(T (S 2 )) and we computed R S 2 e(T (S)) = 2 so R S 2 x = 1 whichchecks with Lemma 2.3.1.If we forget the complex structure on T c (M) when M is holomorphic,then we obtain the real tangent space T (M). Consequently:andT (M) C = T c T cc(T (CP n ) C) =c(T c (CP n ) T c (CP n ))=(1+x) n+1 (1 ; x) n+1 =(1; x 2 ) n+1 :When we take into account the sign changes involved in dening the totalPontrjagin form, we conclude:Lemma 2.3.3.(a) If M = S 2 S 2 has dimension 2n,then R M e(T (M)) = (M) =2n .(b) If M = CP n has dimension 2n, thenc(T c (M)) = (1 + x) n+1 and p(T (M)) = (1 + x 2 ) n+1 :x 2 H 2 (CP n C) is the generator given by x = c 1 (L ) = ;c 1 (L), L isthe tautological line bundle over CP n , and L is the dual, the hyperplanebundle.The projective spaces form a dual basis to both the real and complexcharacteristic classes. Let be a partition of the positive integer k in the

Of Complex Projective Space 113form k = i 1 + + i j where we choose the notation so i 1 i 2 . Welet (k) denote the number of such partitions. For example, if k =4then(4) = 5 and the possible partitions are:4=4 4=3+1 4=2+2 4=2+1+1 4=1+1+1+1:We dene classifying manifolds:M c = CP i1 CP ij and M r = CP 2i1 CP 2ijto be real manifolds of dimension 2k and 4k.Lemma 2.3.4. Let k be a positive integer. Then:(a) Let constants c() be given. There exists a unique polynomial P (A) ofdegree k in the components of a k k complex matrix which is GL(k C)invariant such that the characteristic class dened by P satises:ZM c P (T c (M c )) = c()for every such partition of k.(b) Let constants c() be given. There exists a polynomial P (A) of degree2k in the components of a 2k 2k real matrix which isGL(2k R) invariantsuch that the characteristic class dened by P satises:ZM r P (T (M r )) = c()for every such partition of k. If P 0 is another such polynomial, thenP (A) =P 0 (A) for every skew-symmetric matrix A.In other words, the real and complex characteristic classes are completelydetermined by their values on the appropriate classifying manifolds.Proof: We prove (a) rst. Let P k denote the set of all such polynomialsP (A). We dene c = c i1 ...c ij 2P k , then by Lemma 2.1.3 the fc g forma basis for P k so dim(P k )=(k). The fc g are not a very convenient basisto work with. We will dene instead:H = ch i1 ...ch ij 2P kand show that the matrix:a( )=ZM c H (T c (M c ))is a non-singular matrix. This will prove the H also form a basis for P kand that the M c are a dual basis. This will complete the proof.

114 2.3. Characteristic ClassesThe advantage of working with the Chern character rather than with theChern class is that:ch i (T c (M 1 M 2 )) = ch i (T c M 1 T c M 2 )=ch i (T c M 1 )+ch i (T c M 2 ):Furthermore, ch i (T c M)=0if2i>dim(M). We dene the length `() =jto be the number of elements in the partition . Then the above remarksimply:a( )=0 if `() >`().Furthermore, if `() = `(), then a( ) =0 unless = . We dene thepartial order > if `() > `() and extend this to a total order. Thena( ) is a triangular matrix. To show it is invertible, it suces to showthe diagonal elements are non-zero.We rst consider the case in which = = k. Using the identityT c (CP k )1 =(L L ) (k +1 times), it is clear that ch k (T c CP k )=(k +1)ch k (L ). For a line bundle, ch k (L )=c 1 (L ) k =k!. If x = c 1 (L ) isthe generator of H 2 (CP k C), then ch k (T c CP k )=(k +1)x k =k! which doesnot integrate to zero. If = = fi 1 ...i j g then:a( )=c`()Y=1a(i i )where c is a positive constant related to the multiplicity with which the i appear. This completes the proof of (a).To prove (b) we replace M c by M c 2 = M r and H by H 2 . We computech 2i (T (M)) def= ch 2i (T (M) C) =ch 2i (T c M T c M)= ch 2i (T c M)+ch 2i (T c M)=2ch 2i (T c M):Using this fact, the remainder of the proof of (b) is immediate from thecalculations performed in (a) and this completes the proof.The Todd class and the Hirzebruch L-polynomial were dened usinggenerating functions. The generating functions were chosen so that theywould be particularly simple on the classifying examples:Lemma 2.3.5.(a) Let x j = ; j =2i be the normalized eigenvalues of a complex matrixA. We dene Td(A) =Td(x) = Q j x j=(1;e ;x j) as the Todd class. Then:ZM c Td(T c (M c )) = 1 for all :

Of Complex Projective Space 115(b) Let x j = j =2 where the eigenvalues of the skew-symmetric real matrixA are f 1 ...g. We dene L(A) = L(x) = Q j x j= tanh x j as theHirzebruch L-polynomial. Then R M r L(T (M r )) = 1 for all .We will use this calculation to prove the integral of the Todd class givesthe arithmetic genus of a complex manifold and that the integral of theHirzebruch L-polynomial gives the signature of an oriented real manifold.In each case, we only integrate the part of the total class which is of thesame degree as the dimension of the manifold.Proof: Td is a multiplicative class:Td(T c (M 1 M 2 )) = Td(T c (M 1 ) T c (M 2 )) = Td(T c (M 1 )) Td(T c (M 2 )):Similarly the Hirzebruch polynomial is multiplicative. This shows it sucesto prove Lemma 2.3.5 in the case = k so M = CP k or CP 2k .We use the decomposition T c (CP k ) 1=L L (k +1times)to compute Td(T c (CP k )) = [Td(x)] k+1 where x = c 1 (L ) is the generatorof H 2 (CP k R). Since x k integrates to 1, it suces to show the coecientof x k in Td(x) k+1 is 1 or equivalently to show:If k =0,then:Res x=0 x ;k;1 [Td(x) k+1 ] = Res x=0 (1 ; e ;x ) ;k;1 =1:(1 ; e ;x ) ;1 = x ; 1 ;1 2 x2 + = x1+ ;1 1 2 x + and the result follows. Similarly, ifk =1(1 ; e ;x ) ;2 = x ;2 (1 + x + )and the result follows. For larger values of k, proving this directly would beacombinatorial nightmare so we use instead a standard trick from complexvariables. If g(x) isany meromorphic function, then Res x=0 g 0 (x) =0. Weapply this to the function g(x) =(1; e ;x ) ;k for k 1 to conclude:This implies immediately that:Res x=0 (1 ; e ;x ) ;k;1 e ;x =0:Res x=0 (1 ; e ;x ) ;k;1 = Res x=0 (1 ; e ;x ) ;k;1 (1 ; e ;x )= Res x=0 (1 ; e ;x ) ;k =1by induction which completes the proof of assertion (a).

116 2.3. Characteristic ClassesWenow assume k is even and study CP k . Again, using the decompositionT c (CP k ) 1=L L it follows thatL(T (CP k )) = [L(x)] k+1 =xk+1tanh k+1 x :Since we are interested in the coecient of x k , we must show:Res x=0 tanh ;k;1 x =1if k is evenor equivalentlyRes x=0 tanh ;k x =1if k is odd.We recall thattanh x = ex ; e ;xe x + e ;xtanh ;1 x = 2+x2 + 2x + so the result is clear if k =1. We now proceed by induction. We dierentiatetanh ;k x to compute:This implies(tanh ;k x) 0 = ;k(tanh ;k;1 x)(1 ; tanh 2 x):Res x=0 tanh ;k+1 x =Res x=0 tanh ;k;1 xfor any integer k. Consequently Res x=0 tanh ;k x = 1 for any odd integerk since these residues are periodic modulo 2. (The residue at k even is, ofcourse, zero). This completes the proof.

2.4. The Gauss-Bonnet Theorem.Let P be a self-adjoint elliptic partial dierential operator of order d>0.If the leading symbol of P is positive denite, we derived an asymptoticexpansion for Trfe ;tP g in section 1.7. This is too general a setting in whichto work so we shall restrict attention henceforth to operators with leadingsymbol given by the metric tensor, as this is the natural category in whichto work.Let P : C 1 (V ) ! C 1 (V ) be a second order operator. We choose a localframe for V . Let x = (x 1 ...x m ) be a system of local coordinates. Letds 2 = g ij dx i dx j be the metric tensor and let g ij denote the inverse matrixdx i dx j = g ij is the metric on the dual space T (M). We assume P hasthe form:P = ;g ij @ 2 @I + a j + b@x i @x j @x jwhere we sumover repeated indices. The a j and b are sections to END(V ).We introduce formal variables for the derivatives of the symbol of P . Letg ij= = d x g ij a j= = d x a j b = = d x b:We will also use the notation g ij=kl... , a j=kl... ,andb =kl... for multiple partialderivatives. We emphasize that these variables are not tensorial but dependupon the choice of a coordinate system (and local frame for V ).There is a natural grading on these variables. We dene:ord(g ij= )=jj ord(a j= )=1+jj ord(b = )=2+jj:Let P be the non-commutative polynomial algebra in the variables of positiveorder. We always normalize the coordinate system so x 0 correspondsto the point (0 ... 0) and so that:g ij (x 0 )= ij and g ij=k (x 0 )=0:Let K(t x x) bethe kernel of e ;tP . ExpandK(t x x) X n0t n;m2 e n (x P )where e n (x P ) is given by Lemma 1.7.4. Lemma 1.7.5(c) implies:Lemma 2.4.1. e n (x P ) 2 P is a non-commutative polynomial in thejets of the symbol of P which is homogeneous of order n. If a n (x P ) =Tr e n (x P ) is the ber trace, then a n (x P ) is homogeneous of order n in

118 2.4. The Gauss-Bonnet Theoremthe components (relative to some local frame) of the jets of the total symbolof P .We could also have proved Lemma 2.4.1 using dimensional analysis andthe local nature of the invariants e n (x P ) instead of using the combinatorialargument given in Chapter 1.We specialize to the case where P = p is the Laplacian on p-forms.If p = 0, then 0 = ;g ;1 @=@x i fgg ij @=@x j g g = det(g ij ) 1=2 denesthe Riemannian volume dvol = gdx. The leading symbol is given by themetric tensor the rst order symbol is linear in the 1-jets of the metricwith coecients which depend smoothly on the metric tensor. The 0 thorder symbol is zero in this case. More generally:Lemma 2.4.2. Let x =(x 1 ...x m ) be a system of local coordinates onM. We use the dx I to provide a local frame for (T M). Relative to thisframe, we expand ( p ) = p 2 + p 1 + p 0 . p 2 = jj 2 I. p 1 is linear in the1-jets of the metric with coecients which depend smoothly on the g ij 's.p 0 is the sum of a term which is linear in the 2-jets of the metric and aterm which is quadratic in the 1-jets of the metric with coecients whichdepend smoothly on the g ij 's.Proof: We computed the leading symbol in the rst chapter. The remainderof the lemma follows from the P decomposition = d + d =d d d d. In at space, = ; i @2 =@x 2 i as computed earlier.If the metric is curved, we must also dierentiate the matrix representingthe Hodge operator. Each derivative applied to \" reduces the order ofdierentiation by one and increases the order in the jets of the metric byone.Let a n (x P )=Tr e n (x P ). Then:Tr e ;tP X n0t n;m2ZMa n (x P )dvol(x):For purposes of illustration, we give without proof the rst few terms inthe asymptotic expansion of the Laplacian. We shall discuss such formulasin more detail in the fourth chapter.Lemma 2.4.3.(a) a 0 (x p )=(4) ;1 dim( p )=(4) ;1; m(b) a 2 (x 0 )=(4) ;1 (;R ijij )=6.(c)a 4 (x 0 )= 14 ;12R ijijkk +5R ijij R klkl ; 2R ijik R ljlk +2R ijkl R ijkl:360In this lemma, \ " denotes multiple covariant dierentiation. a 0 , a 2 ,a 4 , and a 6 have been computed for p for all p, but as the formulas areextremely complicated, we shall not reproduce them here.p .

The Gauss-Bonnet Theorem 119We consider the algebra generated by the variables fg ij= g jj2 . IfX is a coordinate system and G a metric and if P is a polynomial inthese variables, we dene P (X G)(x 0 ) by evaluation. We always normalizethe choice of X so g ij (X G)(x 0 ) = ij and g ij=k (X G)(x 0 ) = 0, andwe omit these variables from consideration. We say that P is invariantif P (X G)(x 0 ) = P (YG)(x 0 ) for any two normalized coordinate systemsX and Y . We denote the common value by P (G)(x 0 ). For example, thescalar curvature K = ; 1 2 R ijij is invariant as are the a n (x p ).We let P m denote the ring of all invariant polynomials in the derivativesof the metric for a manifold of dimension m. We dened ord(g ij= )=jjlet P mn be the subspace of invariant polynomials which are homogeneousof order n. This is an algebraic characterization it is also useful to havethe following coordinate free characterization:Lemma 2.4.4. Let P 2P m , then P 2P mn if and only if P (c 2 G)(x 0 )=c ;n P (G)(x 0 ) for every c 6= 0.Proof: Fix c = 0 and let X be a normalized coordinate system for themetric G at the point x 0 . We assume x 0 = (0 ... 0) is the center of thecoordinate system X. Let Y = cX be a new coordinate system, then:@=@y i = c ;1 @=@x id y = c ;jj d xc 2 G(@=@y i @=@y j ) = G(@=@x i @=@x j )This implies that if A is any monomial of P that:g ij= (Yc 2 G)=c ;jj g ij= (X G):A(Yc 2 G)(x 0 )=c ; ord(A) A(X G)(x 0 ):Since Y is normalized coordinate system for the metric c 2 G, P (c 2 G)(x 0 )=P (Yc 2 G)(x 0 ) and P (G)(x 0 )=P (X G)(x 0 ). This proves the Lemma.If P 2 P m we can always decompose P = P 0 + + P n into homogeneouspolynomials. Lemma 2.4.4 implies the P j are all invariant separately.Therefore P m has a direct sum decomposition P m = P m0 P m1 P mn and has the structure of a graded algebra. Using Taylor's theorem,we can always nd a metric with the g ij= (X G)(x 0 )=c ij arbitraryconstants for jj 2 and so that g ij (X G)(x 0 )= ij , g ij=k (X G)(x 0 )=0.Consequently, if P 2 P m is non-zero as a polynomial, then we can alwaysnd G so P (G)(x 0 ) 6= 0soP is non-zero as a formula. It is for this reasonwe work with the algebra of jets. This is a pure polynomial algebra. Ifwe work instead with the algebra of covariant derivatives of the curvaturetensor, we must introduce additional relations which correspond to theBianchi identities as this algebra is not a pure polynomial algebra.We note nally that P mn is zero if n is odd since we may take c = ;1.Later in this chapter, we will let P mnp be the space of p-form valuedinvariants which are homogeneous of order n. A similar argument willshow P mnp is zero if n ; p is odd.Lemmas 2.4.1 and 2.4.2 imply:

120 2.4. The Gauss-Bonnet TheoremLemma 2.4.5. a n (x p ) denes an element of P mn .This is such an important fact that we give another proof based onLemma 2.4.4 to illustrate the power of dimensional analysis embodied inthis lemma. Fix c>0andlet p (c 2 G)=c ;2 p (G) be the Laplacian correspondingto the new metric. Since dvol(c 2 G)=c m dvol(G), we conclude:e ;t p(c 2 G) = e ;tc;2 p(G)K(t x x p (c 2 G)) dvol(c 2 G)=K(c ;2 t x x p (G)) dvol(G)XnK(t x x p (c 2 G)) = c ;m K(c ;2 t x x p (G))Xt n;m2 a n (x p (c 2 G)) c ;m c m c ;n t n;m2 a n (x p (G))na n (x p (c 2 G)) = c ;n a n (x p (G)):We expand a n (x p (G)) = P a np + r in a nite Taylor series aboutg ij = ij in the g ij= variables. Then if a np is the portion which ishomogeneous of order , we use this identity to show a np =0for n 6= and to show the remainder in the Tayor series is zero. This shows a n is ahomogeneous polynomial of order n and completes the proof.Since a n (x p )=0ifn isPodd, in many references the authors replace theasymptotic series by t ; m 2n tn a n (x p ), They renumber this sequencea 0 a 1 . . . rather than a 0 0a 2 0a 4 ... . We shall not adopt this notationalconvention as it makes dealing with boundary problems more cumbersome.H. Weyl's theorem on the invariants of the orthogonal group gives aspanning set for the spaces P mn :Lemma 2.4.6. We introduce formal variables R i1 i 2 i 3 i 4 i 5 ...i kfor the multiplecovariant derivatives of the curvature tensor. The order of such avariable is k +2. We consider the polynomial algebra in these variablesand contract on pairs of indices. Then all possible such expressions generateP m . In particular:f1g spans P m0 fR ijij g spans P m2fR ijijkk R ijij R klkl R ijik R ljlk R ijkl R ijkl g spans P m4 :This particular spanning set for P m4 is linearly independent and formsa basis if m 4. If m =3, dim(P 34 )=3while if m =2, dim(P 24 )=2so there are relations imposed if the dimension is low. The study of theseadditional relations is closely related to the Gauss-Bonnet theorem.There is a natural restriction mapr: P mn !P m;1n

The Gauss-Bonnet Theorem 121which is dened algebraically as follows. We letdeg k (g ij= )= ik + jk + (k)be the number of times an index k appears in the variable g ij= . Letgij= 2Pr(g ij= )=m;1 if deg m (g ij= )=00 if deg m (g ij= ) > 0:We extend r: P m !P m;1 to be an algebra morphism r(P ) is a polynomialin the derivatives of a metric on a manifold of dimension m ; 1. It is clearr preserves the degree of homogeneity.r is the dual of a natural extension map. Let G 0 be a metric on amanifold M 0 of dimension m ; 1. We dene i(G 0 ) = G + d 2 on themanifold M = M 0 S 1 where S 1 is the unit circle with natural parameter. If X 0 is a local coordinate system on M 0 , then i(X 0 ) = (x ) is a localcoordinate system on M. It is clear that:(rP)(X 0 G 0 )(x 0 0)=P (i(X 0 )i(G 0 ))(x 0 0 0 )for any 0 2 S 1 what we have done by restricting to product manifoldsM 0 S 1 with product metrics G 0 + d 2 is to introduce the relation whichsays the metric is at in the last coordinate. Restiction is simply thedual of this natural extension rP is invariant if P is invariant. Thereforer : P mn !P mn;1 .There is one nal description of the restriction map which will be useful.In discussing a H. Weyl spanning set, the indices range from 1 throughm. We dene the restriction by letting the indices range from 1 throughm ; 1. Thus R ijij 2P m2 is its own restriction in a formal sense of courser(R ijij )=0if m =2since there are no non-trivial local invariants over acircle.Theorem 2.4.7.(a) r: P mn !P m;1n is always surjective.(b) r: P mn !P m;1n is bijective if n

122 2.4. The Gauss-Bonnet TheoremTheorem 2.4.8. Let a n (x d + ) = P p (;1)p a n (x p ) 2 P nm be theinvariant of the de Rham complex. We showed in Lemma 1.7.6 that:ZMa n (x d + )dvol(x) =(M) if n = m0 if n 6= m.Then:(a) a n (x d + ) =0if either m is odd or if n

2.5. Invariance Theory and thePontrjagin Classes of the Tangent Bundle.In the previous subsection, we gave in Theorem 2.4.7 an axiomatic characterizationof the Euler class in terms of functorial properties. In thissubsection we will complete the proof of Theorem 2.4.7. We will also givea similar axiomatic characterization of the Pontrjagin classes which wewilluse in our discussion of the signature complex in Chapter 3.Let T : R m ! R m be the germ of a dieomorphism. We assume that:T (0) = 0 and dT (x) =dT (0) + O(x 2 ) for dT (0) 2 O(k):If X is any normalized coordinate system for a metric G, then TX is anothernormalized coordinate system for G. We dene an action of the group ofgerms of dieomorphisms on the polynomial algebra in the fg ij= g, jj 2variables by dening the evaluation:(T P )(X G)(x 0 )=P (TXG)(x 0 ):Clearly P is invariant if and only if T P = P for every such dieomorphismT .Let P be invariant and let A be a monomial. We let c(A P ) be thecoecient of A in P c(A P ) denes a linear functional on P for anymonomial A. We say A is a monomial of P if c(A P ) 6= 0. Let T j be thelinear transformation:T j (x k )=;xj if k = jx k if k 6= j.This is reection in the hyperplane dened by x j =0. Thenfor any monomial A. SinceT j (A) =(;1) deg j (A) AT j P = X (;1) deg j (A) c(A P )A = P = X c(A P )Awe conclude deg j (A) must be even for any monomial A of P . If A has theform:A = g i1 j 1 = 1...g ir j r = rwe dene the length of A to be:`(A) =r:It is clear 2`(A) + ord(A) = P j deg j(A) soord(A) is necessarily even if Ais a monomial of P . This provides another proof P mn =0if n is odd.

124 2.5. Invariance TheoryIn addition to the hyperplane reections, it is convenient to considercoordinate permutations. If is a permutation, then T (A) =A is denedby replacing each index i by (i) in the variables g ij=k... . Since T (P )=P ,we conclude the form of P is invariant under coordinate permutations orequivalently c(A P )=c(A P)forevery monomial A of P .We can use these two remarks to begin the proof of Theorem 2.4.7.Fix P 6= 0 with r(P ) = 0 and P 2 P mn . Let A be a monomial of P .Then r(P ) = 0 implies deg m (A) > 0 is even. Since P is invariant undercoordinate permutations, deg k (A) 2 for 1 k m. We construct thechain of inequalities:2m X kdeg k (A) =2`(A)+ord(A)= X (2 + j j) X 2j j = 2 ord(A) =2n:We have used the fact j j2 for 1 `(A). This implies m n so inparticular P =0if n2.Let P be a polynomial and assume Tz P = P for all z 2 S 1 . Then:(a) If g 12= divides a monomial A of P for some , then g 11= divides amonomial B of P for some .(b) If g ij= divides a monomial of A of P for some (i j), then g kl= dividesa monomial B of P for some and some (k l) where (1) = (1) + (2)and (2)=0.Of course, the use of the indices 1 and 2 is for convenience only. Thislemma holds true for any pair of indices under the appropriate invarianceassumption.We postpone the proof of this lemma for the moment and use it tocomplete the proof of Theorem 2.4.7. Let P 6= 0 2 P mm with r(P ) = 0.

Classes of the Tangent Bundle 125Let A be a monomial of P . We noted deg k (A) =2 for 1 k m and Ais a polynomial in the 2-jets of the metric. We decomposeA = g i1 j 1 =k 1 l 1...g ir j r =k r l rfor 2r = m.By making a coordinate permutation we can choose A so i 1 =1. If j 1 6=1,we can make a coordinate permutation to assume j 1 = 2 and then applyLemma 2.5.1(a) to assume i 1 = j 1 = 1. We let P 1 = P c(A P )A wherethe sum ranges over A of the form:A = g 11=k1 l 1...g ir j r =k r l rwhich are monomials of P . Then P 1 6= 0 and P 1 is invariant under coordinatetransformations which x the rst coordinate. Since deg 1 (A) = 2,the index 1 appears nowhere else in A. Thus k 1 is not 1 so we may makea coordinate permutation to choose A so k 1 =2. If l 2 6=2,then l 1 3 sowe may make a coordinate permutation to assume l 1 =3. We then applyLemma 2.5.1(b) to choose A a monomial of P 1 of the form:A = g 11=22 A 0 :We have deg 1 (A) = deg 2 (A) = 2 so deg 1 (A 0 ) = deg 2 (A 0 ) = 0 so theseindices do not appear in A 0 . We dene P 2 = P c(A P )A where the sumranges over those monomial A of P which are divisible by g 11=22 , thenP 2 6=0.We proceed inductively in this fashion to show nally thatA 0 = g 11=22 g 33=44 ...g m;1m;1=mmis a monomial of P so c(A 0 P) 6= 0. The function c(A 0 P)isa separatinglinear functional on the kernel of r in P mm and thereforedim(f P 2P mm : r(P )=0g) 1:We complete the proof of Theorem 2.4.7(c) by showing that r(E m )=0. IfE m is the Euler form and M = M 0 S 1 , then:E m (M) =E m (M 0 S 1 )=E m;1 (M 0 )E 1 (S 1 )=0which completes the proof dim N(r) =1and E m 2 N(r) spans.Before we begin the proof of Lemma 2.5.1, it is helpful to consider a fewexamples. If we take A = g 11=11 then it is immediate:@=@y 1 = a@=@x 1 + b@=@x 2 @=@y 2 = ;b@=@x 1 + a@=@x 2@=@y k = @=@x kfor k>2.

126 2.5. Invariance TheoryWe compute T z (A) by formally replacing each index 1 by a1+b2 and eachindex 2 by ;b1+a2 and expanding out the resulting expression. Thus, forexample:T z (g 11=11 )=g a1+b2a1+b2=a1+b2a1+b2= a 4 g 11=11 +2a 3 bg 11=12 +2a 3 bg 12=11+ a 2 b 2 g 11=22 + a 2 b 2 g 22=11 +4a 2 b 2 g 12=12+2ab 3 g 12=22 +2ab 3 g 22=12 + b 4 g 22=22 :We note that those terms involving a 3 b arose from changing exactly oneindex to another index (1 ! 2 or 2 ! 1) the coecient reects the multiplicity.Thus in particular this polynomial is not invariant.In computing invariance under the circle action, all other indices remainxed soT z (g 34=11 + g 34=22 )=(a 2 g 34=11 + b 2 g 34=22 +2abg 34=12+ a 2 g 34=22 + b 2 g 34=11 ; 2abg 34=12 )=(a 2 + b 2 )(g 34=11 + g 34=22 )is invariant. Similarly, itis easy to compute:T z (g 11=22 + g 22=11 ; 2g 12=12 )=(a 2 + b 2 ) 2 (g 11=22 + g 22=11 ; 2g 12=12 )so this is invariant. We note this second example is homogeneous of degree4 in the (a b) variables since deg 1 (A)+deg 2 (A) =4.With these examples in mind, we begin the proof of Lemma 2.5.1. LetP be invariant under the action of the circle acting on the rst two coordinates.We decompose P = P 0 + P 1 + where each monomial A of P jsatises deg 1 (A) + deg 2 (A) =j. If A is such a monomial, then T z (A) is asum of similar monomials. Therefore each oftheP j is invariant separatelyso we may assume P = P n for some n. By setting a = b = ;1, we see nmust be even. Decompose:T z (P )=a n P (0) + ba n;1 P (1) + + b n P (n)where P = P (0) . We use the assumption T z (P )=P and replace b by ;bto see:0=T (ab) (P ) ; T (a;b) (P )=2ban;1 P (1) +2b 3 a n;3 P (3) + We divide this equation by b and take the limit as b ; ! 0toshowP (1) =0.(In fact, it is easy to show P (2j+1) = 0 and P (2j) n=2=j P but as we shallnot need this fact, we omit the proof).

Classes of the Tangent Bundle 127We let A 0 denote a variable monomial which will play the same role asthe generic constant C of Chapter 1. We introduce additional notationwe shall nd useful. If A is a monomial, we decompose T (ab) A = an A +a n;1 bA (1) + . If B is a monomial of A (1) then deg 1 (B) = deg 1 (A) 1and B can be constructed from the monomial A by changing exactly oneindex 1 ! 2 or 2 ! 1. If deg 1 (B) = deg 1 (A) +1, then B is obtainedfrom A by changing exactly one index 2 ! 1 c(BA (1) ) is a negativeinteger which reects the multiplicity with which thischange can be made.If deg 1 (B) =deg 1 (A) ; 1, then B is obtained from A by changing exactlyone index 1 ! 2 c(BA (1) ) is a positive integer. We dene:A(1 ! 2) = X c(BA (1) )B summed over deg 1 (B) =deg 1 (A) ; 1A(2 ! 1) = X ;c(BA (1) )B summed over deg 1 (B) =deg 1 (A)+1A (1) = A(1 ! 2) ; A(2 ! 1):For example, if A =(g 12=33 ) 2 g 11=44 then n =6and:A(1 ! 2) = 2g 12=33 g 22=33 g 11=44 +2(g 12=33 ) 2 g 12=44A(2 ! 1) = 2g 12=33 g 11=33 g 11=44 :It is immediate from the denition that:XBc(BA(1 ! 2)) = deg 1 (A)andXBc(BA(2 ! 1)) = deg 2 (A):Finally, it is clear that c(BA (1) ) 6= 0if and only if c(A B (1) ) 6= 0,andthat these twocoecients will be opposite in sign, and not necessarily equalin magnitude.Lemma 2.5.2. Let P be invariant under the action of SO(2) on the rsttwo coordinates and let A be a monomial of P . Let B be a monomial ofA (1) . Then there exists a monomial of A 1 dierent from A so thatc(BA (1) )c(A P )c(BA (1)1 )c(A 1P) < 0:Proof: We know P (1) =0. We decomposeP (1) = X Ac(A P )A (1) = X Bc(A P )c(BA (1) )B:Therefore c(BP (1) )=0impliesXAc(A P )c(BA (1) )=0

128 2.5. Invariance Theoryfor all monomials B. If we choose B so c(BA (1) ) 6= 0then there must besome other monomial A 1 of P which helps to cancel this contribution. Thesigns must be opposite which proves the lemma.This lemma is somewhat technical and formidable looking, but it is exactlywhat we need about orthogonal invariance. We can now complete theproof of Lemma 2.5.1. Suppose rst that A =(g 12= ) k A 0 for k > 0 whereg 12= does not divide A 0 . Assume c(A P ) 6= 0.Let B = g 11= (g 12= ) k;1 A 0then c(BA (1) ) = ;k 6= 0. Choose A 1 6= A so c(A 1 P) 6= 0. Thenc(A 1 B (1) ) 6= 0. If c(A 1 B(2 ! 1)) 6= 0 then A 1 is constructed from Bby changing a 2 ! 1 index so A 1 = g 11= ... has the desired form. Ifc(A 1 B(1 ! 2)) 6= 0,we expand B(1 ! 2) = A+ terms divisible by g 11=for some . Since A 1 6= A, again A 1 has the desired form which proves (a).To prove (b), wechoose g ij= dividing some monomial of P . Let be chosenwith (1)+(2) = (1)+(2) and (k) =(k)fork>2 with (1) maximalso g uv= divides some monomial of P for some (u v). Suppose (2) 6=0, we argue for a contradiction. Set =((1)+1(2);1(3) ...(m)).Expand A = (g uv= ) k A 0 and dene B = g uv= (g uv= ) k;1 A 0 where g uv=does not divide A 0 . Then c(BA (1) ) = ;(2)k 6= 0 so we may chooseA 1 6= A so c(A 1 B (1) ) 6= 0. If c(A 1 B(2 ! 1)) 6= 0 then either A 1 isdivisible by g u0 v 0 = or by g uv= 0 where 0 (1) = (1) + 1, 0 (2) = (2) ; 1,and 0 (j) = (j) for j > 2. Either possibility contradicts the choice of as maximal so c(A 1 B(1 ! 2)) 6= 0. However, B(1 ! 2) = (1)A+ termsdivisible by g u0 v 0 = for some (u 0 v 0 ). This again contradicts the maximalityas A 6= A 1 and completes the proof of Lemma 2.5.1 and thereby ofTheorems 2.4.7 and 2.4.8.If I = f1 i 1 i p mg, let jIj = p and dx I = dx i1 ^ ^dx ip . A p-form valued polynomial is a collection fP I g = P for jIj = p ofpolynomials P P I in the derivatives of the metric. We will also sometimeswrite P = jIj=p P I dx I as a formal sum to represent P . If all the fP I g arehomogeneous of order n, we say P is homogeneous of order n. We dene:P (X G)(x 0 )= X IP I (X G) dx I 2 p (T M)to be the evaluation of such a polynomial. We say P is invariant ifP (X G)(x 0 ) = P (YG)(x 0 ) for every normalized coordinate systems Xand Y as before we denote the common value by P (G)(x 0 ). In analogywith Lemma 2.4.4 we have:Lemma 2.5.3. Let P be p-form valued and invariant. Then P is homogeneousof order n if and only if P (c 2 G)(x 0 ) = c p;n P (G)(x 0 ) for everyc 6= 0.The proof is exactly the same as that given for Lemma 2.4.4. The onlynew feature is that dy I = c p dx I which contributes the extra feature of c pin this equation.

Classes of the Tangent Bundle 129We dene P mnp to be the vector space of all p-form valued invariantswhich are homogeneous of order n on a manifold of dimension m. If P mpdenotes the vector space of all p-form valued invariant polynomials, thenwe have a direct sum decomposition P mp = L n P mnp exactly as in thescalar case p =0.We dene deg k (I) to be 1 if k appears in I and 0 if k does not appearin I.Lemma 2.5.4. Let P 2P mnp with P 6= 0. Then n ; p is even. If A is amonomial then A is a monomial of at most one P I . If c(A P I ) 6= 0then:deg k (A) + deg k (I)is always even.Proof: Let T be the coordinate transformation dened by T (x k )=;x kand T (x j )=x j for j 6= k. ThenXP = T (P )= (;1) deg k (A)+deg k (I) c(A P I ) dx IIAwhich implies deg k (A) + deg k (I) is even if c(A P I ) 6= 0. ThereforeXkdeg k (A)+deg k (I) =2`(A) + ord(A)+p =2`(A)+n + pmust be even. This shows n+p is even if P 6= 0.Furthermore, if c(A P I ) 6=0thenI is simply the ordered collection of indices k so deg k (A) is odd whichshows A is a monomial of at most one P I and completes the proof.We extend the Riemannian P metric to abermetric on (T M) C. Itis clear that P P = I P IP I since the fdx I g form an orthonormal basisat x 0 . P P is a scalar invariant. We use Lemma 2.5.1 to prove:Lemma 2.5.5. Let P be p-form valued and invariant under the action ofO(m). Let A be a monomial of P . Then there is a monomial A 1 of P withdeg k (A 1 )=0for k>2`(A).Proof: Let r = `(A) and let Pr 0 = P`(B)=r c(BP I) dx I 6= 0. Sincethis is invariant under the action of O(m), we may assume without lossof generality P = P r . We construct a scalar invariant by taking the innerproduct Q =(P P). By applying Lemma 2.5.1(a) and making a coordinatepermutation if necessary, we can assume g 11=1 divides some monomial ofQ. We apply 2.5.1(b) to the indices > 1 to assume 1 (k) = 0 for k > 2.g 11=1 must divide some monomial of P . LetXP 1 = c(g 11=1 A 0 P I )g 11=1 A 0 dx I 6=0:A 0 I

130 2.5. Invariance TheoryThis is invariant under the action of O(m;2) on the last m;2 coordinates.We letQ 1 =(P 1 P 1 ) and let g i2 j 2 = 2divide some monomial of Q 1 . If bothi 2 and j 2 are 2 we leave this alone. If i 2 2 and j 2 3 we perform acoordinate permutation to assume j 2 = 3. In a similar fashion if i 2 3and j 2 2, we choose this variable so i 2 =3. Finally, if both indices 3,we apply Lemma 2.5.1(a) to choose this variable so i 2 = j 2 =3. We applyLemma 2.5.1(b) to the variables k 4tochoose this variable so 2 (k) =0for k>4. If A 2 = g 11=1 g i2 j 2 = 2then:deg k (A 2 )=0for k>4 and A 2 divides some monomial of P .We continue inductively to construct A r = g 11=1 ...g ir j r = rso thatdeg k (A r )=0for k>2r and A r divides some monomial of P .Since every monomial of P has length r, this implies A r itself is a monomialof P and completes the proof.Let P j (G) = p j (TM) be the j th Pontrjagin form computed relative tothe curvature tensor of the Levi-Civita connection. If we expand p j interms of the curvature tensor, then p j is homogeneous of order 2j in thefR ijkl g tensor so p j is homogeneous of order 4j in the jets of the metric. Itis clear p j is an invariantly dened 4j-form so p j 2 P m4j4j . The algebragenerated by the p j is called the algebra of the Pontrjagin forms. ByLemma 2.2.2, this is also the algebra of real characteristic forms of T (M).If is a partition of k = i 1 + + i j we dene p = p i1 ...p ij 2P m4k4k .The fp g form a basis of the Pontrjagin 4k forms. By R Lemma 2.3.4, theseare linearly independent ifm =4k since the matrixM r p is non-singular.By considering products of these manifolds with at tori T m;4k we caneasily show that the fp g are linearly independent in P m4k4k if 4k m.We let (k) be the number of partitions of k this is the dimension of thePontrjagin forms.The axiomatic characterization of the real characteristic forms of thetangent space which is the analogue of the axiomatic characterization ofthe Euler class given in Theorem 2.4.7 is the following:Lemma 2.5.6.(a) P mnp =0if n

Classes of the Tangent Bundle 131and let A be a monomial of P . Use Lemma 2.5.5 to nd a monomial A 1 ofsome P I where I = f1 i 1 < 2`(A).Since deg ip(A) isodd (and hence non-zero) and since 2`(A) P j j = nas A is a polynomial in the jets of order 2 and higher, we estimate:p i p 2`(A) n:This proves P =0if np and Adx 1 ^ ^dx p appears in P:There is a natural restriction map r: P mnp ! P m;1np dened in thesame way as the restriction map r: P mn0 ! P m;1n0 discussed earlier.This argument shows r: P mnn ! P m;1nn is injective for n < m sincer(Adx 1 ^ ^dx p )=Adx 1 ^ ^dx p appears in r(P ). The Pontrjaginforms have dimension (k) for n = 4k. By induction, r m;n : P mnn !P nnn is injective so dim P mnn dim P nnn .We shall prove Lemma 2.5.6(b) for the special case n = m. If n isnot divisible by 4k, then dim P nnn = 0 which implies dim P mnn = 0.If n = 4k, then dim P nnn = (k) implies that (k) dim P mnn dim P nnn (k) so dim P mnn = (k). Since the Pontrjagin forms spana subspace of exactly dimension (k) inP mnn this will complete the proofof (b).This lemma is at the heart of our discussion of the index theorem. Weshall givetwo proofs for the case n = m = p. The rst is based on H. Weyl'stheorem for the orthogonal group and follows the basic lines of the proofgiven in Atiyah-Bott-Patodi. The second proof is purely combinatorial andfollows the basic lines of the original proof rst given in our thesis. TheH. Weyl based proof has the advantage of being somewhat shorter butrelies upon a deep theorem we have not proved here while the second proofalthough longer is entirely self-contained and has some additional featureswhich are useful in other applications.We review H. Weyl's theorem briey. Let V be a real vector space witha xed inner product. Let O(V ) denote the group of linear maps of V ! Vwhich preserve this inner product. Let Nk (V )=V V denote thek th tensor prodect of V . If g 2 O(V ), we extend g to act orthogonallyon Nk (V ). We let z 7! g(z) denote this action. Let f: Nk (V ) ! Rbe a multi-linear map, then we say f is O(V ) invariant if f(g(z)) = f(z)for every g 2 O(V ). By letting g = ;1, it is easy to see there are noO(V ) invariant maps if k is odd. We let k = 2j and construct a map

132 2.5. Invariance Theoryf 0 : Nk (V )=(V V ) (V V ) (V V ) ! R using the metricto map (V V ) ! R. More generally, if is any permutation of theintegers 1 through k, we dene z 7! z as a map from Nk (V ) ! Nk (V )and let f (z) =f 0 (z ). This will be O(V )invariant for any permutation .H. Weyl's theorem states that the maps ff g dene a spanning set for thecollection of O(V ) invariant maps.For example, let k = 4. Let fv i g be an orthonormal basis for V andexpress any z 2 N4 (V ) in the form a ijkl v i v j v k v l summed overrepeated indices. Then after weeding out duplications, the spanning set isgiven by:f 0 (z) =a iijj f 1 (z) =a ijij f 2 (z) =a ijjiwhere we sum over repeated indices. f 0 corresponds to the identity permutationf 1 corresponds to the permutation which interchanges the secondand third factors f 2 corresponds to the permutation which interchangesthe second and fourth factors. We note that these need not be linearly independentif dim V = 1 then dim( N4 V ) = 1 and f 1 = f 2 = f 3 . However,once dim V is large enough these become linearly independent.We are interested in p-form valued invariants. We take Nk (V ) wherek ; p is even. Again, there is a natural map we denote byf p (z) =f 0 (z 1 ) ^ (z 2 )where we decompose Nk (V )= N k;p (V ) Np (V ). We let f 0 act on therst k ; p factors and then use the natural map Np (V ) ! p (V ) on thelast p factors. If is a permutation, we setf p (z) =f p (z ). These maps areequivariant in the sense that f p (gz) =gf p (z) wherewe extend g to act on p (V )aswell. Again, these are a spanning set for the space of equivariantmulti-linear maps from Nk (V ) to p (V ). If k = 4 and p = 2, then aftereliminating duplications this spanning set becomes:f 1 (z) =a iijk v j ^ v k f 2 (z) =a ijik v j ^ v k f 3 (z) =a ijki v j ^ v kf 4 (z) =a jiki v j ^ v k f 5 (z) =a jiik v i ^ v k f 6 (z) =a jkii v j ^ v k :Again, these are linearly independent if dim V is large, but there are relationsif dim V is small. Generally speaking, to construct a map fromN k(V ) ! p (V ) we must alternate p indices (the indices j, k in this example)and contract the remaining indices in pairs (there is only one pairi i here).Theorem 2.5.7 (H. Weyl's theorem on the invariants of theorthogonal group). The space of maps ff p g constructed above spanthe space of equivariant multi-linear maps from N k V ! p V .The proof of this theorem is beyond the scope of the book and will beomitted. We shall use it to give a proof of Lemma 2.5.6 along the lines

Classes of the Tangent Bundle 133of the proof given by Atiyah, Bott, and Patodi. We will then give anindependent proof of Lemma 2.5.6 by other methods which does not relyon H. Weyl's theorem.We apply H. Weyl's theorem to our situation as follows. Let P 2P nnnthen P is a polynomial in the 2-jets of the metric. If we letX be a system ofgeodesic polar coordinates centered at x 0 , then the 2-jets of the the metricare expressible in terms of the curvature tensor so we can express P asa polynomial in the fR ijkl g variables which is homogeneous of order n=2.The curvature denes an element R 2 N4 (T (M)) since it has 4 indices.There are, however, relations among the curvature variables:R ijkl = R klij R ijkl = ;R jikl and R ijkl + R iklj + R iljk =0:We let V be the sub-bundle of N4 (T (M)) consisting of tensors satisfyingthese 3 relations.If n = 2, then P : V ! 2 (T (M)) is equivariant while more generally,P : N n=2 (V ) ! n (T (M)) is equivariant under the action of O(T (M)).(We use the metric tensor to raise and lower indices and identify T (M) =T (M)). Since these relations dene an O(T (M)) invariant subspace ofN 2n(T (M)), we extend P to be zero on the orthogonal complement ofN n=2(V )in N 2n (T (M)) to extend P to an equivariant action on the wholetensor algebra. Consequently, we can use H. Weyl's theorem to span P nnnby expressions in which we alternate n indices and contract in pairs theremaining n indices.For example, we compute:p 1 = C R ijab R ijcd dx a ^ dx b ^ dx c ^ dx dfor a suitable normalizing constant C represents the rst Pontrjagin form.In general, we will use letters a, b, c ... for indices to alternate on andindices i, j, k ... for indices to contract on. We let P be such an elementgiven by H.Weyl's theorem. There are some possibilities we can eliminateon a priori grounds. The Bianchi identity states:R iabc dx a ^ dx b ^ dx c = 1 3 (R iabc + R ibca + R icab ) dx a ^ dx b ^ dx c =0so that three indices of alternation never appear in any R ... variable. Sincethere are n=2 R variables and n indices of alternation, this implies each Rvariable contains exactly two indices of alternation. We use the Bianchiidentity again to express:R iajb dx a ^ dx b = 1 2 (R iajb ; R ibja ) dx a ^ dx b= 1 2 (R iajb + R ibaj ) dx a ^ dx b= ; 1 2 R ijba dx a ^ dx b = 1 2 R ijab dx a ^ dx b :

134 2.5. Invariance TheoryThis together with the other curvature identities means we can expressP in terms of R ijab dx a ^ dx b = ij variables. Thus P = P ( ij ) is apolynomial in the components of the curvature matrix where we regard ij 2 2 (T M). (This diers by various factors of 2 from our previousdenitions, but this is irrelevant to the present argument). P () is anO(n) invariant polynomial and thus is a real characteristic form. Thiscompletes the proof.The remainder of this subsection is devoted to giving a combinatorialproof of this lemma in the case n = m = p independent of H. Weyl'stheorem. Since the Pontrjagin forms span a subspace of dimension (k)if n = 4k we must show P nnn = 0 if n is not divisible by 4 and thatdim P 4k4k4k (k) since then equality must hold.We showed n =2j is even and any polynomial depends on the 2-jets ofthe metric. We improve Lemma 2.5.1 as follows:Lemma 2.5.8. Let P satisfy the hypothesis of Lemma 2.5.1 and be apolynomial in the 2-jets of the metric. Then:(a) Let A = g 12= A 0 be a monomial of P . Either by interchanging 1 and2 indices or by changing two 2 indices to 1 indices we can construct amonomial A 1 of the form A 1 = g 11= A 0 0 which is a monomial of P .(b) Let A = g ij=12 A 0 be a monomial of P . Either by interchanging 1and 2 indices or by changing two 2 indices to 1 indices we can construct amonomial of A 1 of the form A 1 = g i0 j 0 =11A 0 0 which is a monomial of P .(c) The monomial A 1 6= A. If deg 1 (A 1 ) = deg 1 (A)+2 then c(A P )c(A 1 P)> 0. Otherwise deg 1 (A 1 ) 6= deg 1 (A) and c(A P )c(A 1 P) < 0.Proof: We shall prove (a) as (b) is the same we will also verify (c). LetB = g 11= A 0 so c(BA(2 ! 1)) 6= 0. Then deg 1 (B) =deg 1 (A)+1. ApplyLemma 2.5.2 to nd A 1 6= A so c(BA (1)1) 6= 0. We noted earlier in theproof of 2.5.1 that A 1 must have the desired form. If c(A 1 B(2 ! 1)) 6= 0then deg 1 (A 1 )=deg 1 (B)+1 = deg 1 (A)+2 so A 1 is constructed from A bychanging two 2 to 1 indices. Furthermore, c(BA (1)1 ) > 0andc(BA(1) ) 0. If, on the other hand, c(A 1 B(1 ! 2)) 6= 0then deg 1 (A 1 ) = deg 1 (B) ; 1 = deg 1 (A) and A changes to A 1 by interchanginga 2 and a 1 index. Furthermore, c(BA (1)1 ) < 0andc(BA(1) ) < 0implies c(A P )c(A 1 P) < 0 which completes the proof.Let P 2 P nnn and express P = P 0 dx 1 ^ ^ dx n . P 0 = P is ascalar invariant which changes sign if the orientation of the local coordinatesystem is reversed. We identify P with P 0 for notational convenience andhenceforth regard P as a skew-invariant scalar polynomial. Thus deg k (A)is odd for every k and every monomial A of P .The indices with deg k (A) =1play a particularly important role in ourdiscussion. We say that an index i touches an index j in the monomial A

Classes of the Tangent Bundle 135if A is divisible by avariable g ij=:: or g ::=ij where \.." indicate two indiceswhich are not of interest.Lemma 2.5.9. Let P 2P nnn with P 6= 0. Then there exists a monomialA of P so(a) deg k (A) 3 all k.(b) If deg k (A) = 1 there exists an index j(k) which touches k in A. Theindex j(k) also touches itself in A.(c) Let deg j A + deg k A = 4. Suppose the index j touches itself and theindex k in A. There is a unique monomial A 1 dierent from A which canbe formed from A by interchanging a j and k index. deg j A 1 +deg k A 1 =4and the index j touches itself and the index k in A 1 . c(A P )+c(A 1 P)=0.Proof: Choose A so the number of indices with deg k (A) = 1 is minimal.Among all such choices, we choose A so the number of indices which touchthemselves is maximal. Let deg k (A) = 1 then k touches some index j =j(k) 6= k in A. Suppose A has the form A = g jk=:: A 0 as the other caseis similar. Suppose rst that deg j (A) 5. Use lemma 2.5.8 to nd A 1 =g kk=:: A 0 . Then deg k (A 1 ) 6= deg k (A) implies deg k (A 1 ) = deg k (A)+2=3.Also deg j (A 1 ) = deg j (A) ; 2 3 so A 1 has one less index with degree1 which contradicts the minimality of the choice of A. We suppose nextdeg j (A) = 1. If T is the coordinate transformation interchanging x j andx k , then T reverses the orientation so T P = ;P . However deg j (A) =deg k (A) =1implies T A = A which contradicts the assumption that A isa monomial of P . Thus deg j (A) =3which proves (a).Suppose j does not touch itself in A. We use Lemma 2.5.8 to constructA 1 = g kk=:: A 0 0. Then deg k (A 1 )=deg k (A)+2=3and deg j (A 1 )=deg j (A) ; 2 = 1. This is a monomial with the same number of indices ofdegree 1 but which has one more index (namely k) which touches itself.This contradicts the maximality ofA and completes the proof of (b).Finally, let A = f k : deg k (A) = g for = 1, 3. The map k 7! j(k)denes an injective map from A 1 !A 3 since no index of degree 3 can touchtwo indices of degree 1 as well as touching itself. The equalities:n = card(A 1 )+card(A 3 ) and 2n = X kdeg k (A) = card(A 1 )+3 card(A 3 )imply 2 card(A 3 ) = n. Thus card(A 1 ) = card(A 3 ) = n=2 and the mapk 7! j(k) is bijective in this situation.(c) follows from Lemma 2.5.8 where j = 1 and k =2. Since deg k (A) =1,A 1 cannot be formed by transforming two k indices to j indices so A 1 mustbe the unique monomial dierent from A obtained by interchanging theseindices. For example, if A = g jj=ab g jk=cd A 0 , then A 1 = g jk=ab g jj=cd A 0 .The multiplicities involved are all 1 so we can conclude c(A P )+c(A 1 P)=0 and not just c(A P )c(A 1 P) < 0.

136 2.5. Invariance TheoryBefore further normalizing the choice of the monomial A, wemust provea lemma which relies on cubic changes of coordinates:Lemma 2.5.10. Let P be a polynomial in the 2-jets of the metric invariantunder changes of coordinates of the form y i = x i + c jkl x j x k x l .Then:(a) g 12=11 and g 11=12 divide no monomial A of P .(b) Let A = g 23=11 A 0 and B = g 11=23 A 0 then A is a monomial of P if andonly if B is a monomial of P . Furthermore, c(A P ) and c(BP) have thesame sign.(c) Let A = g 11=22 A 0 and B = g 22=11 A 0 , then A is a monomial of P if andonly if B is a monomial of P . Furthermore, c(A P ) and c(BP) have thesame sign.Proof: We remark the use of the indices 1, 2, 3 is for notational convenienceonly and this lemma holds true for any triple of distinct indices.Since g ij (X G) = ij + O(x 2 ), g ij =kl(X G)(x 0 )=;g ij=kl (X G)(x 0 ). Furthermore,under changes of this sort, d y (x 0 ) = d x (x 0 ) if jj = 1. Weconsider the change of coordinates:y 2 = x 2 + cx 3 1 dy 2 = dx 2 +3cx 2 1 dx 1y k = k xdy k = dx kotherwiseotherwisewithg 12 (YG)=g 12 (X G)+3cx 2 1 + O(x 4 )g ij (YG)=g ij (X G)+O(x 4 ) otherwiseg 12=11 (YG)(x 0 )=g 12=11 (X G)(x 0 ) ; 6cg ij=kl (YG)(x 0 )=g ij=kl (X G)(x 0 ) otherwise.We decompose A =(g 12=11 ) A 0 . If > 0, T (A) =A;6c(g 12=11 ) ;1 A 0 +O(c 2 ). Since T (P )=P , and since there is no way to cancel this additionalcontribution, A cannot be a monomial of P so g 12=11 divides no monomialof P .Next we consider the change of coordinates:y 1 = x 1 + cx 2 1 x 2dy 1 = dx 1 +2cx 1 x 2 dx 1 + cx 2 1 dx 2y k = x kdy k = dx kotherwiseotherwisewithg 11=12 (YG)(x 0 )=g 11=12 (X G)(x 0 ) ; 4cg 12=11 (YG)(x 0 )=g 12=11 (X G)(x 0 ) ; 2cg ij=kl (YG)(x 0 )=g ij=kl (X G)(x 0 )otherwise:We notedg 12=11 divides no monomial of P . If A =(g 11=12 ) A 0 ,then > 0implies T (A) =A ; 4cv(g 11=12 ) ;1 A 0 + O(c 2 ). Since there would be no

Classes of the Tangent Bundle 137way to cancel such a contribution, A cannot be a monomial of P whichcompletes the proof of (a).Let A 0 0 be a monomial not divisible by anyofthevariables g 23=11 , g 12=13 ,g 13=12 , g 11=23 and let A(p q rs)=(g 23=11 ) p (g 12=13 ) q (g 13=12 ) r (g 11=23 ) s A 0 0.We set c(p q rs) = c(A(p q rs)P) and prove (b) by establishing somerelations among these coecients. We rst consider the change of coordinates,y 2 = x 2 + cx 2 1x 3 dy 2 = dx 2 +2cx 1 x 3 dx 1 + cx 2 1 dx 3 y k = x kdy k = dx kotherwiseotherwisewithg 12=13 (YG)(x 0 )=g 12=13 (X G)(x 0 ) ; 2cg 23=11 (YG)(x 0 )=g 23=11 (X G)(x 0 ) ; 2cg ij=kl (YG)(x 0 )=g ij=kl (X G)(x 0 )otherwise:We compute that:T (A(p q rs))= A(p q rs)+cf;2qA(p q ; 1rs) ; 2pA(p ; 1qrs)g + O(c 2 ):Since T (P )=P is invariant, we concludepc(p q rs)+(q +1)c(p ; 1q+1rs)=0:By interchanging the roles of 2 and 3 in the argument we also conclude:pc(p q rs)+(r +1)c(p ; 1qr+1s)=0:(We set c(p q rs)=0if any of these integers is negative.)Next we consider the change of coordinates:y 1 = x 1 + cx 1 x 2 x 3 dy 1 = dx 1 + cx 1 x 2 dx 3 + cx 1 x 3 dx 2 + cx 2 x 3 dx 1 y k = x kdy k = dx kotherwiseotherwisewithg 11=23 (YG)(x 0 )=g 11=23 (X G)(x 0 ) ; 2cg 12=13 (YG)(x 0 )=g 12=13 (X G)(x 0 ) ; cg 13=12 (YG)(x 0 )=g 13=12 (YG)(x 0 ) ; cso thatT (A(p q rs)) = A(p q rs)+cf;2sA(p q rs; 1); rA(p q r ; 1s) ; qA(p q ; 1rs)g + O(c 2 ):

138 2.5. Invariance TheoryThis yields the identities(q +1)c(p q +1rs; 1)+(r +1)c(p q r +1s; 1)+2sc(p q rs)=0:Let p 6= 0 and let A = g 23=11 A 0 0 be a monomial of P . Then c(p;1q+1rs)and c(p;1qr+1s) are non-zero and have the opposite sign as c(p q rs).Therefore (q +1)c(p ; 1q+1rs)+(r +1)c(p ; 1qr+1s)isnon-zerowhich implies c(p ; 1qrs+1)(s + 1) is non-zero and has the same sign asc(p q rs). This shows g 11=23 A 0 0 is a monomial of P . Conversely, ifA is nota monomial of P , the same argument shows g 11=23 A 0 0 is not a monomial ofP . This completes the proof of (b).The proof of (c) is essentially the same. Let A 0 0 be a monomial notdivisible by the variables fg 11=22 g 12=12 g 22=11 g and letA(p q r)=(g 11=22 ) p (g 12=12 ) q (g 22=11 ) r :Let c(p q r)=c(A(p q r)P). Consider the change of coordinates:y 1 = x 1 + cx 1 x 2 2 dy 1 = dx 1 + cx 2 2 dx 1 +2cx 1 x 2 dx 2 y k = x kdy k = dx kotherwiseotherwisewithg 11=22 (Yg)(x 0 )=g 11=22 (X G)(x 0 ) ; 4cg 12=12 (YG)(x 0 )=g 12=12 (X G)(x 0 ) ; 2cg ij=kl (YG)(x 0 )=g ij=kl (X G)(x 0 )otherwise:This yields the relation 2pc(p q r) +(q + 1)c(p ; 1q + 1r) = 0. Byinterchanging the rolesof1and2we obtain the relation 2rc(p q r)+(q +1)c(p q +1r; 1) = 0 from which (c) follows.This step in the argument is functionally equivalent to the use made ofthe fR ijkl g variables in the argument given previously whichusedH.Weyl'sformula. It makes use in an essential way of the invariance of P under awider group than just rst and second order transformations. For theEuler form, by contrast, we only needed rst and second order coordinatetransformations.We can now construct classifying monomials using these lemmas. Fixn = 2n 1 and let A be the monomial of P given by Lemma 2.5.9. Bymaking a coordinate permutation, we may assume deg k (A) =3fork n 1and deg k (A) = 1 for k > n 1 . Let x(i) = i + n 1 for 1 i n 0 we mayassume the index I touches itself and x(i) in A for i n 0 .We further normalize the choice of A as follows. Either g 11=ij or g ij=11divides A. Since deg 1 (A) = 3 this term is not g 11=11 and by Lemma 2.5.10it is not g 11=1x(1) nor g 1x(1)=11 . By Lemma 2.5.10(b) or 2.5.10(c), we mayassume A = g 11=ij ... for i, j 2. Since not both i and j can have degree

Classes of the Tangent Bundle 1391 in A, by making a coordinate permutation if necessary we may assumethat i = 2. If j = 2, we apply Lemma 2.5.9(c) to the indices 2 and x(2)to perform an interchange and assume A = g 11=2x(2) A 0 . The index 2musttouch itself elsewhere in A. We apply the same considerations to choose Ain the form A = g 11=2x(2) g 22=ij A 0 . If i or j is 1, the cycle closes and weexpress A = g 11=2x(2) g 11=1x(1) A 0 where deg k (A 0 )=0for k =1,2, 1+n 1 ,2+n 1 . If i, j > 2 we continue this argument until the cycle closes. Thispermits us to choose A to be a monomial of P in the form:A = g 11=2x(2) g 22=3x(3) ...g j;1j;1=jx(j) g jj=1x(1) A 0where deg k (A 0 )=0for 1 k j and n 1 +1 k n 1 + j.We wish to show that the length of the cycle involved is even. We applyLemma 2.5.10 to showB = g 2x(2)=11 g 3x(3)=22 ...g 1x(1)=jj A 0satises c(A P )c(BP) > 0. We apply Lemma 2.5.9 a total of j times toseeC = g 221x(1) g 33=2x(2) ...g 11=jx(j) A 0satises c(BP)c(C P)(;1) j > 0. We now consider the even permutation:(1) = j(k) =k ; 1 2 k jk j 0. However A = C so c(A P ) 2 (;1) j > 0which shows j is necessarily even. (This step is formally equivalent to usingthe skew symmetry of the curvature tensor to show that only polynomialsof even degree can appear to give non-zero real characteristic forms).We decompose A 0 into cycles to construct A inductively so that A hasthe form:A =fg 11=2x(2) ...g i1 i 1 =1x(1) gfg i1 +1i 1 +1=i 1 +2x(i 1 +2) ...g i1 +i 2 i 1 +i 2 =i 1 x(i 1 ) g ...where we decompose A into cycles of length i 1 i 2 . . . with n 1 = i 1 + +i j .Since all the cycles must have even length, `(A) = n=2 is even so n isdivisible by 4.We let n =4k and let be a partition of k = k 1 + + k j . We letA bedened using the above equation where i 1 =2k 1 , i 2 =2k 2 ... . By makinga coordinate permutation we can assume i 1 i 2 . We have shown

140 2.5. Invariance Theorythat if P =0,then c(A P)=0 for some . Since there are exactly (k)such partitions, we have constructed a family of (k) linear functionalson P nnn which form a seperating family. This implies dim P nnn < (k)which completes the proof.We conclude this subsection with a few remarks on the proofs we havegiven of Theorem 2.4.7 and Lemma 2.5.6. We know of no other proof ofTheorem 2.4.7 other than the one we have given. H. Weyl's theorem is onlyused to prove the surjectivity of the restriction map r and is inessential inthe axiomatic characterization of the Euler form. This theorem gives animmediate proof of the Gauss-Bonnet theorem using heat equation methods.It is also an essential step in settling Singer's conjecture for the Eulerform as we shall discuss in the fourth chapter. Thefactthatr(E m )=0is,of course, just an invariant statement of the fact E m is an unstable characteristicclass this makes it dicult to get hold of axiomatically in contrastto the Pontrjagin forms which are stable characteristic classes.We have discussed both of the proofs of Lemma 2.5.6 which exist inthe literature. The proof based on H. Weyl's theorem and on geodesicnormal coordinates is more elegant, but relies heavily on fairly sophisticatedtheorems. The second is the original proof and is more combinatorial. Itis entirely self-contained and this is the proof which generalizes to Kaehlergeometry to yield an axiomatic characterization of the Chern forms ofT c (M) for a holomorphic Kaehler manifold. We shall discuss this casein more detail in section 3.7.

2.6. Invariance Theory andMixed Characteristic Classesof the Tangent Space and of a Coecient Bundle.In the previous subsection, we gave in Lemma 2.5.6 an axiomatic characterizationof the Pontrjagin forms in terms of functorial properties. Indiscussing the Hirzebruch signature formula in the next chapter, it willbe convenient to have a generalization of this result to include invariantswhich also depend on the derivatives of the connection form on an auxilarybundle.Let V be a complex vector bundle. We assume V is equipped with aHermitian ber metric and let r be a Riemannian or unitary connectionon V . We let ~s = (s 1 ...s a ...s v ) be a local orthonormal frame for Vand introduce variables ! abi for the connection 1-formr(s a )=! abi dx i s b i.e., r~s = ! ~s:We introduce variables ! abi= = d x (! abi ) for the partial derivatives of theconnection 1-form. We shall also use the notation ! abi=jk... . We use indices1 a b v to index the frame for V and indices 1 i j k m forthe tangent space variables. We dene:ord(! abi= )=1+jj and deg k (! abi= )= ik + (k):We letQ be the polynomial algebra in the f! abi= g variables for jj 1 ifQ 2Qwe dene the evaluation Q(X ~s r)(x 0 ). We normalize the choice offrame ~s by requiring r(~s)(x 0 )=0. We also normalize the coordinate systemX as before so X(x 0 )=0, g ij (X G)(x 0 )= ij , and g ij=k (X G)(x 0 )=0. We say Q is invariant if Q(X ~s r)(x 0 ) = Q(Y~s 0 r)(x 0 ) for any normalizedframes ~s, ~s 0 and normalized coordinate systems X, Y we denotethis common value by Q(r) (although it also depends in principle on themetric tensor and the 1-jets of the metric tensor through our normalizationof the coordinate system X). We letQ mpv denote the space of all invariantp-form valued polynomials in the f! abi= g variables for jj 1 denedon a manifold of dimension m and for a vector bundle of complex berdimension v. We letQ mnpv denote the subspace of invariant polynomialshomogeneous of order n in the jets of the connection form. Exactly as wasdone for the P algebra in the jets of the metric, we can show there is adirect sum decompositionQ mpv = M nQ mnpv and Q mnpv =0 for n ; p odd.Let Q(gAg ;1 ) = Q(A) be an invariant polynomial of order q in thecomponents of a vv matrix. Then Q() denes an elementofQ m2qqv for

142 2.6. Invariance Theoryany m 2q. By taking Q Q we can dene scalar valued invariants and bytaking (Q) we can dene other form valued invariants in Q m2q+12q;1v .Thus there are a great many such form valued invariants.In addition to this algebra, we let R mnpv denote the space of p-formvalued invariants which are homogeneous of order n in the fg ij= ! abk= gvariables for jj 2 and jj 1. The spaces P mnp and Q mnpv are bothsubspaces of R mnpv . Furthermore wedge product gives a natural mapP mnp Q mn0 p 0 v !R mn+n0 p+p 0 v . We say that R 2R mppv is acharacteristicform if it is in the linear span of wedge products of Pontrjaginforms of T (M) and Chern forms of V . The characteristic forms are characterizedabstractly by the following Theorem. This is the generalizationof Lemma 2.5.6 which we shall need in discussing the signature and spincomplexes.Theorem 2.6.1.(a) R mnpv =0if n 2, then ! abi= dividessome other monomial of R where (1) = (1) + (2) and (2) = 0.The proof of this is exactly the same as that given for Lemma 2.5.1 andis therefore omitted.The proof of Theorem 2.6.1 parallels the proof of Lemma 2.5.6 for awhile so we summarize the argument briey. Let 0 6= R 2 R mnpv . Thesame argument given in Lemma 2.5.3 shows an invariant polynomial ishomogeneous of order n if R(c 2 G r) =c p;n R(G r) which gives a invariantdenition of the order of a polynomial. The same argument as givenin Lemma 2.5.4 shows n ; p must be even and that if A is a monomialof R, A is a monomial of exactly one of the R I . If c(A R I ) = 0 thendeg k (A) + deg k (I) isalways even. We decompose A in the form:A = g i1 j 1 = 1...g iq j q = q! a1 b 1 k 1 = 1...! ar b r k r = r= A g A !and dene `(A) = q + r to the length of A. We argue using Lemma2.6.2 to choose A so deg k (A g ) = 0 for k > 2q. By making a coordinatepermutation we can assume that the k 2q + r for 1 r. We

And Mixed Characteristic Classes 143apply Lemma 2.6.2(c) a total of r times to choose the i so 1 (k) =0fork>2q + r +1, 2 (k) = 0 for k>2q + r +2 ..., r (k) = 0 for k>2q +2r.This chooses A so deg k (A) = 0 for k > 2`(A). If A is a monomial of R Ifor I = f1 i 1 < < i p mg then deg ip (A) is odd. We estimatep i p 2`(A) P j j + P (j j +1)=n so that P mnpv = f0g if np= n and Adx 1 ^^dx p appears in R. There is a natural restrictionmapr: R mnpv !R m;1npvand our argument shows r: R mnnv !R m;1nnv is injective for n 0 for k sA otherwise.The only additional relations imposed are to set g ij=kl = 0 if any of theseindices exceeds s and to set ! abi=j =0if either i or j is less than or equalto s.We use these projections to reduce the proof of Theorem 2.6.1 to thecase in which R 2 Q tttv . Let 0 6= R 2 R nnnv and let A = A g A ! be amonomial of R. Let s = 2`(A g )=ord(A g ) and let t = n ; s = 2`(A ! )=

144 2.6. Invariance Theoryord(A ! ). We choose A so deg k (A g )=0fork>s. Since deg k (A) 1mustbe odd for each index k, we can estimate:X X Xt deg k (A) = deg k (A ! ) k>sk>skdeg k (A ! )=ord(A ! )=t:As all these inequalities must be equalities, we conclude deg k (A ! ) = 0for k s and deg k (A ! ) = 1 for k > s. This shows in particular that (st) (R) 6= 0for some (s t) so thatMs+t=n (st) : R nnnv !Ms+t=nP sss Q tttvis injective.We shall prove that Q tttv consists of characteristic forms of V . Weshowed earlier that P sss consists of Pontrjagin forms of T (M). The characteristicforms generated by the Pontrjagin forms of T (M) and of V areelements of R nnnv and (st) just decomposes such products. Thererfore is surjective when restricted to the subspace of characteristic forms. Thisproves is bijective and also that R nnnv is the space of characteristicforms. This will complete the proof of Theorem 2.6.1.We have reduced the proof of Theorem 2.6.1 to showing Q tttv consistsof the characteristic forms of V . We noted that 0 6= Q 2 Q tttv is apolynomial in the f! abi=j g variables and that if A is a monomial of Q,then deg k (A) = 1 for 1 k t. Since ord(A) = t is even, we concludeQ tttv =0if t is odd.The components of the curvature tensor are given by: abij = ! abi=j ; ! abi=j and ab = X abij dx i ^ dx jup to a possible sign convention and factor of 1 which play no role in this2discussion. If A is a monomial of P , we decompose:A = ! a1 b 1 i 1 =i 2...! au b u i t;1 =i twhere 2u = t.All the indices i are distinct. If is a permutation of these indices, thenc(A P )=sign()c(A P). This implies we can express P in terms of theexpressions:A =(! a1 b 1 i 1 =i 2; ! a1 b 1 i 2 =i 1)...(! au b u i t;1 =i t; ! au b u i t =i t;1)= a1 b 1 i 1 i 2... au b u i t;1 i tdx i1 ^ ^dx itdx i1 ^ ^dx itAgain, using the alternating nature of these expression, we can express Pin terms of expressions of the form: a1 b 1^ ^ au b u

And Mixed Characteristic Classes 145so that Q = Q() is a polynomial in the components ab of the curvature.Since the value of Q is independent of the frame chosen, Q is the invariantunder the action of U(v). Using the same argument as that given in theproof of Lemma 2.1.3 we see that in fact Q is a characteristic form whichcompletes the proof.We conclude this subsection with some uniqueness theorems regardingthe local formulas we have been considering. If M is a holomorphic manifoldof real dimension m and if V is a complex vector bundle of berdimension v, we let R chmmpv denote the space of p-form valued invariantsgenerated by the Chern forms of V and of T c (M). There is a suitable axiomaticcharacterization of these spaces using invariance theory for Kaehlermanifolds which we shall discuss in section 3.7. The uniqueness result weshall need in proving the Hirzebruch signature theorem and the Riemann-Roch theorem is the following:Lemma 2.6.3.(a) Let 0 6= R 2 R mmmv then there exists (MV ) so MRis oriented andMR(G r) 6= 0.(b) Let 0 6= R R 2R chmmmv then there exists (MV ) so M is a holomorphicmanifold andMR(G r) 6= 0.Proof: We prove (a) rst. Let = f1 i 1 i g be a partitionk() =i 1 + +i for 4k = s m. Let fM r g be the collection of manifoldsof dimension s discussed in Lemma 2.3.4. Let P be the corresponding realcharacteristic form soZM r P (G) = :The fP g forms a basis for P mss for any s m. We decomposeR = X P Q for Q 2Q mttk , where t + s = m.This decomposition is, of course, nothing but the decomposition dened bythe projections (st) discussed in the proof of the previous lemma.Since R 6= 0, at least one of the Q 6=0. We choose so k() is maximalwith Q 6= 0. We consider M = M r M 2 and (Vr) = 2(V 2 r 2 )where (V 2 r 2 ) is a bundle over M 2 which will be specied later. Then wecompute:ZMP (G)Q (r) =0unless ord(Q ) dim(M 2 ) since r is at along M. r This implies k() k() so if this integral is non-zero k() =k() by the maximalityof. ThisimpliesZZZP (G)Q (r) = P (G) Q (r 2 )= Q (r 2 ):MZM 2 M 2M r

146 2.6. Invariance TheoryThis shows thatZMR(G r) =ZM 2Q (r 2 )and reduces the proof of this lemma to the special case Q 2Q mmmv .Let A be a v v complex matrix and let fx 1 ...x v g be the normalizedeigenvalues of A. If 2k = m and if is a partition of k we denex = x i 11...x i then x is a monomial of Q(A) for some . We let M = M 1 M with dim(M j ) = 2i j and let V = L 1 L 1 v; where the L j areline bundles over M j . If c(x Q)isthe coecient of x in Q, then:ZMQ(r) =c(x Q)Y Zj=1M jc 1 (L j ) i j:This is, of course, nothing but an application of the splitting principle.This reduces the proof of this lemma to the special case Q 2Q mmm1 .If Q = c k 1 we take M = S 2 S 2 to be the k-fold product of twodimensional spheres. We let L j be a line bundle over the j th factor of S 2and let L = L 1 L k . Then c 1 (L) =c 1 (L j ) soZMc 1 (L) k = k!kY Zj=1S 2 c 1 (L j )which reduces the proof to the case m = 2 and k = 1. We gave an examplein Lemma 2.1.5 of a line bundle over S 2 so R S 2 c 1 (L) = 1. Alternatively,if we use the Gauss-Bonnet theorem with L = T c (S 2 ), thenRS 2 c 1 (T c (S 2 )) = R S 2 e 2 (T (S 2 )) = (S 2 ) = 2 6= 0 completes the proof of(a). The proof of (b) is the same where we replace the real manifolds M r bythe corresponding manifolds M c and the real basis P by the correspondingbasis P c of characteristic forms of T c (M). The remainder of the proofis the same and relies on Lemma 2.3.4 exactly as for (a) and is thereforeomitted in the interests of brevity.

CHAPTER 3THE INDEX THEOREMIntroductionIn this the third chapter, we complete the proof of the index theorem forthe four classical elliptic complexes. We give a proof of the Aityah-Singertheorem in general based on the Chern isomorphism between K-theory andcohomology (which is not proved). Our approach is to use the results of therst chapter to show there exists a suitable formula with the appropriatefunctorial properties. The results of the second chapter imply it mustbeacharacteristic class. The normalizing constants are then determined usingthe method of universal examples.In section 3.1, we dene the twisted signature complex and prove theHirzebruch signature theorem. We shall postpone until section 3.4 thedetermination of all the normalizing constants if we take coecients in anauxilary bundle. In section 3.2, we introduce spinors as a means of connectingthe de Rham, signature and Dolbeault complexes. In section 3.3, wediscuss the obstruction to putting aspinstructure on a real vector bundlein terms of Stieel-Whitney classes. We computethecharacteristic classesof spin bundles.In section 3.4, we discuss the spin complex and the ^A genus. In section3.5, we use the spin complex together with the spin c representationto discuss the Dolbeault complex and to prove the Riemann-Roch theoremfor almost complex manifolds. In sections 3.6 and 3.7 we give anothertreatment of the Riemann-Roch theorem based on a direct approach forKaehler manifolds. For Kaehler manifolds, the integrands arising from theheat equation can be studied directly using an invariant characterizationof the Chern forms similar to that obtained for the Euler form. These twosubsections may be deleted by a reader not interested in Kaehler geometry.In section 3.8, wegive the preliminaries we shall need to provetheAtiyah-Singer index theorem in general. The only technical tool we will use whichwe do not prove is the Chern isomorphism between rational cohomology andK-theory. We give a discussion of Bott periodicity using Cliord algebras.In section 3.9, weshow that the index can be treated as a formula in rationalK-theory. We use constructions based on Cliord algebras to determine thenormalizing constants involved. For these two subsections, some familaritywith K-theory is helpful, but not essential.Theorems 3.1.4 and 3.6.10 were also derived by V. K. Patodi using acomplicated cancellation argument asa replacement of the invariance theorypresented in Chapter 2. A similar although less detailed discussionmay also be found in the paper of Atiyah, Bott and Patodi.

3.1. The Hirzebruch Signature Formula.The signature complex is best described using Cliord algebras as theseprovide a unied framework in which to discuss (and avoid) many of the signs which arise in dealing with the exterior algebra directly. Thereader will note that we are choosing the opposite sign convention for ourdiscussion of Cliord algebras from that adopted in the example of Lemma2.1.5. This change in sign convention is caused by the p ;1 present indiscussing the symbol of a rst order operator.Let V be a real vector with a positive denite inner product. The Cliordalgebra CLIF(V ) is the universal algebra generated by V subject to therelationsv v +(v v) =0for v 2 V:If the fe i g are an orthonormal basis for V and if I = f1 i 1 <

Hirzebruch Signature Formula 149so the map w 7! c(w)1 denes a vector space isomorphism (which is not, ofcourse an algebra morphism) between CLIF(V ) and (V ). Relative to anorthonormal frame, we simply replace Cliord multiplication by exteriormultiplication.Since these constructions are all independent of the basis chosen, theyextend to the case in which V is a real vector bundle over M with aber metric which is positive denite. We dene CLIF(V ), (V ), andc: CLIF(V ) ! END((V )) as above. Since we only want to deal withcomplex bundles, we tensor with C at the end to enable us to view thesebundles as being complex. We emphasize, however, that the underlyingconstructions are all real.Cliord algebras provide a convenient way to describe both the de Rhamand the signature complexes. Let (d + ): C 1 ((T M)) ! C 1 ((T M))be exterior dierentiation plus its adjoint as discussed earlier. The leadingsymbol of (d+) is p ;1(ext();int()) = p ;1c(). We use the followingdiagram to dene an operator A letr be covariant dierentiation. Then:A: C 1 ((T M)) r ! C 1 (T M (T M)) c ! C 1 ((T M)):A is invariantly dened. If fe i g is a local orthonormal frame for T (M)which we identify with T (M), then:A(!) = X i(ext(e i ) ; int(e i ))r ei (!):Since the leading symbol of A is p ;1c(), these two operators have thesame leading symbol so (d+);A = A 0 is an invariantly dened 0 th orderoperator. Relative to a coordinate frame, we can express A 0 as a linearcombination of the 1-jets of the metric with coecients which are smoothin the fg ij g variables. Given any point x 0 , we can always choose a frameso the g ij=k variables vanish at x 0 so A 0 (x 0 ) = 0 so A 0 0. This provesA = (d + ) is dened by this diagram which gives a convenient way ofdescribing the operator (d + ) in terms of Cliord multiplication.This trick will be useful in what follows. If A and B are natural rstorder dierential operators with the same leading symbol, then A = Bsince A ; B is a 0 th order operator which is linear in the 1-jets of themetric. This trick does not work in the holomorphic category unless weimpose the additional hypothesis that M is Kaehler. This makes the studyof the Riemann-Roch theorem more complicated as we shall see later sincethere are many natural operators with the same leading symbol.We let 2 END((T M)) be dened by:(! p )=(;1) p ! pfor ! p 2 p (T M):

150 3.1. Hirzebruch Signature FormulaIt is immediate thatext() = ; ext() and int() = ; int()so that ;(d + ) and (d + ) have the same leading symbol. This implies(d + ) =;(d + ):We decompose (T M)= e (T M) o (T M)into the dierential formsof even and odd degree. This decomposes (T M)into the 1 eigenspacesof . Since (d + ) anti-commutes with , we decompose(d + ) eo : C 1 ( eo (T M)) ! C 1 ( eo (T M))where the adjoint of(d+) e is (d+) o . This is, of course, just the de Rhamcomplex, and the index of this elliptic operator is (M).If dim(M) = m is even and if M is oriented, there is another naturalendomorphism 2 END((T M)). It can be used to dene an ellipticcomplex over M called the signature complex in just the same way thatthe de Rham complex was dened. Let dvol 2 m (T M) be the volumeform. If fe i g is an oriented local orthonormal frame for T M,thendvol =e 1 ^ ^e m . We can also regard dvol = e 1 e m 2 CLIF(T M) andwe deneWe compute: =( p ;1) m=2 c(dvol) =( p ;1) m=2 c(e 1 )...c(e m ): 2 =(;1) m=2 c(e 1 e m e 1 e m )=(;1) m=2 (;1) m+(m;1)++1=(;1) (m+(m+1)m)=2 =1:Because m is even, c() = ;c() so anti-commutes with the symbol of(d + ). If we decompose (T M) = + (T M) ; (T M) into the 1eigenvalues of , then (d + ) decomposes to dene:(d + ) : C 1 ( (T M)) ! C 1 ( (T M))where the adjoint of (d + ) + is (d + ) ; . We dene:signature(M) = index(d + ) +to be the signature of M. (This is also often refered to as the index ofM, but we shall not use this notation as it might be a source of someconfusion).

Hirzebruch Signature Formula 151We decompose the Laplacian = + ; sosignature(M) = dim N( + ) ; dim N( ; ):Let a s n(x orn) = a n (x + ) ; a n (x ; ) be the invariants of the heat equationthey depend on the orientation orn chosen. Although we have complexiedthe bundles (T M) and CLIF(T M), the operator (d+) is real.If m 2 (4), then is pure imaginary. Complex conjugation denes anisomorphism + (T M) ' ! ; (T M) and + ' ! ; :This implies signature(M) = 0 and a s n(x orn) = 0 in this case. We canget a non-zero index if m 2 (4) if we take coecients in some auxiliarybundle as we shall discuss shortly.If m 0 (4), then is a real endomorphism. In general, we compute:(e 1 ^ ^e p )= ( p ;1) m=2 e 1 e m e 1 e p=( p ;1) m=2 (;1) p(p;1)=2 e p+1 ^ ^e m :If \" is the Hodge star operator discussed in the rst sections, then p =( p ;1) m=2+p(p;1) pacting on p-forms. The spaces p (T M) m;p (T M)areinvariant under. If p 6= m ; p there is a natural isomorphism p (T M) ! ( p (T M) m;p (T M)) by ! p 7! 1 2 (! p ! p ):This induces a natural isomorphism from p' ! on ( p (T M) m;p (T M)) so these terms all cancel o in the alternating sum and the only contributionis made in the middle dimension p = m ; p.If m = 4k and p = 2k then = . We decompose N( p ) = N( + p ) N( ; p ) so signature(M) = dim N( + p ) ; dim N( ; p ). There is a naturalsymmetric bilinear form on H 2k (T M C) = N( p ) dened byI( 1 2 )=ZM 1 ^ 2 :If we use the de Rham isomorphism to identify de Rham and simplicialcohomology, then this bilinear form is just the evaluation of the cup product

152 3.1. Hirzebruch Signature Formulaof two cohomology classes on the top dimensional cycle. This shows I canbe dened in purely topological terms.The index of a real quadratic form is just the number of +1 eigenvaluesminus the number of ;1 eigenvalues when it is diagonalized over R. Sincewe seeI( ) =ZM ^ =ZM( )dvol = ( ) L 2 index(I) = dimf+1 eigenspace of on H p g; dimf;1 eigenspace of on H p g= dim N( + ) ; dim N( ; ) = signature(M):This gives a purely topological denition of the signature of M in termsof cup product. We note that if we reverse the orientation of M, then thesignature changes sign.Example: Let M = CP 2k be complex projective space. Let x 2 H 2 (M C)be the generator. Since x k is the generator of H 2k (M C) and since x k ^x k = x 2k is the generator of H 4k (M C), we conclude that x k = x k .dim N( + 2k )=1and dim N(; 2k )=0so signature(CP 2k) =1.An important tool in the study of the de Rham complex was its multiplicativeproperties under products. Let M i be oriented even dimensionalmanifolds and let M = M 1 M 2 with the induced orientation. Decompose:(T M)=(T M 1 ) (T M 2 )CLIF(T M) = CLIF(T M 1 ) CLIF(T M 2 )as graded non-commutative algebras|i.e.,(! 1 ! 2 ) (! 0 1 ! 0 2)=(;1) deg ! 2deg ! 0 1 (!1 ! 0 1) (! 2 ! 0 2)for = either ^ or . Relative to this decomposition, we have:This implies that: = 1 2 where the i commute. + (M)= + (T M 1 ) + (T M 2 ) ; (T M 1 ) ; (T M 2 ) ; (M)= ; (T M 1 ) + (T M 2 ) + (T M 1 ) ; (T M 2 )N( + )=N( + 1 ) N(+ 2 ) N(; 1 ) N(; 2 )N( ; )=N( ; 1 ) N(+ 2 ) N(+ 1 ) N(; 2 )signature(M) = signature(M 1 ) signature(M 2 ):

Hirzebruch Signature Formula 153Example: Let be a partition of k = i 1 + + i j and M r = CP 2i1 CP 2ij . Then signature(M) r = 1. Therefore if L k is the HirzebruchL-polynomial,by Lemma 2.3.5.signature(M r )=ZM p L k (T (M r ))We can now begin the proof of the Hirzebruch signature theorem. Weshall use the same argumentaswe used to prove the Gauss-Bonnet theoremwith suitable modications. Let m =4k and let a s n(x orn) =a n (x + ) ;a n (x ; ) be the invariants of the heat equation. By Lemma 1.7.6:Z a s 0if n 6= mn(x orn) =signature(M) if n = mMso this gives a local formula for signature(M). We can express functoriallyin terms of the metric tensor. We can nd functorial local frames for relative to any oriented coordinate system in terms of the coordinateframes for (T M). Relative to such a frame, we express the symbol of functorially in terms of the metric. The leading symbol is jj 2 I therst order symbol is linear in the 1-jets of the metric with coecientswhich depend smoothly on the fg ij g variables the 0 th order symbol islinear in the 2-jets of the metric and quadratic in the 1-jets of the metricwith coecients which depend smoothly on the fg ij g variables. By Lemma2.4.2, we conclude a s n(x orn) is homogeneous of order n in the jets of themetric.It is worth noting that if we replace the metric G by c 2 G for c>0, thenthe spaces are not invariant. On p we have:(c 2 G)(! p )=c 2p;m (G)(! p ):However, in the middle dimension we have is invariant as are the spaces p for 2p = m. Clearly p (c 2 G) = c ;2 p (G). Since a s n(x orn) onlydepends on the middle dimension, this provides another proof that a s n ishomogeneous of order n in the derivatives of the metric sincea s n(x orn)(c 2 G)=a n (x c ;2 + p ) ; a n (x c ;2 ; p )= c ;n a n (x + p ) ; c ;n a n (x ; p )=c ;n a s n(x orn)(G):If we reverse the orientation, weinterchange the roles of + and ; so a s nchanges sign if we reverse the orientation. This implies a s n can be regardedas an invariantly dened m-form a s n(x) =a s n(x orn) dvol 2P mnm .

154 3.1. Hirzebruch Signature FormulaTheorem 3.1.1. Let a s n = fa n (x + );a n (x ; )g dvol 2P mnm then:(a) a s n =0if either m 2 (4) or if n

Hirzebruch Signature Formula 155on M 1 # M 2 then signature(M 1 # M 2 ) = signature(M 1 ) + signature(M 2 ),if m 0 (4).Proof: (a) is an immediate consequence of the fact wehave local formulasfor the Euler characteristic and for the signature. To prove (b), we notethat the two disks we are punching out glue together to form a sphere. Weuse the additivity of local formulas to prove the assertion about X. Thesecond assertion follows similarly if we note signature(S m )=0.This corollary has topological consequences. Again, we present just oneto illustrate the methods involved:Corollary 3.1.3. Let F ! CP 2j ! M be a nite covering. ThenjF j =1and M = CP 2j .Proof: (CP 2j )=2j +1soas2j +1=jF j(M), we conclude jF j mustbe odd. Therefore, this covering projection is orientation preserving so M isorientable. The identity 1 = signature(CP 2j ) = jF j signature(M) impliesjF j =1and completes the proof.If m 2 (4), the signature complex does not give a non-trivial index.We twist by taking coecients in an auxiliary complex vector bundle V toget a non-trivial index problem in any even dimension m.Let V be a smooth complex vector bundle of dimension v equipped witha Riemannian connection r. Wetake the Levi-Civita connection on T (M)andon(T M) and let r be the tensor product connection on (T M)V .We dene the operator (d + ) V on C 1 ((T M) V ) using the diagram:(d + ) V : C 1 ((T M) V ) r ! C(T M (T M) V )c1;;! C 1 ((T M) V ):We have already noted that if V = 1 is the trivial bundle with at connection,then the resulting operator is (d + ).We dene V = 1, then a similar argument to that given for thesignature complex shows V 2 = 1 and V anti-commutes with (d + ) V . The1 eigenspaces of V are (T M) V and the twisted signature complexis dened by the diagram:(d + ) V : C1 ( (T M) V ) ! C 1 ( (T M) V )where as before (d + ) ; V is the adjoint of (d + )+ V . We let Vassociated Laplacians and dene:be thesignature(MV ) = index((d + ) + V ) = dim N(+ V ) ; dim N(; V )ZMa s n(x V )=fa n (x + V ) ; a n(x ; V )g dvol 2 ma s n(x V ) = signature(MV ):

156 3.1. Hirzebruch Signature FormulaThe invariance of the index under homotopies shows signature(MV ) isindependent of the metric on M, of the ber metric on V , and of theRiemannian connection on V . If wedonot choose a Riemannian connectionon V , we can still compute signature(MV ) = index((d + ) + V ), but then(d + ) ; V is not the adjoint of(d + )+ V and as n is not an invariantly denedm form.Relative to a functorial coordinate frame for , the leading symbol of is jj 2 I. The rst order symbol is linear in the 1-jets of the metricand the connection form on V . The 0 th order symbol is linear in the 2-jetsof the metric and the connection form and quadratic in the 1-jets of themetric and the connection form. Thus a s n(X V ) 2 R mnmv . Theorem2.6.1 implies a s n =0for n

Hirzebruch Signature Formula 157Since the integrals are additive, we apply the uniqueness of Lemma 2.6.3to conclude the local formulas must be additive. This also follows fromLemma 1.7.5 so that:a s n(x V 1 V 2 )=a s n(x V 1 )+a s n(x V 2 ):Let fP g jj=s be the basis for P m4s4s and expand:a s m(x V )=X4jj+2t=mP ^ Q mtvfor Q mtv 2Q m2t2tva characteristic form of V . Then the additivity under direct sum implies:Q mtv (V 1 V 2 )=Q mtv1 (V 1 )+Q mtv2 (V 2 ):If v = 1, then Q mt1 (V 1 )=c c 1 (V ) t since Q m2t2t1 is one dimensional.If A is diagonal matrix, then the additivity implies:Q mtv (A) =Q mtv () =c X j t j = c ch t (A):Since Q is determined by its values on diagonal matrices, we conclude:Q mtv (V )=c(m t )ch t (V )where the normalizing constant does not depend on the dimension v. Therefore,we expand a s m in terms of ch t (V ) to express:a s m(x V )=X4s+2t=mP ms ^ 2 t ch t (V ) for P ms 2P m4s4s :We complete the proof of the theorem by identifying P ms = L s we

158 3.1. Hirzebruch Signature FormulaA similar decomposition of the Laplacians yields:signature(MV ) = signature(M 1 V 1 ) signature(M 2 V 2 )= 2 signature(M 1 V 1 )by Lemma 3.1.4. Since the signatures are multiplicative, the local formulasare multiplicative by the uniqueness assertion of Lemma 2.6.3. This alsofollows by using Lemma 1.7.5 thata s m(x V )=Xp+q=ma s p(x 1 V 1 )a s q(x 2 V 2 )and the fact a p =0for p

3.2. Spinors and their Representations.To dene the signature complex, we needed to orient the manifold M.For any Riemannian manifold, by restricting to local orthonormal frames,we can always assume the transition functions of T (M) are maps g : U \U ! O(m). If M is oriented, by restricting to local orthonormal frames,we can choose the transition functions of T (M) to lie in SO(m) and toreduce the structure group from GL(m R) to SO(m). The signature complexresults from the represntation of SO(m) it cannot be dened interms of GL(m R) or O(m). By contrast, the de Rham complex resultsfrom the representation eo which is a representation of GL(m R) so thede Rham complex is dened for non-orientable manifolds.To dene the spin complex, which is in some sense a more fundamentalelliptic complex than is either the de Rham or signature complex, we willhave to lift the transition functions from SO(m) to SPIN(m). Just as everymanifold is not orientable, in a similar fashion there is an obstruction todening a spin structure.If m 3, then 1 (SO(m)) = Z 2 .Abstractly, we dene SPIN(m) tobe the universal cover of SO(m). (If m = 2, then SO(2) = S 1 and we letSPIN(2) = S 1 with the natural double cover Z 2 ! SPIN(2) ! SO(2) givenby 7! 2). To discuss the representations of SPIN(m), it is convenientto obtain a more concrete representation of SPIN(m) in terms of Cliordalgebras.Let V be a real vector space of dimension v 0 (2). LetN V = R V (V V ) N k V be the complete tensor algebra of V . We assume V is equipped with asymmetric positive denite bilinear form. Let I be the two-sided idealof N V generated by fv v + jvj 2 g v2V , then the real Cliord algebraCLIF(V ) = N V mod I. (Of course, we will always construct the correspondingcomplex Cliord algebra by tensoring CLIF(V ) with the complexnumbers). There is a natural transpose dened on N V by:(v 1 v k ) t = v k v 1 :Since this preserves the ideal I, it extends to CLIF(V ). If fe 1 ...e v g arethe orthonormal basis for V , then e i e j + e j e i = ;2 ij and(e i1 e ip ) t =(;1) p(p;1)=2 e i1 e ip :If V is oriented, we let fe i g be an oriented orthonormal basis and dene: =( p ;1) v=2 e 1 e v :

160 3.2. SpinorsWe already computed that: 2 =(;1) v=2 e 1 e v e 1 ...e v=(;1) v(v;1)=2 e 1 e v e 1 e v= e 1 e v e v e 1 =(;1) v =1:We let SPIN(V ) be the set of all w 2 CLIF(V ) such that w can bedecomposed as a formal product w = v 1 v 2j for some j where thev i 2 V are elements of length 1. It is clear that w t is such an element andthat ww t = 1 so that SPIN(V ) forms a group under Cliord multiplication.We dene(w)x = wxw t for x 2 CLIFw 2 CLIF(V ):For example, if v 1 = e 1 is the rst element ofourorthonormal basis, then:e 1 e i e 1 =;e1 i =1e i i 6= 1:The natural inclusion of V in N V induces an inclusion of V in CLIF(V ).(e 1 ) preserves V and is reection in the hyperplane dened by e 1 . If w 2SPIN(V ), then (w): V ! V is a product of an even number of hyperplanereections. It is therefore in SO(V ) so: SPIN(V ) ! SO(V )is a group homomorphism. Since any orthogonal transformation of determinantone can be decomposed as a product of an even numberofhyperplanereections, is subjective.If w 2 CLIF(V ) is such thatwvw t = v all v 2 V and ww t =1then wv = vw for all v 2 V so w must be in the center of CLIF(V ).Lemma 3.2.1. If dim V = v is even, then the center of CLIF(V ) is onedimensional and consists of the scalars.Proof: Let fe i g be an orthonormal basis for V and let fe I g be the correspondingorthonormal basis for CLIF(V ). We compute:e i e i1 e ip e i = e i1 e ipfor p 1if (a) p is even and i is one of the i j or (b) p is odd and i is not oneof the i j . Thus given I we can choose i so e i e I = ;e I e i . Thus

And their Representations 161e i ( P I c Ie I ) = P I c Ie I e i for all i implies c I = 0 for jIj > 0 whichcompletes the proof.(We note that this lemma fails if dim V is odd since the center in that caseconsists of the elements a+be 1 e v and the center is two dimensional).If (w) = 1 and w 2 SPIN(V ), this implies w is scalar so w = 1. Byconsidering the arc in SPIN(V ) given by w() = ((cos )e 1 + (sin )e 2 ) (;(cos )e 1 +(sin)e 2 ), we note w(0) = 1 and w( ) = ;1 so SPIN(V ) is2connected. This proves we have an exact sequence of groups in the form:Z 2 ! SPIN(V ) ! SO(V )and shows that SPIN(V ) is the universal cover of SO(V ) for v>2.We note that is it possible to dene SPIN(V ) using Cliord algebraseven if v is odd. Since the center of CLIF(V )istwo dimensional for v odd,more care must be used with the relevant signs which arise. As we shallnot need that case, we refer to Atiyah-Bott-Shapiro (see bibliography) forfurther details.SPIN(V ) acts on CLIF(V ) from the left. This is an orthogonal action.Lemma 3.2.2. Let dim V = 2v 1 . We complexify CLIF(V ). As a leftSPIN(V ) module, this is not irreducible. We can decompose CLIF(V ) as adirect sum of left SPIN(V ) modules in the formCLIF(V )=2 v 1:This representation is called the spin representation. It is not irreduciblebut further decomposes in the form= + ; :If we orient V and let be the orientation form discussed earlier, then leftmultiplication by is 1 on so these are inequivalent representations.They are irreducible and act on a representation space of dimension 2 v 1;1and are called the half-spin representations.Proof: Fix an oriented orthonormal basis fe i g for V and dene: 1 = p ;1 e 1 e 2 2 = p ;1 e 3 e 4 ... v1 = p ;1 e v;1 e vas elements of CLIF(V ). It is immediate that = 1 ... v1and: 2 i =1 and i j = j i :We let the f i g act on CLIF(V ) from the right and decompose CLIF(V )into the 2 v 1simultaneous eigenspaces of this action. Since right and leftmultiplication commute, each eigenspace is invariant as a left SPIN(V )

162 3.2. Spinorsmodule. Each eigenspace corresponds to one of the 2 v 1possible sequencesof + and ; signs. Let " be such a string and let " be the correspondingrepresentation space.Since e 1 1 = ; 1 e 1 and e 1 i = i e 1 for i > 1, multiplication on theright by e 1 transforms " ! " 0,where"(1) = ;" 0 (1) and "(i) =" 0 (i) fori > 1. Since this map commutes with left multiplication by SPIN(V ), wesee these two representations are isomorphic. A similar argument on theother indices shows all the representation spaces " are equivalent so wemay decompose CLIF(V )=2 v 1 as a direct sum of 2 v 1equivalent representationspaces . Since dim(CLIF(V )) = 2 v = 4 v 1, the representationspace has dimension 2 v 1. We decompose into 1 eigenspaces underthe action of to dene . Since e 1 e v is in the center of SPIN(V ),these spaces are invariant under the left action of SPIN(V ). We note thatelements of V anti-commute with so Cliord multiplication on the leftdenes a map:cl: V ! so both of the half-spin representations have the same dimension. Theyare clearly inequivalent. We leave the proof that they are irreducible tothe reader as we shall not need this fact.We shall use the notation to denote both the representations andthe corresponding representation spaces. Form the construction we gave,it is clear the CLIF(V ) acts on the left to preserve the space so is arepresentation space for the left action by the whole Cliord algebra. Sincev = ;v for v 2 V , Cliord multiplication on the left by an element of Vinterchanges + and ; .Lemma 3.2.3. There is a natural map given by Cliord multiplication ofV ! . This map induces a map on the representations involved: 7! . If this map is denoted by v w then v v w = ;jvj 2 w.Proof: We already checked the map on the spaces. We check:wvw t wx 7! wvw t wx = wvxto see that the map preserves the relevant representation. We emphasizethat are complex representations since we must complexify CLIF(V )to dene these representation spaces (ww t =1for w 2 SPIN).There is a natural map : SPIN(V ) ! SO(V ). There are natural representations, , eo of SO(V ) on the subspaces of (V ) = CLIF(V ). Weuse to extend these representations of SPIN(V )aswell. They are relatedto the half-spin representations as follows:Lemma 3.2.4.(a) = .(b) ( + ; ; )=( + ; ; ) ( + + ; ).

And their Representations 163(c) ( e ; o )=( + ; ; ) ( + ; ; )(;1) v=2 :These identities are to be understood formally (in the sense of K-theory).They are shorthand for the identities: + =( + + ) ( + ; ) and ; =( ; + ) ( ; ; )and so forth.Proof: If we let SPIN(V ) act on CLIF(V ) by right multiplication byw t = w ;1 , then this representation is equivalent to left multiplication sowe get the same decomposition of CLIF(V )=2 v 1asright spin modules.Since (w)x = wxw t , it is clear that = where one factor is viewed asa left and the other as a right spin module. Since is the decompositionof SPIN(V ) under the action of from the left, (b) is immediate. (c)follows similarly once the appropriate signs are taken into considerationw 2 CLIF(V ) even if and only ifw t =(;1) v=2 wwhich proves (c).It is helpful to illustrate this for the case dim V =2. Let fe 1 e 2 g be anoriented orthonormal basis for V . We compute that:; (cos )e1 + (sin )e 2; (cos )e1 + (sin )e 2= ; ; cos cos ; sin sin + ; cos sin ; sin cos e 1 e 2so elements of the form cos +(sin )e 1 e 2 belong to SPIN(V ). We compute:; cos +(sin)e1 e 2; cos + (sin )e1 e 2 = cos( + )+sin( + )e1 e 2so spin(V ) is the set of all elements of this form and is naturally isomorphicto the circle S 1 =[0 2] with the endpoints identied. We compute that(w)(e 1 )= ; cos + (sin )e 1 e 2e1; cos + (sin )e2 e 1= ; cos 2 ; sin 2 e 1 + 2(cos sin )e 2= cos(2)e 1 + sin(2)e 2(w)(e 2 ) = cos(2)e 2 ; sin(2)e 1so the map : S 1 ! S 1 is the double cover 7! 2.We construct the one dimensional subspaces V i generated by the elements:v 1 =1+v 1 = v 1 v 1 = v 1 v 2 =1; v 2 = ;v 2 v 2 = ;v 2 v 3 =(1+)e 1 v 3 = v 3 v 3 = ;v 3 v 4 =(1; )e 1 v 4 = ;v 4 v 4 = v 4 :

164 3.2. SpinorsWhen we decompose CLIF(V ) under the left action of SPIN(V ),V 1 ' V 3 ' + and V 2 ' V 4 ' ; :When we decompose CLIF(V ) under the right action of SPIN(V ), andreplace by ; = t acting on the right:V 2 ' V 3 ' + and V 1 ' V 4 ' ; :From this it follows that as SO(V ) modules we have:V 1 = + ; V 2 = ; + V 3 = + + V 4 = ; ;from which it is immediate that: e = V 1 V 2 = + ; ; + o = V 3 V 4 = + + ; ; + = V 1 V 3 = + ; + + ; = V 2 V 4 = ; + ; ; :If V is a one-dimensional complex vector space, we let V r be the underlyingreal vector space. This denes a natural inclusion of U(1) ! SO(2).If we let J : V ! V be complex multiplication by p ;1 then J(e 1 ) = e 2and J(e 2 ) = ;e 1 so J is equivalent to Cliord multiplication by e 1 e 2 onthe left. We dene a complex linear map from V to ; ; by:T (v) =v ; iJ(v) 2 V 4by computing T (e 1 )=(e 1 ; ie 2 )=(1; ie 1 e 2 )(e 1 ) and T (e 2 )=e 2 + ie 1 =(i)(e 1 ; ie 2 ).Lemma 3.2.5. Let U(1) be identied with SO(2) in the usual manner. IfV is the underlying complex 1-dimensional space corresponding to V r thenV ' ; ; and V ' + + as representation spaces of SPIN(2).Proof: We have already veried the rst assertion. The second followsfrom the fact that V was made into a complex space using the map ;Jinstead of J on V r . This takes us into V 3 instead of into V 4 . It is alsoclear that ; = ( + ) if dim V = 2. Of course, all these statements areto be interpreted as statements about representations since they are trivialas statements about vector spaces (since any vector spaces of the samedimension are isomorphic).

3.3. Spin Structures on Vector Bundles.We wish to apply the constructions of 3.2 to vector bundles over manifolds.We rst review some facts regarding principal bundles and Stieel-Whitney classes which we shall need.Principal bundles are an extremely convenient bookkeeping device. If Gis a Lie group, a principal G-bundle is aberspace : P G ! M with berG such that the transition functions are elements of G acting on G by leftmultiplication in the group. Since left and right multiplication commute,we can dene a right action of G on P G which is ber preserving. Forexample, let SO(2k) and SPIN(2k) bethegroups dened by R 2k with thecannonical inner product. Let V be an oriented Riemannian vector bundleof dimension 2k over M and let P SO be the bundle of oriented framesof V . P SO is an SO(2k) bundle and the natural action of SO(2k) fromthe right which sends an oriented orthonormal frame s = (s 1 ...s 2k ) tos g =(s 0 1 ...s0 2k ) is dened by:s 0 i = s 1 g 1i + + s 2k g 2ki :The ber of P SO is SO(V x ) where V x is the ber of V over the point x.This isomorphism is not natural but depends upon the choice of a basis.It is possible to dene the theory of characteristic classes using principalbundles rather than vector bundles. In this approach, a connection is asplitting of T (P G )into vertical and horizontal subspaces in an equivariantfashion. The curvature becomes a Lie algebra valued endomorphism ofT (P G ) which is equivariant under the right action of the group. We referto Euguchi, Gilkey, Hanson for further details.Let fU g be a cover of M so V is trivial over U and let ~s be localoriented orthonormal frames over U . On the overlap, we express ~s =g ~s where g : U \ U ! SO(2k). These satisfy the cocycle condition:g g g = I and g = I:The principal bundle P SO of oriented orthonormal frames has transitionfunctions g acting on SO(2k) from the left.A spin structure on V is a lifting of the transition functions to SPIN(2k)preserving the cocyle condition. If the lifting is denoted by g 0 , then weassume:(g 0 ) =g g 0 g 0 g 0 = I and g 0 = I:This is equivalent to constructing a principal SPIN(2k) bundle P SPIN togetherwith a double covering map : P SPIN ! P SO which preserves thegroup action|i.e.,(x g 0 )=(x) (g 0 ):

166 3.3. Spin StructuresThe transition functions of P SPIN are just the g 0 acting on the left.Attempting to nd a spin structure on V is equivalent to taking a squareroot in a certain sense which we will make clear later. There is an obstructionto dening a spin structure which is similar to that to dening anorientation on V . These obstructions are Z 2 characteristic classes calledthe Stieel-Whitney classes and are most easily dened in terms of Cechcohomology. To understand these obstructions better, we review the constructionbriey.We x a Riemannian structure on T (M) to dene a notion of distance.Geodesics on M are curves which locally minimize distance. A set U issaid to be geodesically convex if (a) given x y 2 U there exists a uniquegeodesic in M joining x to y with d(x y) = length() and (b) if is anysuch geodesic then is actually contained in U. It is immediate that theintersection of geodesically convex sets is again geodesically convex andthat every geodesically convex set is contractible.It is a basic theorem of Riemannian geometry that there exist opencovers of M by geodesically convex sets. Acover fU g issaidtobesimpleif the intersection of any number of sets of the cover is either empty or iscontractible. A cover of M by open geodesically convex sets is a simplecover.We x such a simple cover hence forth. Since U is contractible, anyvector bundle over M is trivial over U . Let Z 2 be the multiplicativegroup f1g. A Cech j-cochain is a function f( 0 ... j ) 2 Z 2 denedfor j + 1-tuples of indices where U 0 \ \ U j 6= which is totallysymmetric|i.e.,f( (0) ... (j) )=f( 0 ... j )for any permutation . If C j (M Z 2 ) denotes the multiplicative group ofall such functions, the coboundary : C j (MZ 2 ) ! C j+1 (M Z 2 ) is denedby:(f)( 0 ... j+1 )=j+1Yi=0f( 0 ... b i ... j+1 ):The multiplicative identity of C j (M Z 2 ) is the function 1 and it is aneasy combinatorial exercise that 2 f =1. For example, if f 0 2 C 0 (M Z 2 )and if f 1 2 C 1 (M Z 2 ), then:(f 0 )( 0 1 )=f( 1 )f( 0 )(f 1 )( 0 1 2 )=f( 1 2 )f( 0 1 )f( o 2 ):Z 2 is a particularly simple coecient group to work with since every elementis its own inverse in dening the Cech cohomology with coecients

On Vector Bundles 167in other abelian groups, more care must be taken with the signs whicharise.Let H j (M Z 2 )=N( j )= R( j;1 ) be the cohomology group it is an easyexercise to show these groups are independent of the particular simple coverchosen. There is a ring structure on H (M Z 2 ), but we shallonlyusethe\additive" group structure (which we shall write multiplicatively).Let V be a real vector bundle, not necessarily orientable. Since the U are contractible, V is trivial over the U and we can nd a local orthonormalframe ~s for V over U . We let ~s = g ~s and dene the 1-cochain:f() = det(g )=1:This is well dened since U \ U is contractible and hence connected.Since f( ) = f(), this denes an element of C 1 (M Z 2 ). Since thefg g satisfy the cocycle condition, we compute:f( ) = det(g g g ) = det(I) =1so (f) = 1 and f denes an element of H 1 (M Z 2 ). If we replace ~s by~s 0 = h ~s the new transition functions become g 0 = h g h ;1 so iff 0 () = det(h ),f 0 ( ) = det(h g h ;1 )=det(h )f( )det(h )=(f 0 )fand f changes by a coboundary. This proves the element in cohomologydened by f is independent of the particular frame chosen and we shalldenote this element by w 1 (V ) 2 H 1 (M Z 2 ):If V is orientable, we can choose frames so det(g ) = 1 and thusw 1 (V ) = 1 represents the trivial element in cohomology. Conversely, ifw 1 (V ) is trivial, then f = f 0 . If we choose h so det(h ) = f 0 (), thenthe new frames ~s 0 = h ~s will have transition functions with det(g 0 )=1and dene an orientation of V . Thus V is orientable if and only if w 1 (V )istrivial and w 1 (V ), which is called the rst Stieel-Whitney class, measuresthe obstruction to orientability.If V is orientable, we restrict henceforth to oriented frames. Let dim V=2k be even and let g 2 SO(2k) be the transition functions. We chooseany lifting ~g to SPIN(2k) so that:(~g )=g and ~g ~g = Isince the U are contractible such lifts always exist. We have g g g = I so (~g ~g ~g )=I and hence ~g ~g ~g = I = f( )I wheref( ) 2 Z 2 . V admits a spin structure if and only if we can choosethe lifting so f( ) = 1. It is an easy combinatorial exercise to show

168 3.3. Spin Structuresthat f is symmetric and that f =0. Furthermore, if we change the choiceof the ~s or change the choice of lifts, then f changes by a coboundary.This implies f denes an element w 2 (V ) 2 H 2 (M Z 2 ) independent of thechoices made. w 2 is called the second Stieel-Whitney class and is trivialif and only if V admits a spin structure.We suppose w 1 (V )=w 2 (V ) = 1 are trivial so V is orientable and admitsa spin structure. Just as there are two orientations which can be chosenon V , there can be several possible inequivalent spin structures (i.e., severalnon-isomorphic principal bundles P SPIN ). It is not dicult to seethat inequivalent spin structures are parametrized by representations ofthe fundamental group 1 (M) ! Z 2 just as inequivalent orientations areparametrized by maps of the components of M into Z 2 .To illustrate the existence and non-existence of spin structures on bundles,we take M = S 2 . S 2 = CP 1 is the Riemann sphere. There is anatural projection from C 2 ; 0 to S 2 given by sending (x w) 7! z=w S 2is obtained by identifying (z w) = (z w) for (z w) 6= (0 0) and 6= 0.There are two natural charts for S 2 :U 1 = fz : jzj 1g and U 2 = fz : jzj 1g [ f1gwhere w =1=z gives coordinates on U 2 .CP 1 is the set of lines in C 2 we let L be the natural line bundle overS 2 this is also refered to as the tautological line bundle. We have naturalsections to L over U i dened by:s 1 =(z 1) over U 1 and s 2 =(1w) over U 2 :these are related by the transition function:s 1 = zs 2so on U 1 \ U 2 we have g 12 = e i . The double cover SPIN(2) ! SO(2) isdened by 7! 2 so this bundle does not have a spin structure since thistransition function cannot be lifted to SPIN(2).This cover is not a simple cover of S 2 so we construct the cover given in'$the followingV 4&%J V 1V 2J V 3

On Vector Bundles 169(where we should \fatten up" the picture to have an open cover). If wehave sections ~s i to L over V i , then we can choose the transition functionsso ~s j = e i ~s 4 for 1 j 3 and ~s 1 = ~s 2 = ~s 3 . We letand dene:0 2=3 parametrize V 1 \ V 42=3 4=3 parametrize V 2 \ V 44=3 2 parametrize V 3 \ V 4~g 14 = e i=2 ~g 24 = e i=2 ~g 34 = e i=2and all the ~g jk =1for 1 j, k 3 then we compute:so the cochain dening w 2 satises:~g 13 ~g 34 ~g 41 = e 0 e i = ;1f(1 3 4) = ;1 with f(i j k)=1 if i or j or k=2.We compute this is non-trivial in cohomology as follows: suppose f = f 1 .Then:1=f(1 2 3) = f 1 (1 2)f 1 (1 3)f 1 (2 3)and multiplying together:1=f(1 2 4) = f 1 (1 2)f 1 (1 4)f 1 (2 4)1=f(2 3 4) = f 1 (2 3)f 1 (2 4)f 1 (3 4)1=f 1 (1 2) 2 f 1 (2 3) 2 f 1 (2 4) 2 f 1 (1 4)f 1 (1 3)f 1 (3 4)= f 1 (1 4)f 1 (1 3)f 1 (3 4) = f(1 3 4) = ;1which is a contradiction. Thus we have computed combinatorially thatw 2 6=1is non-trivial.Next we let V = T (M) be the real tangent bundle. We identify U(1)with SO(2) to identify T (M) withT C (M). Since w =1=z we have:ddz = dwdzddw = ;z;2so the transition function on the overlap is ;e ;2i . The minus sign can beeliminated by using ; ddinstead ofdw dw as a section over U 2 to make thetransition function be e ;2i . Since the double cover of SPIN(2) ! SO(2)is given by 7! 2, this transition function lifts and T (M) has a spinstructure. Since S 2 is simply connected, the spin structure is unique. Thisddw

170 3.3. Spin Structuresalso proves T C (M) =L L since the two bundles have the same transitionfunctions.There is a natural inclusion of SPIN(V ) and SPIN(W ) into subgroupsof SPIN(V W ) which commute. This induces a map from SPIN(V ) SPIN(W ) ! SPIN(V W ). Using this map, it is easy to compute w 2 (V W ) = w 2 (V )w 2 (W ). To dene w 2 (V ), we needed to choose a xed orientationof V . It is not dicult to show that w 2 (V ) is independent of theorientation chosen and of the ber metric on V . (Our denition does dependupon the fact that V is orientable. Although it is possible to denew 2 (V ) more generally, the identity w 2 (V W ) = w 2 (V )w 2 (W ) fails if Vand W are not orientable in general). We summarize the properties of w 2we have derived:Lemma 3.3.1. Let V be a real oriented vector bundle and let w 2 (V ) 2H 2 (M Z 2 ) be the second Stieel-Whitney class. Then:(a) w 2 (V ) = 1 represents the trivial cohomology class if and only if Vadmits a spin structure.(b) w 2 (V W )=w 2 (V )w 2 (W ).(c) If L is the tautological bundle over S 2 then w 2 (L) is non-trivial.We emphasize that (b) is written in multiplicative notation since wehave chosen the multiplicative version of Z 2 . w 2 is also functorial underpull-backs, but as we have not dened the Cech cohomology as a functor,we shall not discuss this property. We also note that the Stieel-Whitneyclasses can be dened in general w(V ) = 1+w 1 (V )+ 2 H (M Z 2 )is dened for any real vector bundle and has many of the same propertiesthat the Chern class has for complex bundles.We can use Lemma 3.3.1 to obtain some other results on the existenceof spin structures:Lemma 3.3.2.(a) If M = S m then any bundle over S m admits a spin structure for m 6= 2.(b) If M = CP k and if L is the tautological bundle over M, then L doesnot admit a spin structure.(c) If M = CP k and if V = T (M) is the tangent space, then V admits aspin structure if and only if k is odd.(d) If M = QP k is quaternionic projective space, then any bundle over Madmits a spin structure.Proof: H 2 (M Z 2 )=f1g in (a) and (d) so w 2 must represent the trivialelement. To prove (b), we suppose L admits a spin structure. S 2 is embeddedin CP k for k 2 and the restriction of L to S 2 is the tautologicalbundle over S 2 . This would imply L admits a spin structure over S 2 whichis false. Finally, we use the representation:T c (CP j ) 1=L L

so thatOn Vector Bundles 171T (CP j ) 1 2 =(L r) (L r)| {z }j+1 timeswhere L r denotes the real vector bundle obtained from L by forgettingthe complex structure. Then w 2 (L r) = w 2 (L r ) since these bundles areisomorphic as real bundles. Furthermore:w 2 (T (CP j )) = w 2 (L r ) j+1and this is the trivial class if j +1 is even (i.e., j is odd) while it is w 2 (L r )and non-trivial if j +1is odd (i.e., j is even).Let M be a manifold and let V be a real vector bundle over M. IfV admits a spin structure, we shall say V is spin and not worry for themoment about the lack of uniqueness of the spin structure. This will notbecome important until we discuss Lefschetz xed point formulas.If V is spin, we dene the bundles (V ) to have transition functions (~g ) where we apply the representation to the lifted transitionfunctions. Alternatively, ifP SPIN is the principal spin bundle dening thespin structure, we dene: (V )=P SPIN where the tensor product across a representation is dened to be P SPIN module the identication (p g) z = p ( (g)z) for p 2 P SPIN ,g 2 P SPIN (2k), z 2 . (The slight confusion of notation is caused byour convention of using the same symbol for the representation and therepresentation space.)Let b be a xed unitary frame for . If ~s is a local oriented orthonormalframe for V , we let ~s i be the two lifts of ~s 2 P SO to P SPIN . ~s 1 = ;~s 2but there is no natural way to distinguish these lifts, although the pair iscannonically dened. If r is a Riemannian connection on V , let r ~s = !~sbe the connection 1-form. ! is a skew symmetric matrix of 1-forms and isa 1-form valued element of the Lie algebra of SO(2k). Since the Lie algebraof SO(2k) and SPIN(2k) coincide, we can also regard ! as an elementwhich is 1-form valued of the Lie algebra of SPIN(2k) and let (!) acton (V ). We dene bases ~s i b for (V ) and dene:r(~s i b )=~s i (!)b to dene a natural connection on (V ). Since the same connection isdened whether ~s 1 or ~s 2 = ;~s 1 is chosen, the Z 2 ambiguity is irrelevantand r is well dened.Lemma 3.2.4 and 3.2.5 extend to:

172 3.3. Spin StructuresLemma 3.3.3. Let V be a real oriented Riemannian vector bundle ofeven ber dimension v. Suppose V admits a spin structure and let (V )be the half-spin bundles. Then:(a) (V ) ' (V ) (V ).(b) ( + ; ; )(V ) ' ( + ; ; ) ( + ; ; )(V ).(c) ( e ; o )(V ) ' (;1) v=2 ( + ; ; ) ( + ; ; )(V ).(d) If v =2and if V is the underlying real bundle of a complex bundle V cthen V c ' ; ; and Vc ' + + .(e) If r is a Riemannian connection on V and if we extend r to (V ),then the isomorphisms given are unitary isomorphisms which preserve r.Spinors are multiplicative with respect to products. It is immediate fromthe denitions we have given that:Lemma 3.3.4. Let V i be real oriented Riemannian vector bundles of evenber dimensions v i with given spin structures. Let V 1 V 2 have the naturalorientation and spin structure. Then:( + ; ; )(V 1 V 2 ) 'f( + ; ; )(V 1 )gf( + ; ; )(V 2 )g:If r i are Riemannian connections on V i and if we dene the natural inducedconnections on these bundles, then the isomorphism is unitary andpreserves the connections.A spin structure always exists locally since the obstruction to a spinstructure is global. Given any real oriented Riemannian bundle V of evenber dimension with a xed Riemannian connection r, we dene (V )and(V ) = + (V ) ; (V ). With the natural metrics and connections, werelate the connection 1-forms and curvatures of these 3 bundles:Lemma 3.3.5. Let fe i g be a local oriented orthonormal frame for V . Welet re j = ! jk s k represent the connection 1-form and e j = jk e k be thecurvature matrix for jk = d! jk ; ! jl ^ ! lk . Let e and e denote thenatural orthonormal frames on (V ) and (V ). Relative to these frameswe compute the connection 1-forms and curvature matrices of (V ) and(V ) by:(a)(b)! = ! jk ext(e k )int(e j ) = jk ext(e k )int(e j )! = 1 4 ! jke j e k = 1 4 jke j e k :Proof: We sum over repeated indices in these expressions. We note (a)is true by denition on 1 (V ) = V . Since both ! and extend to act asderivations on the exterior algebra, this implies (a) is true on forms of alldegree.Let so(n) =fA 2 nn real matrices with A+A t =0g be the Lie algebraof SO(n). This is also the Lie algebra of SPIN(n) so we must identify this

On Vector Bundles 173with an appropriate subset of the Cliord algebra in order to prove (b).We choose a representative element of so(n) and denei.e.,Ae 1 = e 2 Ae 2 = ;e 1 Ae j =0 for j > 2A 12 =1 A 21 = ;1 A jk =0 otherwise.If we let g(t) 2 SO(n) be dened byg(t)e 1 = (cos t)e 1 + (sin t)e 2 g(t)e 2 = (cos t)e 2 ; (sin t)e 1 g(t)e j = e jj > 2then g(0) = I and g 0 (0) = A. We must lift g(t) from SO(n) to SPIN(n).Dene:h(t) = ; cos(t=4)e 1 + sin(t=4)e 2;; cos(t=4)e1 +sin(t=4)e 2= cos(t=2) + sin(t=2)e 1 e 2 2 SPIN(n)Then (h) 2 SO(n) is dened by:(h)e j = ; cos(t=2) + sin(t=2)e 1 e 2ej; cos(t=2) ; sin(t=2)e1 e 2so that (h)e j = e j for j > 2. We compute:(h)e 1 = ; cos(t=2)e 1 + sin(t=2)e 2; cos(t=2) ; sin(t=2)e1 e 2= ; cos 2 (t=2) ; sin 2 (t=2) e 1 + 2 sin(t=2) cos(t=2)e 2(h)e 2 = ; cos(t=2)e 2 ; sin(t=2)e 1; cos(t=2) ; sin(t=2)e1 e 2= ; cos 2 (t=2) ; sin 2 (t=2) e 2 ; 2sin(t=2) cos(t=2)e 1so that (h) = g. This gives the desired lift from SO(n) to SPIN(n). Wedierentiate to geth 0 (0) = 1 2 e 1e 2 = 1 4 A jke j e k :This gives the lift of a matrix in this particular form. Since the wholeLie algebra is generated by elements of this form, it proves that the lift ofA jk in general is given by 1 A 4 jke j e k . Since SPIN(n) acts on the Cliordalgebra by Cliord multiplication on the left, this gives the action of thecurvature and connection 1-form on the spin representations and completesthe proof.

174 3.3. Spin StructuresWe can dene ch( (V )) as SO characteristic forms. They can be expressedin terms of the Pontrjagin and Euler forms. We could use theexplicit representation given by Lemma 3.3.5 to compute these forms itis most easy, however, to compute in terms of generating functions. Weintroduce formal variables fx j g for 1 j (dim V )=2 so thatp(V )= Y j(1 + x 2 j ) and e(V )= Y jx j :All these computations are really induced from the corresponding matrixidentities and the fx j g arise from putting a skew-symmetric matrix A inblock form.Lemma 3.3.6. Let V be an oriented real Riemannian vector bundle ofdimension n 0 (2). Let ch( (V )) be a real characteristic form theseare well dened even if V does not admit a global Q spin structure. Then:(a) ch( + (V )) + ch( ; (V )) = ch((V )) = fe xj=2 + e ;x j =2 g.j(b)(;1) n=2 fch( + (V )) ; ch( ; (V ))g = Y jfe x j=2 ; e ;x j =2 g= e(V )(1 + higher order terms):Proof: This is an identity among invariant polynomials in the Lie algebraso(n). We may therefore restrict to elements of the Lie algebra whichsplit in block diagonal form. Using the multiplicative properties of theChern character and Lemma 3.3.4, it suces to prove this lemma for thespecial case that n = 2 so (V ) are complex line bundles. Let V c bea complex line bundle and let V be the underlying real bundle. Thenx 1 = x = c 1 (V c )=e(V ). Since:we conclude:V c = ; ; and V c = + +x =2c 1 ( ; ) and ; x =2c 1 ( + )which shows (a) and the rst part of (b). We expand:e x=2 ; e ;x=2 = x + 124 x3 + to see ch( ; ) ; ch( + )=e(V )(1 + 124 p 1(V )+) to complete the proofof (b).

On Vector Bundles 175There is one nal calculation in characteristic classes which will provehelpful. We dened L and ^A by the generating functions:L(x)= Y j= Y j^A(x)= Y jx jtanh x jx je x j+ e ;x je x j ; e ;x jx jsinh(x j =2) = Y jx je x j=2; e ;x j =2 :Lemma 3.3.7. Let V be an oriented Riemannian real vector bundle ofdimension n =2n 0 . If we compute the component of the dierential formwhich is in n (T M) then:fL(V ))g n = fch((V )) ^ ^A(V )gn :Proof: It suces to compute the component of the corresponding symmetricfunctions which are homogeneous of order n 0 . If we replace x j byx j =2 then:fL(x j )gn 0 = f2 n 0L(x j =2)g n0== Yj8< Y:jx j tanh(x j =2)x je x=2 + e ;x=2n 0e x j =2; e ;x j =29=n 0which completes the proof.= fch((V )) ^ ^A(V )gn0

3.4. The Spin Complex.We shall use the spin complex chiey as a formal construction to link thede Rham, signature, and Dolbeault complexes. Let M be a Riemannianmanifold of even dimension m. Let T (M) be the real tangent space. Weassume that M is orientable and that T (M) admits a spin structure. Welet (M) bethe half-spin representations. There is a natural map givenby Lemma 3.2.3 from the representations ofT (M) ! HOM( (M) (M))which we will call c() (since it is essentially Cliord multiplication) suchthat c() 2 = ;jj 2 I. We extend the Levi-Civita connection to act naturallyon these bundles and dene the spin complex by the diagram:A : C 1 ( (M)) r ! C 1 (T M (M)) c ! C 1 ( (M))to be the operator with leading symbol c. (A + ) = A ; and A is ellipticsince c() 2 = ;jj 2 I this operator is called the Dirac operator.Let V be a complex bundle with a Riemannian connection r. We denethe spin complex with coecients in r by using the diagram:A V : C1 ( (M) V ) ! C 1 (T M (M) V )c1;;! C 1 ( (M) V ):This is completely analogous to the signature complex with coecients inV . We dene:index(Vspin) = index(A + V )a priori this depends on the particular spin structure chosen on V ,butweshall show shortly that it does not depend on the particular spin structure,although it does depend on the orientation. We leta spinn (x V )denote the invariants of the heat equation. If we reverse the orientation ofM, we interchange the roles of + and of ; so a spinn changes sign. Thisimplies a spinn can be regarded as an invariantly dened m-form since thescalar invariant changes sign if we reverse the orientation.Let X be an oriented coordinate system. We apply the Gramm-Schmidtprocess to the coordinate frame to construct a functorial orthonormal frame~s (X) for T (M) = T (M). We lift this to dene two local sections ~s i (X)to the principal bundle P SPIN with ~s 1 (X) =;~s 2 (X). There is, of course,no cannonical way to prefer one over the other, but the pair is invariantlydened. Let b be xed bases for the representation space and let

The Spin Complex 177b i (X) = ~s i(X) b provide local frames for (M). If we x i = 1or i = 2, the symbol of the spin complex with coecients in V can befunctorially expressed in terms of the 1-jets of the metric on M and interms of the connection 1-form on V . The leading symbol is given byCliord multiplication the 0 th order term is linear in the 1-jets of themetric on M and in the connection 1-form on V with coecients which aresmooth functions of the metric on M. If we replace b 1 by b 2 = ;b 1 thenthe local representation of the symbol is unchanged since multiplication by;1 commutes with dierential operators. Thus wemay regard a spin n (x V ) 2R mnmdim V as an invariantly dened polynomial which is homogeneousof order n in the jets of the metric and of the connection form on V whichis m-form valued.This interpretation denes a spinn (x V ) even if the base manifold M isnot spin. We can always dene the spin complex locally as the Z 2 indeterminacyin the choice R of a spin structure will not aect the symbol of theoperator. Of course,M aspin m (x V ) can only be given the interpretation ofindex(Vspin) if M admits a spin structure. In particular, if this integralis not an integer, M cannot admit a spin structure.a spin n (x V ) is a local invariant which is not aected by the particularglobal spin structure chosen. Thus index(Vspin) is independent of theparticular spin structure chosen.We use exactly the same arguments based on the theorem of the secondchapter and the multiplicative nature of the twisted spin complex as wereused to prove the Hirzebruch signature theorem to establish:Lemma 3.4.1.(a) a R spinn =0if n

178 3.4. The Spin ComplexLemma 3.4.2. Let U be an open contractible subset of M. Over U, wedene the signature, de Rham, and spin complexes. Then:(a) ( e ; o ) V ' (;1) m=2 ( + ; ; ) ( + ; ; ) V .(b) ( + ; ; ) V ' ( + ; ; ) ( + + ; ) V .(c) These two isomorphisms preserve the unitary structures and the connections.They also commute with Cliord multiplication on the left.(d) The natural operators on these complexes agree under this isomorphism.Proof: (a){(c) follow from previous results. The natural operators onthese complexes have the same leading symbol and therefore must be thesame since they are natural rst order operators. This proves (d).We apply this lemma in dimension m = 2 with V = the trivial bundleand M = S 2 :(S 2 ) = 2 = index(d + ) = index( ; ; + spin)=Z= cZS 2 cfc 1 ( ; ) ; c 1 ( + )gS 2 e(TM)=2cby Lemmas 2.3.1, 3.3.5, and 3.3.4. This establishes that the normalizingconstant of Lemma 3.4.1 must be X1 so that:a spinm = ^A 0 ^ ch t (V ):4s+2t=mWe apply this lemma to the twisted signature complex in dimension m =2with V a non-trivial line bundle over S 2 to conclude:signature(S 2 V) = index(( + ; ) Vspin)=ZS 2 ch(( + ; ) V )=ZS 2 2 c 1 (V ):This shows that the normalizing constant of Lemma 3.1.4 for the twistedsignature complex is 2 and completes the proof of Lemma 3.1.4. This thereforecompletes the proof of the Hirzebruch signature theorem in general.The de Rham complex with coeecients in V is dened by the diagram:C 1 ( eo (M) V ) r ! C 1 (T M eo (M) V ) c1;;! C 1 ( oe (M) V )and we shall denote the operator by (d + ) e V . The relations given byLemma 3.4.2 give rise to relations among the local formulas:a n (x (d + ) e V )=a n (x A + (;1) m=2 ( + ; ; )V )a n (x (d + ) + V )=a n(x A + ( + ; ; )V )

The Spin Complex 179where (d + ) + V is the operator of the twisted signature complex. Theserelations are well dened regardless of whether or not M admits a spinstructrue on T (M).We deal rst with the de Rham complex. Using Lemma 3.4.1 and thefact that the normalizing constant c is 1, we conclude:a n (x (d + ) e V )= ^A0 ^ ch((;1) m=2 ( + ; ; ))) ^ ch(V ):Since ch((;1) m=2 ( + ; ; )) = e(M) is already a top dimensional form byLemma 3.3.5(b), we conclude a n (x (d + ) e V ) = (dim V )e(M). This proves:Theorem 3.4.3. Let (d + ) e V be the de Rham complex with coecientsin the bundle V and let a n (x (d + ) e V ) be the invariants of the heat equation.Then:(a) a n (x (d + ) e V )=0if n

3.5. The Riemann-Roch TheoremFor Almost Complex Manifolds.So far, we have discussed three of the four classical elliptic complexes,the de Rham, the signature, and the spin complexes. In this subsection, wedene the Dolbeault complex for an almost complex manifold and relate itto the spin complex.Let M be a Riemannian manifold of dimension m = 2n. An almostcomplex structure on M is a linear map J: T (M) ! T (M) with J 2 =;1. The Riemannian metric G is unitary if G(X Y )=G(JXJY) for allX Y 2 T (M) we can always construct unitary metrics by averaging overthe action of J. Henceforth we assume G is unitary. We extend G to beHermitian on T (M) C, T (M) C, and (M) C.Since J 2 = ;1, we decompose T (M) C = T 0 (M) T 00 (M) into thei eigenspaces of J. This direct sum is orthogonal with respect to themetric G. Let 10 (M) and 01 (M) be the dual spaces in T (M) C toT 0 and T 00 . We choose a local frame fe j g for T (M) so that J(e j ) = e j+nfor 1 i n. Let fe j g be the corresponding dual frame for T (M). Then:T 0 (M) = span C fe j ; ie j+n g n j=1 T 00 (M) = span C fe j + ie j+n g n j=1 10 (M) = span C fe j + ie j+n g n j=1 01 (M) = span C fe j ; ie j+n g n j=1 :These four vector bundles are all complex vector bundles over M. Themetric gives rise to natural isomorphisms T 0 (M) ' 01 (M) and T 00 (M) = 10 (M). We will use this isomorphism to identify e j with e j for much ofwhat follows.If we forget the complex structure on T 0 (M), then the underlying realvector bundle is naturally isomorphic to T (M). Complex multiplication byi on T 0 (M) isequivalent to the endomorphism J under this identication.Thus we may regard J as giving a complex structure to T (M).The decomposition:gives rise to a decomposition:T (M) C = 10 (M) 01 (M)(T M) C = M pq pq (M)for pq (M) = p ( 10 (M)) q ( 01 (M)):Each of the bundles pq is a complex bundle over M and this decompositionof (T M)C is orthogonal. Henceforth we will denote these bundlesby T 0 , T 00 , and pq when no confusion will arise.

The Riemann-Roch Theorem 181If M is a holomorphic manifold, we let z =(z 1 ...z n ) be a local holomorphiccoordinate chart. We expand z j = x j + iy j and dene:@@z j= 1 2 @@x j; i @@y j@@z j= 1 2 @@x j+ i @@y jdz j = dx j + idy j dz j = dx j ; idy j :We dene:@(f) = X j@fXdz j and @f@(f) = dz j :@z j @z jjThe Cauchy-Riemann equations imply that a function f is holomorphicif and only if @(f) = 0. If w = (w1 ...w n ) is another holomorphiccoordinate system, then:@@w j= X k@z k@w j@@z k@@ w j= X k@z k@ w j@@z kdw j = X k@w j@z kdz k d w j = @ w j@z kdz k :We dene:T 0 @ n(M) =span@z j j=1T 00 @ n(M) = span@z j j=1 10 (M)=spanfdz j g n j=1 01 (M) = spanfdz j g n j=1then these complex bundles are invariantly dened independent of thechoice of the coordinate system. We also note@: C 1 (M) ! C 1 ( 10 (M)) and @: C 1 (M) ! C 1 ( 01 (M))are invariantly dened and decompose d = @ + @.There is a natural isomorphism of T 0 (M) with T (M) as real bundlesin this example and we let J be complex multiplication by i under thisisomorphism. Equivalently:J @@x j= @@y jandJ @@y j= ; @@x j:T 0 (M) = T c (M) is the complex tangent bundle in this example we shallreserve the notation T c (M) for the holomorphic case.Not every almost complex structure arises from a complex structurethere is an integrability condition. If J is an almost complex structure, we

182 3.5. The Riemann-Roch Theoremdecompose the action of exterior dierentiation d on C 1 () with respectto the bigrading (p q) todene:@: C 1 ( pq ) ! C 1 ( p+1q ) and @: C 1 ( pq ) ! C 1 ( pq+1 ):Theorem 3.5.1 (Nirenberg-Neulander). The following are equivalentand dene the notion of an integrable almost complex structure:(a) The almost complex structure J arises from a holomorphic structureon M.(b) d = @ + @. (c) @ @ =0.(d) T 0 (M) is integrable|i.e., given X Y 2 C 1 (T 0 (M)), then the Liebracket [X Y ] 2 C 1 (T 0 (M)) where we extend [ ] to complex vectorelds in the obvious fashion.Proof: This is a fairly deep result and we shall not give complete details.It is worth, however, giving a partial proof of some of the implicationsto illustrate the concepts we will be working with. Suppose rst M isholomorphic and let fz j g be local holomorphic coordinates on M. Dene:dz I = dz i1 ^ ^dz ip and dz J = dz j1 ^ ^dz jqthen the collection fdz I ^ dz J g gives a P local frame for pq which isclosed.If ! 2 C 1 ( pq ), we decompose ! = f IJ dz I ^ dz J and compute:d! = d XfIJ dz I ^ dz J = X df IJ ^ dz I ^ dz J :On functions, we decompose d = @ + @. Thus d! 2 C 1 ( p+1q pq+1 )has no other components. Therefore d! = @! + @! so (a) implies (b).We use the identity d 2 = 0 to compute (@ + @) 2 = (@ 2 )+(@ @ + @@)+( @ 2 ) = 0. Using the bigrading and decomposing this we conclude (@ 2 ) =(@ @ + @@)=( @ 2 )=0so (b) implies (c). Conversely, suppose that @ @ =0on C 1 (M). We must show d: C 1 ( pq ) ! C 1 ( p+1q pq+1 ) has noother components. Let fe j g be a local frame for 10 and let fe j g be thecorresponding local frame for 01 . We decompose:Then we compute:d Xfj e jde j = @e j + @e j + A jfor A j 2 02de j = @e j + @e j + A j for A j 2 20 .= X df j ^ e j + f j de j= X f@f j ^ e j + @f j ^ e j + f j @e j + f j @ej + f j A j g= @ Xfj e j+ @ Xfj e j+ X f j A j :

SimilarlyFor Almost Complex Manifolds 183 X X X Xd fj e j = @ fj e j + @ fj e j + f jAj :If A and A denote the 0 th order operators mapping 10 ! 02 and 01 ! 20 then we compute:d(!) =@(!)+ @(!)+A! for ! 2 C 1 ( 10 )d(!) =@(!)+ @(!)+ A! for ! 2 C 1 ( 01 ):Let f 2 C 1 (M) and compute:0=d 2 f = d(@f + @f)=(@ 2 + A @)f +(@ @ + @@)f +( @ 2 + A@)fwhere we have decomposed the sum using the bigrading. This implies that( @ 2 +A@)f =0soA@f =0. Since f@fg spans 10 , this implies A = A =0since A isa0 th order operator. Thus de j = @e j + @e j and de j = @e j + @e j .We compute d(e I ^ e J ) has only (p +1q) and (p q +1) components sod = @ + @. Thus (b) and (c) are equivalent.It is immediate from the denition that X 2 T 0 (M) if and only if !(X) =0 for all ! 2 01 . If X Y 2 C 1 (T M), then Cartan's identity implies:d!(X Y )=X(!Y ) ; Y (!X) ; !([X Y ]) = ;!([X Y ]):If (b) is true then d! has no component in 20 so d!(X Y ) = 0 whichimplies !([X Y ]) = 0 which implies [X Y ] 2 C 1 (T 0 M) which implies (d).Conversly, if (d) is true, then d!(X Y ) = 0 so d! has no component in 20 so d = @ + @ on 01 . By taking conjugates, d = @ + @ on 10 as wellwhich implies as noted above that d = @ + @ in general which implies (c).We have proved that (b){(d) are equivalent and that (a) implies (b). Thehard part of the theorem is showing (b) implies (a). We shall not give thisproof as it is quite lengthy and as we shall not need this implication of thetheorem.As part of the previous proof, we computed that d;(@+ @)isa0 th orderoperator (whichvanishes if and only if M is holomorphic). Wenow computethe symbol of both @ and @. We use the metric to identify T (M) =T (M).Let fe j g be a local orthonormal frame for T (M) such that J(e j ) = e j+nfor 1 j n. We extend ext and int to be complex linear maps fromT (M) C ! END((T M) C).Lemma 3.5.2. Let @ and @ be dened as before and let 0 and 00 be theformal adjoints. Then:

184 3.5. The Riemann-Roch Theorem(a)@: C 1 ( pq ) ! C 1 ( p+1q )and L (@)(x )= i 2@: C 1 ( pq ) ! C 1 ( pq+1 )and L ( @)(x ) = i 2 0 : C 1 ( pq ) ! C 1 ( p;1q )and L ( 0 )(x )=; i 2 00 : C 1 ( pq ) ! C 1 ( pq;1 )XjnXjnand L ( 00 )(x )=; i 2( j ; i j+n )ext(e j + ie j+n )( j + i j+n )ext(e j ; ie j+n )XjnXjn( j + i j+n )int(e j ; ie j+n )( j ; i j+n )int(e j + ie j+n ):(b) If 0 c =(@ + 0 ) 2 and 00 c =( @ + 00 ) 2 then these are elliptic on with L ( 0 c)= l ( 00 c )= 1 2 jj2 I.P Proof: We know L (d)(x )=i j j ext(e j ). We deneA() = 1 2Xjn( j+n ) ext(e j + ie j+n )then A(): pq ! p+1q and A(): pq ! pq+1 . Since iA()+i A() = L (d), we conclude that iA and i A represent the decomposition of L (d)under the bigrading and thus dene the symbols of @ and @. The symbolof the adjoint is the adjoint of the symbol and this proves (a). (b) is animmediate consequence of (a).If M is holomorphic, the Dolbeault complex is the complex f@ 0q g andthe index of this complex is called the arithmetic genus of M. If M is notholomorphic, but only has an almost complex structure, then @ 2 6= 0 sowe can not dene the arithmetic genus in this way. Instead, we use a trickcalled \rolling up" the complex. We dene: 0+ = M q 02q and 0; = M q 02q+1to dene a Z 2 grading on the Dolbeault bundles. (These are also oftendenoted by 0even and 0odd in the literature). We consider the two termelliptic complex:( @ + 00 ) : C 1 ( 0 ) ! C 1 ( 0 )

For Almost Complex Manifolds 185and dene the arithmetic genus of M to be the index of (@ + 00 ) + . Theadjoint of (@ + 00 ) + is (@ + 00 ) ; and the associated Laplacian is just 00 c restricted to 0 so this is an elliptic complex. If M is holomorphic,then the index of this elliptic complex is equal to the index of the complex(@ 0q ) by the Hodge decomposition theorem.We dene:c 0 () =( p 2) ;1 X jnf( j + i j+n )ext(e j ; ie j+n )then it is immediate that:; ( j ; i j+n )int(e j + ie j+n )gc 0 ()c 0 () =;jj 2 and L ( @ + 00 )=ic 0 ()= p 2:Let r be a connection on 0 , then we dene the operator A (r) bythediagram:A (r): C 1 ( 0 ) r ! C 1 (T M 0 ) c0 = p 2;;;! C 1 ( 0 )This will have the same leading symbol as (@ + 00 ) . There exists aunique connection so A (r) = (@ + 00 ) but as the index is constantunder lower order perturbations, we shall not need this fact as the index ofA + (r) = index( @ + 00 ) + for any r. We shall return to this point in thenext subsection.We now let V be an arbitrary coecient bundle with a connection r anddene the Dolbeault complex with coecients in V using the diagram:A : C 1 ( 0 V ) r ! C 1 (T M 0 V ) c0 = p 21;;;;;! C 1 ( 0 V )and we dene index(VDolbeault) to be the index of this elliptic complex.The index is independent of the connections chosen, of the ber metricschosen, and is constant under perturbations of the almost complex structure.If M is holomorphic and if V is a holomorphic vector bundle, then wecan extend @: C 1 ( 0q V ) ! C 1 ( 0q+1 V ) with @ @ =0. Exactly aswas true for the arithmetic genus, the index of this elliptic complex is equalto the index of the rolled up elliptic complex so our denitions generalizethe usual denitions from the holomorphic category to the almost complexcategory.We will compute a formula for index(VDolbeault) using the spin complex.There is a natural inclusion from U( m ) = U(n) into SO(m), but2this does not lift in general to SPIN(m). We saw earlier that T (CP k )

186 3.5. The Riemann-Roch Theoremdoes not admit a spin structure if k is even, even though it does admita unitary structure. We dene SPIN c (m) = SPIN(m) S 1 =Z 2 where wechoose the Z 2 identication (g ) = (;g ;). There is a natural map c : SPIN c (m) ! SO(m) S 1 induced by the map which sends (g ) 7!((g) 2 ). This map is a Z 2 double cover and is a group homomorphism.There is a natural map of U( m ) into SO(m) 2 S 1 dened by sendingg 7! (g det(g)). The interesting thing is that this inclusion does lift wecan dene f:U( m ) ! SPIN 2 c(m) so the following diagram commutes:SPIN c(m)f?y; ;;;;;%U( m 2 ) 1 det;;;;;;;;;! SO(m) S 1 c = 2We dene the lifting as follows. For U 2 U( m ), we choose a unitary2basis fe j g n j=1 so that U(e j)= j e j . We dene e j+n = ie j so fe j g 2nj=1is anorthogonal basis for R m = C n . Express j = e i jand dene:f(U) =nYj=1fcos( j =2) + sin( j =2)e j e j+n gnYj=1e i j=2 2 SPIN c (m):We note rst that e j e j+n is an invariant of the one-dimensional complexsubspace spanned by e j and does not change if we replace e j by ze j forjzj =1. Since all the factors commute, the order in which the eigenvaluesis taken does not aect the product. If there is a multiple eigenvalue, thisproduct is independent of the particular basis which is chosen. Finally,if we replace j by j +2 then both the rst product and the secondproduct change sign. Since (g ) =(;g ;) inSPIN c (m), this element isinvariantly dened. It is clear that f(I) =I and that f is continuous. It iseasily veried that c f(U) =i(U) det(U) where i(U) denotes the matrixU viewed as an element of SO(m) where we have forgotten the complexstructure on C n . This proves that f is a group homomorphism near theidentity and consequently f is a group homomorphism in general. c is acovering projection.The cannonical bundle K is given by:K = n0 = n (T 0 M)

For Almost Complex Manifolds 187so K = n (T 0 M) is the bundle with transition functions det(U ) wherethe U are the transitions for T 0 M. If we have a spin structure on M, wecan split:P SPIN c= P SPIN P S 1 =Z 2where P S 1 represents a line bundle L 1 over M. From this description, itis clear K = L 1 L 1 . Conversely, if we can take the square root of K(or equivalently of K ), then M admits a spin structure so the obstructionto constructing a spin structure on an almost complex manifold is theobstruction to nding a square root of the cannonical bundle.Let V = C n = R m with the natural structures. We extend torepresentations of SPIN c (m) in the natural way. We relate this representationto the Dolbeault representation as follows:Lemma 3.5.3. There is a natural isomorphism between 0 and cwhich denes an equivalence of these two representations of U( m ). Under2this isomorphism, the action of V by Cliord multiplication on the left ispreserved.Proof: Let fe j g be an orthonormal basis for R m with Je j = e j+n for1 j n. Dene: j = ie j e j+n j = e j + ie j+n j = e j ; ie j+n for 1 j n:We compute: j j = ; j and j k = k j for k 6= j:We dene = 1 ... n then spans n0 and j = ;, 1 j n. Wedene: 0 = M q 0q and n = M q nq = 0 :Since dim( 0 ) = 2 n , we conclude that 0 is the simultaneous ;1eigenspace for all the j and that therefore: c = 0 as a left representation space for SPIN(m). Again, we compute: j j = j j j = ; j j k = k j j k = k j for j 6= kso that if x 2 0q then since = 1 ... n we have x =(;1) q x so that c = 0 :

188 3.5. The Riemann-Roch TheoremWe now study the induced representation of U(n). LetWe compute:Consequently:g = ; cos(=2) + sin(=2)e 1 Je 1 e i=2 2 SPIN c (m):g 1 = fcos(=2) ; i sin(=2)ge i=2 1 = 1g 1 = fcos(=2) + i sin(=2)ge i=2 1 = e i 1g k = k g and g k = k g for k>1:g j = j if j 1 > 1e i j if j 1 =1.A similar computation goes for all the other indices and thus we computethat if Ue j = e i je j is unitary that:f(U) J = e i j 1 ...ei j q J which is of course the natural represenation of U(n) on 0q .Finally we compare the two actions of V by Cliord multiplication. Weassume without loss of generality that = (1 0 ... 0) so we must studyCliord multiplication by e 1 on c andon 0 . We compute:fext(e 1 ; ie 1+n ) ; int(e 1 + ie 1+n )g= p 2=c 0 (e 1 )e 1 1 = ;1+ie 1 e 2 =(e 1 ; ie 2 )(e 1 + ie 2 )=2 = 1 1 =2e 1 1 1 = ;2 1 e 1 k = ; k e 1for k>1:From this it follows immediately that:e 1 J =(;1) q 1 J =2 if j 1 > 12 J 0 where J 0 = fj 2 ...j q g if j 1 =1:Similarly, we compute:c 0 (e 1 ) J =1 J = p 2 if j 1 > 1; p 2 J 0 for J 0 = fj 2 ...j q g if j 1 =1:From these equations, it is immediate that if we dene T ( J ) = J ,then T will not preserve Cliord multiplication. We leta(q) be a sequenceof non-zero constants and dene T : 0 ! c by:T ( J )=a(jJj) J :

For Almost Complex Manifolds 189Since the spaces 0q are U(n)invariant, this denes an equivalence betweenthese two representations of U(n). T will induce an equivalence betweenCliord multiplication if and only if we have the relations:(;1) q a(q)=2 =a(q +1)= p 2 andp2a(q ; 1) = (;1) q 2 a(q):These give rise to the inductive relations:a(q +1)=(;1) q a(q)= p 2 and a(q) =(;1) q;1 a(q ; 1)= p 2:These relations are consistent and we set a(q) = ( p 2) ;q (;1) q(q;1)=2dene the equivalence T and complete the proof of the Lemma.From this lemma, we conclude:Lemma 3.5.4. There is a natural isomorphism of elliptic complexes( + c ; ; c ) V ' ( 0+ ; 0; ) V which takes the operator of theSPIN c complex to an operator which has the same leading symbol as theDolbeault complex (and thus has the same index). Furthermore, we canrepresent the SPIN c complex locally in terms of the SPIN complex in theform: ( + c ; ; c ) V =( + ; ; ) L 1 V where L 1 is a local squareroot of n (T 0 M).We use this sequence of isomorphisms to dene an operator on the Dolbeaultcomplex with the same leading symbol as the operator ( @ + @ 00 )which is locally isomorphic to the natural operator of the SPIN complex.The Z 2 ambiguity inthedenition of L 1 does not aect this construction.(This is equivalent to choosing an appropriate connection called the spinconnection on 0 .) We can compute index(VDolbeault) using this operator.The local invariants of the heat equation for this operator are thelocal invariants of the twisted spin complex and therefore arguing exactlyas we did for the signature complex, we compute:index(VDolbeault) =ZM^A(TM) ^ ch(L 1 ) ^ ch(V )where ch(L 1 ) is to be understood as a complex characteristic class ofT 0 (M).Theorem 3.5.5 (Riemann-Roch). Let Td(T 0 M) be the Todd classdened earlier by the generating function Td(A) = Q x =(1;e ;x ). Thenindex(VDolbeault) =ZMTd(T 0 M) ^ ch(V ):Proof: We must simply identify ^A(TM) ^ ch(L1 ) with Td(T 0 M). Weperform a computation in characteristic classes using the splitting principal.to

190 3.5. The Riemann-Roch TheoremWe formally decompose T 0 M = ~ L1 ~ Ln as the direct sum of linebundles. Then n (T 0 M) = ~ L1 ~ Ln so c 1 ( n T 0 M) = c 1 (~ L1 )++ c 1 (~ Ln ) = x 1 + + x n . Since L 1 L 1 = n T 0 M, we concludec 1 (L 1 )= 1 2 (x 1 + + x n ) so that:ch(L 1 )= Y e x =2 :Therefore:^A(M) ^ ch(L 1 )= Y = Y e x =2x e x =2; e ;x =2x 1 ; e ;x = Td(T 0 M):Of course, this procedure is only valid if the given bundle does in factsplit as the direct sum of line bundles. We can make this procedure acorrect way of calculating characteristic classes by using ag manifolds orby actually calculating on the group representaton and using the fact thatthe diagonizable matrices are dense.It is worth giving another proof of the Riemann-Roch formulatoensurethat we have not made a mistake of sign somewhere in all our calculations.Using exactly the same multiplicative considerations as we used earlier andusing the qualitative form of the formala for index(VDolbeault) given bythe SPIN c complex, it is immediate that there is some formula of the form:index(VDolbeault) =ZXN s+t=nTd 0 s(T 0 M) ^ c t ch t (V )where c is some universal constant to be determined and where Td 0 s is somecharacteristic form of T 0 M.If M = CP j , then we shall show in Lemma 3.6.8 that the arithmeticgenus of CP j is 1. Using the multiplicative property of the Dolbeaultcomplex, we conclude the arithmetic genus of CP j1 CP jk is 1 aswell. If M c are the manifolds of Lemma 2.3.4 then if we take V = 1 weconclude:1 = index(1 Dolbeault) = arithmetic genus of CP j1 CP jk=ZM c Td 0 (T 0 M):We veried in Lemma 2.3.5 that Td(T 0 M) also has this property so bythe uniqueness assertion of Lemma 2.3.4 we conclude Td = Td 0 . We takem =2and decompose: 0 2 = 00 11 and 1 = 10 01

For Almost Complex Manifolds 191so that e ; o =( 00 ; 01 ) ( 00 ; 10 ):In dimension 2 (and more generally if M is Kaehler), it is an easy exerciseto compute that = 2( @ 00 + 00 @) so that the harmonic spaces are thesame therefore:(M) = index( 00 Dolbeault) ; index( 10 Dolbeault)= ;cZMch( 10 )=cZMch(T 0 M)=cZMe(M):The Gauss-Bonnet theorem (or the normalization of Lemma 2.3.3) impliesthat the normalizing constant c =1and gives another equivalent proof ofthe Riemann-Roch formula.There are many applications of the Riemann-Roch theorem. We presenta few in dimension 4 to illustrate some of the techniques involved. Ifdim M =4,then we showed earlier that:c 2 (T 0 M)=e 2 (TM) and (2c 2 ; c 2 1)(T 0 M)=p 1 (T 0 M):The Riemann-Roch formula expresses the arithmetic genus of M in termsof c 2 and c 2 1. Consequently, there is a formula:Z Zarithmetic genus = a 1 e 2 (TM)+a 2 p 1 (TM)=3MM= a 1 (M)+a 2 signature(M)where a 1 and a 2 are universal constants. If we consider the manifoldsS 2 S 2 and CP 2 we derive the equations:1=a 1 4+a 2 0 and 1=a 1 3+a 2 1so that a 1 = a 2 =1=4 which proves:Lemma 3.5.6. If M is an almost complex manifold of real dimension 4,then:arithmetic genus(M) =f(M) + signature(M)g=4:Since the arithmetic genus is always an integer, we can use this result toobtain some non-integrability results:Corollary 3.5.7. The following manifolds do not admit almost complexstructures:(a) S 4 (the four dimensional sphere).(b) CP 2 with the reversed orientation.

192 3.5. The Riemann-Roch Theorem(c) M 1 # M 2 where the M i are 4-dimensional manifolds admitting almostcomplex structures (# denotes connected sum).Proof: We assume the contrary in each of these cases and attempt tocompute the arithmetic genus:a:g:(S 4 )= 1 4 (2 + 0) = 1 2a:g:(;CP 2 )= 1 4 (3 ; 1) = 1 2a:g:(M 1 # M 2 )= 1 4 ((M 1 # dM 2 ) + sign(M 1 # M 2 )= 1 4 ((M 1)+(M 2 ) ; 2 + sign(M 1 ) + sign(M 2 ))=a:g:(M 1 )+a:g:(M 2 ) ; 1 2 :In none of these examples is the arithmetic genus an integer which showsthe impossibility ofconstructing the desired almost complex structure.

3.6. A Review of Kaehler Geometry.In the previous subsection, we proved the Riemann-Roch theorem usingthe SPIN c complex. This was an essential step in the proof even if themanifold was holomorphic.Let M be holomorphic and let the coecient bundle V be holomorphic(i.e., the transition functions are holomorphic). We can dene the Dolbeaultcomplex directly by dening:@ V : C 1 ( 0q V ) ! C 1 ( 0q+1 V )by @ V (! s) = @! swhere s is a local holomorphic section to V . It is immediate @ V @V = 0 andthat this is an elliptic complex the index of this elliptic complex is justindex(VDolbeault) as dened previously.We let a n (x V Dolbeault) be the invariant of the heat equation for thiselliptic complex we use the notation a j (x Dolbeault) when V is the trivialbundle. Then:Remark 3.6.1. Let m = 2n >2. Then there exists a unitary Riemannianmetric on the m-torus (with its usual complex structure) and a pointx such that:(a) a j (x Dolbeault) 6= 0for j even and j n,(b) a m (x Dolbeault) 6= Td n where both are viewed as scalar invariants.The proof of this is quite long and combinatorial and is explained elsewherewe simply present the result to demonstrate that it is not in generalpossible to prove the Riemann-Roch theorem directly by heat equationmethods.The diculty is that the metric and the complex structure do not ttogether properly. Choose local holomorphic coordinates z = (z 1 ...z n )and extend the metric G to be Hermitian on T (M) C so that T 0 and T 00are orthogonal. We dene:g j k = G(@=@z j @=@z k )then the matrix g j k is a positive denite Hermitian matrix which determinesthe original metric on T (M):ds 2 =2 X jkg j k dzj dz kG(@=@x j @=@x k ) = G(@=@y j @=@y k )=(g j k + g k| )G(@=@x j @=@y k ) = ;G(@=@y j @=@x k )= 1 i (g jk ; g k| )We use this tensor to dene the X Kaehler 2-form:=i g j k dzj ^ dz k :jk

194 3.6. Kaehler GeometryThis is a real 2-form which is dened by the identity:[X Y ]=;G(X JY ):(This is a slightly dierent sign convention from that sometimes followed.)The manifold M is said to be Kaehler ifisclosed and the heat equationgives a direct proof of the Riemann-Roch theorem for Kaehler manifolds.We introduce variables:g j k=l and g j k= lfor the jets of the metric. On a Riemannian manifold, we can always nda coordinate system in which all the 1-jets of the metric vanish at a pointthis concept generalizes as follows:Lemma 3.6.2. Let M be a holomorphic manifold and let G be a unitarymetric on M. The following statements are equivalent:(a) The metric is Kaeher (i.e., d =0).(b) For every z 0 2 M there is a holomorphic coordinate system Z centredat z 0 so that g j k (Z G)(z 0)= jk and g j k=l (Z G)(z 0)=0.(c) For every z 0 2 M there is a holomorphic coordinate system Z centredat z 0 so g j k (Z G)(z 0)= jk and so that all the 1-jets of the metric vanishat z 0 .Proof: We suppose rst the metric is Kaehler and compute:d =i X jkfg j k=l dzl ^ dz j ^ dz k + g j k= l dzl ^ dz j ^ dz k g:Thus Kaehler is equivalent to the conditions:g j k=l ; g l k=j = g j k= l ; g j l= k =0:By making a linear change of coordinates, we can assume that the holomorphiccoordinate system is chosen to be orthogonal at the center z 0 . Weletz 0 j = z j + X c j kl z kz l (where c j kl = cj lk )and compute therefore:Xdzj 0 = dz j +2 c jX kl z ldz k@=@zj 0 = @=@z j ; 2 c k jlz l @=@z k + O(z 2 )g 0 j k = g j k ; 2X c k jl z l + terms in z + O(jzj 2 )g 0 j k=l = g j k=l ; 2ck jl at z 0 :

Kaehler Geometry 195We dene c k jl = 1 g 2 jk=l and observe that the Kaehler condition shows thisis symmetric in the indices (jl) so (a) implies (b). To show (b) implies (c)we simply note that g j k= l = g k|=l = 0 at z 0 . To prove (c) implies (a) weobserve d(z 0 )=0so since z 0 was arbitrary, d 0:If M is Kaehler, then is linear in the 1-jets of the metric when wecompute with respect to a holomorphic coordinate system. This impliesthat = 0 so is harmonic. We compute that k is also harmonic for1 k n and that n = c dvol is a multiple of the volume form (and inparticular is non-zero). This implies that if x is the element in H 2 (M C)dened by using the Hodge decomposition theorem, then 1x...x nall represent non-zero elements in the cohomology ring of M. This canbe used to show that there are topological obstructions to constructingKaehler metrics:Remark 3.6.3. If M = S 1 S m;1 for m even and m 4 then thisadmits a holomorphic structure. No holomorphic structure on M admitsa Kaehler metric.Proof: Since H 2 (M C) =0itisclearM cannot admit a Kaehler metric.We construct holomorphic structures on M as follows: let 2 C withjj > 1 and let M = fC n ; 0g= (where we identify z and w if z = k wfor some k 2 Z). The M are all topologically S 1 S m;1 for example if 2 R and we introduce spherical coordinates (r)onC n = R m then weare identifying (r) and ( k r). If we let t = log r, then we are identifyingt with t + k log so the manifold is just [1 log ] S m;1 where we identifythe endpoints of the interval. The topological identication of M for other is similar.We now turn to the problem of constructing a Kaehler metric on CP n .Let L be the tautological line bundle over CP n and let x = ;c 1 (L) 2 11 (CP n ) be the generator of H 2 (CP n Z) discussed earlier. We expandx in local coordinates in the form:x =i2Xjkg j k dzj ^ dz kand dene G(@=@z k @=@z k )=g j k . This gives an invariantly dened formcalled the Fubini-Study metric on T c (CP n ). We will show G is positivedenite and denes a unitary metric on the real tangent space. Since theKaehler form of G, =2x, is a multiple of x, we conclude d =0so Gwill be a Kaehler metric.The 2-form x is invariant under the action of U(n+1). Since U(n+1) actstransitively on CP n , it suces to show G is positive denite and symmetricat a single point. Using the notation of section 2.3, let U n = fz 2 CP n :z n+1 (z) 6= 0g. We identify U n with C n by identifying z 2 C n with the line

196 3.6. Kaehler Geometryin C n+1 through the point (z 1). Then:x =i2 @ @ log(1 + jzj 2 )so at the center z =0we compute that:x =i2Xjdz j ^ dz j so that g j k = j k :If G 0 is any other unitary metric on CP n which is invariant under theaction of U(n +1), then G 0 is determined by its value on T c (CP n ) at asingle point. Since U(n) preserves the origin of U n = C n , we concludeG 0 = cG for some c>0. This proves:Lemma 3.6.4.(a) There exists a unitary metric G on CP n which is invariant under theaction of U(n +1)such that the Kaehler form of G is given by = 2xwhere x = ;c 1 (L) (L is the tautological bundle over CP n ). G is a Kaehlermetric.(b) R CP nx n =1.(c) If G 0 is any other unitary metric on CP n which is invariant under theaction of U(n +1)then G 0 = cG for some constant c>0.A holomorphic manifold M is said to be Hodge if it admits a Kaehlermetric G such that the Kaehler form has the form = c 1 (L) for someholomorphic line bundle L over M. Lemma 3.6.4 shows CP n is a Hodgemanifold. Any submanifold of a Hodge manifold is again Hodge wherethe metric is just the restriction of the given Hodge metric. Thus everyalgebraic varietyisHodge. The somewhat amazing fact is that the converseis true:Remark 3.6.5. A holomorphic manifold M is an algebraic variety (i.e.,is holomorphically equivalent to a manifold dened by algebraic equationsin CP n for some n) if and only if it admits a Hodge metric.We shall not prove this remark but simply include it for the sake of completeness.We also note in passing that there are many Kaehler manifoldswhich are not Hodge. The Riemann period relations give obstructions oncomplex tori to those tori being algebraic.We now return to our study of Kaehler geometry. We wishtorelatethe@ cohomology of M to the ordinary cohomology. Dene:H pq (M) =ker @=im @in bi-degree (p q):Since the Dolbeault complex is elliptic, these groups are nite dimensionalusing the Hodge decomposition theorem.

Kaehler Geometry 197P Lemma 3.6.6. Let M be Kaehler and let = i g j k dzj ^ dz k be theKaehler form. Let d = @ + @ and let = 0 + 00 . Then we have theidentities:(a) @ int() ; int() @ = i 0 .(b) @ 0 + 0 @ = @ 00 + 00 @ =0.(c) d + d =2(@ 0 + 0 @)=2(@ 00 + 00 @). (d) We can decompose the de Rham cohomology in terms of the Dolbeaultcohomology:H n (M C) =MH pq (M C):p+q=n(e) There are isomorphisms H pq ' H qp ' H n;pn;q ' H n;qn;p .Proof: We rst prove (a). We dene:A = @ int() ; int() @ ; i 0 :If we can show that A is a 0 th order operator, then A will be a functoriallydened endomorphism linear in the 1-jets of the metric. This impliesA = 0 by Lemma 3.6.2. Thus we must check L (A) = 0. We choose anorthonormal basis fe j g for T (M) =T (M) so Je j = e j+n , 1 j n. ByLemma 3.5.2, it suces to show that:ext(e j ; ie j+n )int() ; int() ext(e j ; ie j+n )=;i int(e j ; ie j+n )for 1 j n. Since g j k = 1 2 jk , is given by= i 2Xj(e j + ie j+n ) ^ (e j ; ie j+n )= X je j ^ e j+n :The commutation relations:int(e k )ext(e l ) + ext(e l )int(e k )= klimply that int(e k ^ e k+n ) = ; int(e k )int(e k+n ) commutes with ext(e j ;ie j+n ) for k 6= j so these terms disappear from the commutator. We mustshow:ext(e j ; ie j+n )int(e j ^ e j+n ) ; int(e j ^ e j+n )ext(e j ; ie j+n )= ;i int(e j ; ie j+n ):This is an immediate consequence of the previous commutation relationsfor int and ext together with the identityint(e j )int(e k )+int(e k )int(e j )=0.This proves (a).

198 3.6. Kaehler GeometryFrom (a) we compute:@ 0 + 0 @ = i(; @ int() @)+i( @ int() @)=0and by taking complex conjugates @ 00 + 00 @ =0. This shows (b). Thus=(d+) 2 = d +d = f@ 0 + 0 @g+f @ 00 + 00 @g = f@ + 0 g 2 +f @ + 00 g 2since all the cross terms cancel. We apply (a) again to compute:which yields the identity:@ @ int() ; @ int() @ = i@ 0@ int()@ ; int() @@ = i 0 @i(@ 0 + 0 @)=@ @ int() ; @ int() @ + @ int()@ ; int() @@:Since is real, we take complex conjugate to conclude:;i( @ 00 + 00 @)= @@int() ; @ int()@ + @ int() @ ; int()@ @= ;i(@ 0 + 0 @):This shows ( @ 00 + 00 @)=(@ 0 + 0 @) so =2( @ 00 + 00 @) which proves(c).H n (M C) denotes the de Rham cohomology groups of M. The Hodgedecomposition theorem identies these groups with the null-space of (d+).Since N(d + ) = N( @ + 00 ), this proves (d). Finally, taking complexconjugate and applying induces the isomorphisms:H pq ' H qp ' H n;qn;p ' H n;pn;q :In particular dim H 1 (M C) iseven if M admits a Kaehler metric. Thisgives another proof that S 1 S 2n;1 does not admit a Kaehler metric forn>1.The duality operations of (e) can be extended to the Dolbeault complexwith coecients in a holomorphic bundle V . If V is holomorphic,the transition functions are holomorphic and hence commute with @. Wedene:@: C 1 ( pq V ) ! ( pq+1 V )by dening @(! s) = @! s relative to any local holomorphic frame forV . This complex is equivalent to the one dened in section 3.5 and denescohomology classes:H pq (V )=ker @=im @ on C 1 ( pq V ):

Kaehler Geometry 199It is immediate from the denition that:H pq (V )=H 0q ( p0 V ):We deneXindex(V@) = index(VDolbeault) = (;1) q dim H 0q (V )to be the index of this elliptic complex.There is a natural duality (called Serre duality) which is conjugate linear: H pq (V ) ! H n;pn;q (V ), where V is the dual bundle. We put aunitary ber metric on V and dene:T (s 1 )(s 2 )=s 2 s 1so T : V ! V is conjugate linear. If : k ! m;k is the ordinary Hodgeoperator, then it extends to dene : pq ! n;qn;p . The complex conjugateoperator denes : pq ! n;pn;q . We extend this to have coecientsin V by dening:(! s) =! Tsso : pq V ! n;pn;q V is conjugate linear. Then:Lemma3.6.7. Let 00V be the adjoint of @ V on C 1 ( p V ) with respectto a unitary metric on M and a Hermitian metric on V . Then(a) 00 = ; @ V .(b) induces a conjugate linear isomorphism between the groups H pq (V )and H n;pn;q (V ).Proof: It is important tonote that this lemma, unlike the previous one,does not depend upon having a Kaehler metric. We suppose rst V isholomorphically trivial and the metric on V is at. Then V = 1 kfor some k and we may suppose k = 1 for the sake ofsimplicity. We notedin section 1.5. This decomposes = ;d = ;d = 0 + 00 = ;@; @:Since : pq ! n;pn;q , we conclude 0 = ;@ and 00 = ; @. Thiscompletes the proof if V is trivial. More generally, we deneA = 00V + @:

200 3.6. Kaehler GeometryThis must be linear in the 1-jets of the metric on V . We can alwaysnormalize the choice of local holomorphic frame so the 1-jets vanish ata basepoint z 0 and thus A = 0 in general which proves (a). We identifyH pq (V ) with N( @ V )\N(V 00 ) using the Hodge decomposition theorem. Thus: H pq (V ) ! H n;pn;q (V ). Since 2 = 1, this completes the proof.We dened x = ;c 1 (L) 2 11 (CP n ) for the standard metric on thetautological line bundle over CP n . In section 2.3 we showed that x washarmonic and generates the cohomology ring of CP n . Since x denes aKaehler metric, called the Fubini-Study metric, Lemma 3.6.6 lets us decomposeH n (CP n C) = L p+q=n Hpq (CP n ). Since x k 2 H kk this shows:Lemma 3.6.8. Let CP n be given the Fubini-Study metric. Let x =;c 1 (L). Then H kk (M) ' C is generated by x k for 1 k n. H pq (M) =0 for p 6= q. Thus in particular,index( @)= arithmetic genus = X (;1) k dim H 0k =1 for CP n :Since the arithmetic genus is multiplicative withrespect to products,index( @)=1 for CP n1 CP nk :We used this fact in the previous subsection to derive the normalizingconstants in the Riemann-Roch formula.We can now present another proof of the Riemann-Roch theorem forKaehler manifolds which is based directly on the Dolbeault complex andnot on the SPIN c complex. Let Z be a holomorphic coordinate system andletg uv = G(@=@z u @=@z v )represent the components of the metric tensor. We introduce additionalvariables:g uv= = d z d z g uvfor 1 u v n, =( 1 ... n ), and =( 1 ... n ):These variables represent the formal derivatives of the metric.We must also consider the Dolbeault complex with coecients in a holomorphicvector bundle V . We choose a local ber metric H for V . Ifs =(s 1 ...s k ) is a local holomorphic frame for V ,weintroduce variables:h pq = H(s p s q ) h pq= = d z d z h pq for 1 p q k =dimV:If Z is a holomorphic coordinate system centered at z 0 and if s is a localholomorphic frame, we normalize the choice so that:g uv (Z G)(z 0 )= uv and h pq (s H)(z 0 )= pq :

Kaehler Geometry 201The coordinate frame f@=@z u g and the holomorphic frame fs p g for V areorthogonal at z 0 . Let R be the polynomial algebra in these variables. IfP 2 R we can evaluate P (Z G s H)(z 0 ) once a holomorphic frame Z ischosen and a holomorphic frame s is chosen. We say P is invariant ifP (Z G s H)(z 0 ) = P (G H)(z 0 ) is independent of the particular (Z s)chosen let R c nk denote the sub-algebra of all invariant polynomials. Asbefore, we dene:ord(g uv= ) = ord(s pq= )=jj + jj:We need only consider those variables of positive order as g uv = j v ands pq = pq at z 0 . If R c nk denotes the subspace of invariant polynomialshomogenous of order in the jets of the metrics (G H), then there is adirect sum decomposition:R c nk = M R c nk :In the real case in studying the Euler form, we considered the restrictionmap from manifolds of dimension m to manifolds of dimension m ; 1. Inthe complex case, there is a natural restriction mapr: R c nk !R c n;1kwhich lowers the complex dimension by one (and the real dimension bytwo). Algebraically, r is dened as follows: letWe dene:deg k (g uv= )= ku + (k)deg k (h pq= )=(k)deg k (g uv= )= kv + (k)deg k (h pq= )=(k):r(g uv= )= guv= if deg n (g uv= ) + deg n(g uv= )=00 if deg n (g uv= ) + deg n(g uv= ) > 0r(h pq= )= hpq= if deg n (h pq= ) + deg n(h pq= )=00 if deg n (h pq= ) + deg n(h pq= ) > 0:This denes a map r: R n !R n;1 which is an algebra morphism we simplyset to zero those variables which do not belong to R n;1 . It is immediatethat r preserves both invariance and the order of a polynomial. Geometrically,we are considering manifolds of the form M n = M n;1 T 2 whereT 2 is the at two torus, just as in the real case we considered manifoldsM m = M m;1 S 1 .Let R chnpk be the space of p-forms generated by the Chern forms ofT c (M) andby the Chern forms of V . We take the holomorphic connectiondened by the metrics G and H on T c (M) and on V . If P 2R chnmk thenP is a scalar invariant. Since P vanishes on product metrics of the formM n;1 T 2 which are at in one holomorphic direction, r(P ) = 0. Thisis the axiomatic characterization of the Chern forms which we shall need.

202 3.6. Kaehler GeometryTheorem 3.6.9. Let the complex dimension of M be n and let the berdimension of V be k. Let P 2R c nk and suppose r(P )=0. Then:(a) If < 2n then P (G H) =0for all Kaehler metrics G.(b) If = 2n then there exists a unique Q 2 R chnmk so P (G H) =Q(G H) for all Kaehler metrics G.As the proof of this theorem is somewhat technical, we shall postpone theproof until section 3.7. This theorem is false if we don't restrict to Kaehlermetrics. There is a suitable generalization to form valued invariants. Wecan apply Theorem 3.6.9 to give a second proof of the Riemann-Roch theorem:Theorem 3.6.10. Let V be a holomorphic bundle over M with bermetric H and let G be a Kaehler metric on M. Let @ V : C 1 ( 0q V ) !C 1 ( 0q+1 V ) denote the Dolbeault complex with coecients in V andlet a (x @ V ) be the invariants of the heat equation. a 2R c nk andZa (x @ 0if 6= 2nV )dvol(x) =index( @ V ) if =2n:MThen(a) a (x @ V )=0for < 2n.(b) a 2n (x @ V )=fTd(T c M) ^ ch(V )g m .Proof: We note this implies the Riemann-Roch formula. By remark3.6.1, we note this theorem is false in general if the metric is not assumed tobe Kaehler. The rst assertions of the theorem (including the homogeneity)follow from the results of Chapter 1, so it suces to prove (a) and (b).The Dolbeault complex is multiplicative under products M = M n;1 T 2 .If we take the product metric and assume V is the pull-back of a bundleover M n;1 then the natural decomposition: 0q V ' 0q (M n;1 ) V 0q;1 (M n;1 ) Vshows that r(a ) = 0. By Theorem 3.6.9, this shows a = 0 for < 2nproving (a).In the limiting case = 2n, we conclude a 2n is a characteristic 2nform. We rst suppose m =1. Any one dimensional complex manifold isKaehler. ThenZZindex( @ V )=a 1 c 1 (T c (M)) + a 2 c 1 (V )Mwhere a 1 and a 2 are universal constants to be determined. Lemma 3.6.6 impliesthat dim H 00 =dimH 0 =1=dimH 2 =dimH 11 while dim H 10 =dim H 01 = g where g is the genus of the manifold. Consequently:(M) =2; 2g = 2 index( @):M

Kaehler Geometry 203We specialize to the case M = S 2 and V = 1 is the trivial bundle. Theng =0andZ1 = index( @ V )=a 1 c 1 (T c (M)) = 2a 1so a 1 = 1 . We now take the line bundle V 2 =10 soZindex( @ V )=g ; 1=;1 =1+a 2 c 1 (Tc )=1; 2a 2so that a 2 =1. Thus the Riemann-Roch formula in dimension 1 becomes:index( @ V )= 1 2ZMMMc 1 (T c (M)) +ZMc 1 (V ):We now use the additive nature of the Dolbeault complex with respectto V and the multiplicative nature with respect to products M 1 M 2 toconclude that the characteristic m-form a m must have the form:fT 0 d(T c M) ^ ch(V )gwhere we use the fact the normalizing constant for c 1 (V ) is 1 if m = 1.We now use Lemma 2.3.5(a) together with the fact that index( @ V )=1forproducts of complex projective spaces to show Td 0 = Td. This completesthe proof.

3.7. An Axiomatic CharacterizationOf the Characteristic Forms forHolomorphic Manifolds with Kaehler Metrics.In subsection 3.6 we gave a proof of the Riemann-Roch formula forKaehler manifolds based on Theorem 3.6.9 which gave an axiomatic characterizationof the Chern forms. This subsection will be devoted to givingthe proof of Theorem 3.6.9. The proof is somewhat long and technical sowe break it up into a number of steps to describe the various ideas whichare involved.We introduce the notation:g u0 v 0 =u 1 ...u

Characteristic Forms 2051 jj ; 2 and dene a new coordinate Xsystem Z by setting:w u = z u + C u z jj=where the constants C u remain to be determined. Since this is the identitytransformation up to order , the ; 2 jets of the metric are unchangedso conditions (a) and (b) are preserved for jj

206 3.7. Characteristic FormsWe adopt the notational conventions that indices (p q) run from 1 throughk and index a frame for V . We compute:Xh pq (s H) =h pq (s 0 H)+ C q z jj=+ terms in z+ terms vanishing to order +1h pq= (x H)(z 0 )=h pq= (s 0 H)(z 0 )+!C qfor jj = and derivatives of lower order are not disturbed. This determines theC q uniquely so h pq= (s H)(z 0 )=0and taking complex conjugate yieldsh q p= (s H)(z 0 )=0which completes the proof of the lemma.Using these two normalizing lemmas, we restrict henceforth to polynomialsin the variables fg( )h pq=1 1g where jj 2, jj 2, j 1 j1,j 1 j 1. We also restrict to unitary transformations of the coordinatesystem and of the ber of V .If P is a polynomial in these variables and if A is a monomial, let c(A P )be the coecient of A in P . A is a monomial of P if c(A P ) 6= 0. Lemma2.5.1 exploited invariance under the group SO(2) and was central to ouraxiomatic characterization of real Pontrjagin forms. The natural groupsto study here are U(1), SU(2) and the coordinate permutations. If we set@=@w 1 = a@=@z 1 , @=@ w 1 = a@=@z 1 for aa = 1 and if we leave the otherindices unchanged, then we compute:A(W ) =a deg 1 (A) a deg 1 (A) A(Z ):If A is a monomial of an invariant polynomial P , then deg 1 (A) = deg 1(A)follows from this identity and consequently deg u (A) = deg u (A) for all1 u n.This is the only conclusion which follows from U(1) invariance so we nowstudy the group SU(2). We consider the coordinate transformation:@=@w 1 = a@=@z 1 + b@=@z 2 @=@ w 1 =a@=@z 1 + b@=@z 2 @=@w u = @=@z u@=@w 2 = ; b@=@z 1 +a@=@z 2 @=@ w 2 = ;b@=@z 1 + a@=@z 2 for u>2@=@ w u = @=@z u for u>2aa + b b =1:We let j = deg 1 (A)+deg 2 (A) =deg 1(A) + deg 2(A) and expandA(W )=a j a j A(Z )+a j;1 a j bA(1 ! 2 or 2 ! 1)(Z )+ a j a j;1 bA(2 ! 1 or 1 ! 2)(Z )+ other terms:

For Kaehler Manifolds 207The notation \A(1 ! 2 or 2 ! 1)" indicates all the monomials (with multiplicity)of this polynomial constructed by either changing a single index1 ! 2 or a single index 2 ! 1. This plays the same role as the polynomialA (1) of section 2.5 where we must now also consider holomorphic andanti-holomorphic indices.Let P be U(2) invariant. Without loss of generality, we may assumedeg 1 (A) +deg 2 (A) = j is constant for all monomials A of P since thiscondition is U(2) invariant. (Of course, the use of the indices 1 and 2 isfor notational convenience only as similar statements will hold true for anypairs of indices). We expand:P (W ) =a j a j P (Z )+a j;1 a j bP (1 ! 2 or 2 ! 1)Since P is invariant, we conclude:+ a j a j;1 bP (2 ! 1 or 1 ! 2) + O(b b) 2 :P (1 ! 2 or 2 ! 1) = P (2 ! 1 or 1 ! 2) = 0:We can now study these relations. Let B be an arbitrary monomial andexpand:B(1 ! 2) = c 0 A 0 + + c k A k B(2 ! 1) = ;(d 0 A 0 0 + + d k A 0 k)where the c's and d's are positiveintegers with P c =deg 1 (B) and P d =deg 2(B): Then it is immediate that the fA j g and the fA 0 jg denote disjointcollections of monomials since deg 1 (A) = deg 1 (B) ; 1 while deg 1 (A) 0 =deg 1 (B). We compute:A (2 ! 1) = ;c 0 B +othertermsA0 (1 ! 2) = d 0 B +othertermswhere again the c's and d's are positive integers (related to certain multiplicities).Since P (2 ! 1 or 1 ! 2) is zero, if P is invariant, we concludean identity:c(BP(2 ! 1 or 1 ! 2)) = X ;c 0 c(A P)+ X d 0 c(A0 P)=0:By varying the creating monomial B (and also interchanging the indices1 and 2), we can construct many linear equations among the coecients.We note that in practice B will never be a monomial of P since deg 1 (B) 6=deg 1(B). We use this principle to prove the following generalization ofLemmas 2.5.1 and 2.5.5:

208 3.7. Characteristic FormsLemma 3.7.3. Let P be invariant under the action of U(2) and let A bea monomial of P .(a) If A = g( )A 0 0, then by changing only 1 and 2 indices and 1 and2 indices we can construct a new monomial A 1 of P which has the formA 1 = g( 1 1 )A 00 0 where (1) + (2) = 1 (1) and where 1 (2) = 0.(b) If A = h pq= A0 0 then by changing only 1 and 2 indices and 1 and2 indices we can construct a new monomial A 1 of P which has the formA 1 = h pq=1 1A 00 0 where (1) + (2) = 1 (1) and where 1 (2) = 0.Proof: We prove (a) as the proof of (b) is the same. Choose A of thisform so (1) is maximal. If (2) is zero, we are done. Suppose the contrary.Let B = g( 1 )A 0 0 where 1 = ((1) + 1(2) ; 1(3) ...(n)). Weexpand:B(1 ! 2) = 1 (1)A + monomials divisible by g( 1 )B(2 ! 1) = monomials divisible by g( 1 1 ) for some 1 .Since A is a monomial of P , we use the principle described previously toconclude there is some monomial of P divisible by g( 1 1 ) for some 1 .This contradicts the maximality of and shows (2) = 0 completing theproof.We now begin the proof of Theorem 3.6.9. Let 0 6= P 2 R c nk be ascalar valued invariant homogeneous of order with r(P )=0. Let A be amonomial of P . DecomposeA = g( 1 1 )...g( r r )h p1 q 1 = r+1 r+1...h ps q s = r+s r+s:Let `(A) = r + s be the length of A. We show `(A) n as follows.Without changing (rsj j j j) we can choose A in the same form sothat 1 (u) =0for u>1. We now x the index 1 and apply Lemma 3.7.3to the remaining indices to choose A so 2 (u) =0for > 2. We continuein this fashion to construct such an A so that deg u (A) =0for u>r + s.Since deg n (A) = deg n (A) > 0, we conclude therefore that r + s n. Weestimate:X = ord(P )= fj j + j j;2g +r 2r +2s 2n = mXr

For Kaehler Manifolds 209This holds true for every monomial of P since the construction of A 1 fromA involved the use of Lemma 3.7.3 and does not change any of the ordersinvolved. By Lemma 3.7.1 and 3.7.2 we assumed all the purely holomorphicand purely anti-holomorphic derivatives vanished at z 0 and consequently:j j = j j =2 for r and j j = j j =1 for r< s + r.Consequently, P is a polynomial in the fg(i 1 i 2 | 1 | 2 )h pq=i| g variables itonly involves the mixed 2-jets involved.We wish to choose a monomial of P in normal form to begin countingthe number of possible such P . We begin this process with:Lemma 3.7.4. Let P satisfy the hypothesis of Theorem 3.6.8(b). Thenthere exists a monomial A of P which has the form:where t + s = n.A = g(11 | 1 | 0 1)...g(tt| t | 0 t )h p1 q 1 =t+1| t+1...h ps q s =m| nProof: We let denote indices which are otherwise unspecied and letA 0 be a generic monomial. We have deg n (A) > 0 for every monomial Aof P so deg u (A) > 0 for every index u as well. We apply Lemma 3.7.3 tochoose A of the form:A = g(11 | 1 | 0 1)A 0 :Suppose r>1. If A = g(11 )g(11 : )A 0 , then we could argue as beforeusing Lemma 3.7.3 that we could choose a monomial A of P so deg k (A) =0for k > 1+`(A 0 ) = r + s ; 1 = n ; 1 which would be false. Thus A =g(11 )g(jk )A 0 where not both j and k are 1. Wemay apply a coordinatepermutation to choose A in the form A = g(11 )g(2j )A 0 . If j 2we can apply Lemma 3.7.3 to choose A = g(11 )g(22 )A 0 . Otherwisewe suppose A = g(11 )g(12 )A 0 . We let B = g(11 )g(22 )A 0 andcompute:B(2 ! 1) = A + terms divisible by g(11 )g(22 )B(1 ! 2) = terms divisible by g(11 )g(22 )so that we conclude in any event we can choose A = g(11 )g(22 )A 0 . Wecontinue this argument with the remaining indices to construct:A = g(11 )g(22 )...g(tt )h p1 q 1 =u 1 v 1...h ps q s =u s v s:We have deg u (A) 6= 0 for all u. Consequently, the indices ft +1 ...t+ sgmust appear among the indices fu 1 ...u s g. Since these two sets haves elements, they must coincide. Thus by rearranging the indices we canassume u = + t which completes the proof. We note deg u (A) = 2 foru t and deg u (A) =1for u>t.This lemma does not control the anti-holomorphic indices, we furthernormalize the choice of A in the following:

210 3.7. Characteristic FormsLemma 3.7.5. Let P satisfy the hypothesis of Theorem 3.6.8(b). Thenthere exists a monomial A of P which has the form:forA = g(11 | 1 | 0 1)...g(tt| t | 0 t )h p1 q 1 =t+1t+1 ...h p s q s =nn1 j j 0 t:Proof: If A has this form, then deg j (A) = 1 for j > t. Thereforedeg | (A) = 1 for j > t which implies 1 j j 0 t automatically. Wesay that | touches itself in A if A is divisible by g(| | ) for some . Wesay thatj touches | in A if A is divisible by h pq=j | for some (p q). ChooseA of the form given in Lemma 3.7.4 so the number of indices j >twhichtouch | in A is maximal. Among all such A, choose A so the number of| t which touch themselves in A is maximal. Suppose A does not satisfythe conditions of the Lemma. Thus A must be divisible by h pq=uv foru 6= v and u > t. We suppose rst v > t. Since deg v (A) = deg v (A) = 1,v does not touch v in A. We let A = h pq=uv A 0 and B = h pq=uu A 0 thendeg v (B) =0. We compute:B(u ! v) =A + A 1 and B(v ! u) =A 2where A 1 is dened by interchanging u and v and where A 2 is dened byreplacing both v and v by u and u in A. Thus deg v (A) 2 =0so A 2 is nota monomial of A. Thus A 1 = h pq=uu A 0 0 must be a monomial of P . Onemore index (namely u) touches its holomorphic conjugate in A 1 than in A.This contradicts the maximality ofA and consequently A = h pq=uv A 0 0 foru > t, v t. (This shows h p0 q 0 =u 0 v 0does not divide A for any u 0 6= v 0and v 0 > t.) We have deg u (A) =1so the anti-holomorphic index u mustappear somewhere in A. It cannot appear in a h variable and consequentlyA has the form A = g(u w)h pq=uv A 00 0. We dene B = g( w w)h pq=uv A 00 0and compute:B(w ! u) =A + terms divisible by g( w w)B(u ! w) =g( w w)h pq=wv A 000:Since g( w w)h pq=wv A 00 0 does not have the index u, it cannot be a monomialof P . Thus terms divisible by g( w w) must appear in P . We constructthese terms by interchanging a w and u index in P so the maximality ofindices u 0 touching u 0 for u 0 >tis unchanged. Since w does not touch w inA, we are adding one additional index of this form which again contradictsthe maximality ofA. This nal contradiction completes the proof.This constructs a monomial A of P which has the form degu (AA = A 0 A 1 for0 ) = deg u (A 0 )=0 u>tdeg u (A 1 ) = deg u (A 1 )=0 u t:

For Kaehler Manifolds 211A 0 involves only the derivatives of the metric g and A 1 involves only thederivatives of h. We use this splitting in exactly the same way we useda similar splitting in the proof of Theorem 2.6.1 to reduce the proof ofTheorem 3.6.9 to the following assertions:Lemma 3.7.6. Let P satisfy the hypothesis of Theorem 3.6.9(b).(a) If P is a polynomial in the fh pq=uv g variables, then P = Q for Q aChern m form of V .(b) If P is a polynomial in the fg(u 1 u 2 v 1 v 2 )g variables, then P = Q forQ a Chern m form of T c M.Proof: (a). If A is a monomial of P , then deg u (A) =deg u (A) = 1 for allu so we can expressA = h p1 q 1 =1u 1...h pn q n =nu nwhere the fu g are a permutation of the indices i through n.A = h =1u1 h =2u2 A 0 0 and B = h =1u1 h =2u1 A 0 0. Then:We letB(u 1 ! u 2 )=A + A 1B(u 2 ! u 1 )=A 2for A 1 = h =1u2 h =2u1 A 0 0for deg u2(A 2 )=0:Therefore A 2 is not a monomial of P . Thus A 1 is a monomial of P andfurthermore c(A P )+c(A 1 P)=0. This implies when we interchange u 1and u 2 that we change the sign of the coecient involved. This impliesimmediately we can express P is terms of expressions:( V p 1 q 1 ^ ^ V p n q n)= X ~u~v(h p1 q 1 =u 1 v 1du 1 ^ dv 1 = n! X ^ h pn q n =u n v ndu n ^ dv n )sign()h p1 q 1 =1(1) ...h pn q n =n(n) + where isapermutation. This implies P can be expressed as an invariantpolynomial in terms of curvature which implies it must be a Chern formas previously computed.The remainder of this section is devoted to the proof of (b).Lemma 3.7.7. Let P satisfy the hypothesis of Lemma 3.7.6(b). Then wecan choose a monomial A of P which has the form:A = g(11 u 1 u 1 )...g(nnu n u n ):This gives a normal form for a monomial. Before proving Lemma 3.7.7,we use this lemma to complete the proof of Lemma 3.7.6(b). By making a

212 3.7. Characteristic Formscoordinate permutation if necessary we can assume A has either the formg(11 11)A 0 0 or g(11 22)A 0 0. In the latter case, we continue inductively toexpress A = g(11 22)g(22 33)...g(u ; 1u; 1 uu)g(uu 11)A 0 0 until thecycle closes. If we permit u = 1 in this decomposition, we can also includethe rst case. Since the indices 1 through u appear exactly twice in Athey do not appear in A 0 0. Thus we can continue to play the same gameto decompose A into cycles. Clearly A is determined by the length ofthe cycles involved (up to coordinate permutations) the number of suchclassifying monomials is (n), the number of partitions of n. This showsthat the dimension of the space of polynomials P satisfying the hypothesisof Lemma 3.7.6(b) is (n). Since there are exactly (n) Chern formsthe dimension must be exactly (n) and every such P must be aChernmform as claimed.We give an indirect proof to complete the proof of Lemma 3.7.7. ChooseA of the form given by Lemma 3.7.5 so the number of anti-holomorphicindices which touch themselves is maximal. If every anti-holomorphic indextouches itself, then A has the form of Lemma 3.7.7 and we are done. Wesuppose the contrary. Since every index appears exactly twice, every antiholomorphicindex which does not touch itself touches another index whichalso does not touch itself. Every holomorphic index touches only itself. Wemay choose the notation so A = g( 12)A 0 0. Suppose rst 1 does not touch2 in A 0 0. Then we can assume A has the form:A = g( 12)g( 13)g( 2 k)A 0 0where possibly k = 3 in this expression. The index 1 touches itself in A.The generic case will be:A = g(11 )g( 12)g( 13)g( 2 k)A 0 0 :The other cases in which perhaps A = g(11 12) . . . or g(11 13) . . . org(11 2 k) . . . are handled similarly. The holomorphic and anti-holomorphicindices do not interact. In exactly which variable they appear does notmatter. This can also be expressed as a lemma in tensor algebras.We suppose k 6= 3we will never let the 2 and 3 variables interact so thecase in which k =3is exactly analogous. Thus A has the form:A = g(11 )g( 12)g( 13)g( 2 k)g( 3| )A 0 0where possibly j = k. Set B = (11 )g( 22)g( 13)g( 2 k)g( 3| )A 0 0then:B(2 ! 1) =2A + A 00B(1 ! 2) = 2A 1

forFor Kaehler Manifolds 213A 00 = g(11 )g( 22)g( 13)g( 1 k)g( 3| )A 0 0A 1 = g(12 )g( 22)g( 13)g( 2 k)g( 3| )A 0 0 :A 00 is also a monomial of the form given by Lemma 3.7.5. Since one moreanti-holomorphic index touches itself in A 00 then does in A, the maximalityof A shows A 00 is not a monomial of P . Consequently A 1 is a monomial ofP . Set B 1 = g(12 )g( 22)g( 33)g( 2 k)g( 3| )A 0 0 then:forB 1 (3 ! 1) = 2A 1 + A 2B 1 (1 ! 3) = A 000A 2 = g(12 )g( 22)g( 33)g( 2 k)g( 1| )A 0 0A 000 = g(32 )g( 22)g( 33)g( 2 k)g( 3| )A 0 0:However, deg 1 (A 000 )=0. Since r(P )=0,A 000 cannot be a monomial of Pso A 2 is a monomial of P . Finally we setB 2 = g(11 )g( 22)g( 33)g( 2 k)g( 1| )A 0 0soB 2 (1 ! 2) = A 2forB 2 (2 ! 1) = A 3 +2A 4A 3 = g(11 )g( 22)g( 33)g( 1 k)g( 1| )A 0 0A 4 = g(11 )g( 12)g( 33)g( 2 k)g( 1| )A 0 0:This implies either A 3 or A 4 is monomial of P . Both these have every holomorphicindex touching itself. Furthermore, one more anti-holomorphicindex (namely 3) touches itself. This contradicts the maximality ofA.In this argument it was very important that 2 6= 3 as we let the index 1interact with each of these indices separately. Thus the nal case we mustconsider is the case in which A has the form:A = g(11 )g(22 )g( 12)g( 12)A 0 :So far we have not had to take into account multiplicities or signs in computingA(1 ! 2) etc we have been content to conclude certain coecientsare non-zero. In studying this case, we must be more careful in our analysisas the signs involved are crucial. We clear the previous notation and dene:A 1 = g(12 )g(22 )g( 12)g( 22)A 0A 2 = g(12 )g(22 )g( 22)g( 12)A 0A 3 = g(11 )g(22 )g( 11)g( 22)A 0A 4 = g(22 )g(22 )g( 22)g( 22)A 0 :

214 3.7. Characteristic FormsWe note that A 3 is not a monomial of P by the maximality of A. A 4is not a monomial of P as deg 1 (A 4 ) = 0 and r(P ) = 0. We let B =g(11 )g(22 )g( 12)g( 22)A 0 . ThenB(2 ! 1) = 2A + A 3 B(1 ! 2) = 2A 1so that A 1 must be a monomial of P since A is a monomial of P and A 4 isnot. We now pay more careful attention to the multiplicities and signs:A(1 ! 2) = B + and A 1 (2 ! 1) = B + However 1 ! 2 introduces a b while 2 ! 1 introduces a ; b so using theargument discussed earlier we conclude not only c(A 1 P) 6= 0 but thatc(A P ) ; c(A 1 P) = 0 so c(A 1 P) = c(A P ). A 2 behaves similarly sothe analogous argument using B 1 = g(11 )g(22 )g( 22)g( 12) showsc(A 2 P) = c(A P ). We now study B 2 = g(12 )g(22 )g( 22)g( 22)and compute:B 2 (1 ! 2) = A 4 B 2 (2 ! 1) = 2A 1 +2A 2A 4 is not a monomial of P as noted above. We again pay careful attentionto the signs:A 1 (1 ! 2) = B 2 and A 2 (1 ! 2) = B 2This implies c(A 1 P)+c(A 2 P)=0. Since c(A 1 P)=c(A 2 P)=c(A P )this implies 2c(A P ) = 0 so A was not a monomial of P . This nal contradictioncompletes the proof.This proof was long and technical. However, it is not a theorem basedon unitary invariance alone as the restriction axiom plays an importantrole in the development. We know of no proof of Theorem 3.6.9 whichis based only on H. Weyl's theorem in the real case, the correspondingcharacterization of the Pontrjagin classes was based only on orthogonalinvariance and we gave a proof based on H. Weyl's theorem in that case.

3.8. The Chern Isomorphism and Bott Periodicity.In section 3.9 we shall discuss the Atiyah-Singer index theorem in generalusing the results of section 3.1. The index theorem gives a topologicalformula for the index of an arbitrary elliptic operator. Before beginingthe proof of that theorem, we must rst review briey the Bott periodicitytheorem from the point ofCliordmodules. We continue our considerationof the bundles over S n constructed in the second chapter. LetGL(k C) =f A : A is a k k complex matrix with det(A) 6= 0gGL 0 (k C) =f A 2 GL(k C) : det(A ; it) 6= 0for all t 2 R gU(k) =f A 2 GL(k C) :A A = I gS(k) =f A 2 GL(k C) :A 2 = I and A = A gS 0 (k) =f A 2 S(k) :Tr(A) =0g:We note that S 0 (k) is empty if k is odd. U(k) is compact and is adeformation retract of GL(k C) S(k) is compact and is a deformationretract of GL 0 (k C). S 0 (k) is one of the components of S(k).Let X be a nite simplicial complex. The suspension X is dened byidentifying X f g to a single point N and X 2 f; g to single point S in2the product X ; 2 2 . Let D (X) denote the northern and southern\hemispheres" in the suspension the intersection D + (X) \ D ; (X) = X:[Reprinter's note:A figure depicting the suspension of X belongs here.]We note both D + (X) and D ; (X) are contractible. (X) is a nite simplicialcomplex. If S n is the unit sphere in R n+1 , then (S n ) = S n+1 .Finally, ifW is a vector bundle over some base space Y , then we choose aber metric on W and let S(W ) be the unit sphere bundle. (W ) is theberwise suspension of S(W )over Y . This can be identied with S(W 1).It is beyond the scope of this book to develop in detail the theory ofvector bundles so we shall simply state relevant facts as needed. We letVect k (X) denote the set of isomorphism classes of complex vector bundles

216 3.8. The Chern Isomorphismover X of ber dimension k. We let Vect(X) = S k Vect k(X) be the set ofisomorphism classes of complex vector bundles over X of all ber dimensions.We assume X is connected so that the dimension of a vector bundleover X is constant.There is a natural inclusion map Vect k (X) ! Vect k+1 (X) dened bysending V 7! V 1 where 1 denotes the trivial line bundle over X.Lemma 3.8.1. If 2k dim X then the map Vect k (X) ! Vect k+1 (X) isbijective.We let F (Vect(X)) be the free abelian group on these generators. Weshall let (V ) denote the element ofK(X) dened by V 2 Vect(X). K(X) isthe quotient modulo the relation (V W )=(V )+(W ) for VW 2 Vect(X).K(X) is an abelian group. The natural map dim:Vect(X) ! Z + extendsto dim: K(X) ! Z e K(X) is the kernel of this map. We shall say thatan element of K(X) is a virtual bundle e K(X) is the subgroup of virtualbundles of virtual dimension zero.It is possible to dene a kind of inverse in Vect(X):Lemma 3.8.2. Given V 2 Vect(X), there exists W 2 Vect(X) so V W =1 j is isomorphic to a trivial bundle of dimension j =dimV + dim W .We combine these two lemmas to give a group structure to Vect k (X) for2k dim X. Given VW 2 Vect k (X), then V W 2 Vect 2k (X). By Lemma3.8.1, the map Vect k (X) ! Vect 2k (X) dened by sending U 7! U 1 k isbijective. Thus there is a unique element we shall denote by V + W 2Vect k (X) so that (V + W ) 1 k = V W . It is immediate that this isan associative and commutative operation and that the trivial bundle 1 kfunctions as the unit. We use Lemma 3.8.2 to construct an inverse. GivenV 2 Vect k (X) there exists j and W 2 Vect j (X) so V W = 1 j+k . Weassume without loss of generality that j k. By Lemma 3.8.1, we chooseW 2 Vect k (X) so that V W 1 j;k = V W =1 j+k . Then the bijectivityimplies V W =1 2k so W is the inverse of V . This shows Vect k (X) is agroup under this operation.There is a natural map Vect k (X) ! e K(X) dened by sending V 7!(V ) ; (1 k ). It is immediate that:(V + W ) ; 1 k =(V + W )+(1 k ) ; (1 2k )=((V + W ) 1 k ) ; (1 2k )=(V W ) ; (1 2k )=(V )+(W ) ; (1 2k )=(V ) ; (1 k )+(W ) ; (1 k )so the map is a group homomorphism. e K(X) is generated by elements ofthe form (V ) ; (W ) for VW 2 Vect j X for some j. If we choose W soW W = 1 v then (V ) ; (W ) = (V W) ; (1 v ) so e K(X) is generatedby elements of the form (V ) ; (1 v ) for V 2 Vect v (X). Again, by adding

And Bott Periodicity 217trivial factors, we may assume v k so by Lemma 3.8.1 V = V 1 1 j;kand (V ) ; (1 j ) = (V 1 ) ; (1 k ) for V 1 2 Vect k (X). This implies the mapVect k (X) ! e K(X) is subjective. Finally, we note that that in fact K(X)is generated by Vect k (X) subject to the relation (V )+(W )=(V W )=(V + W )+(1 k )sothatthis map is injective and we may identify e K(X) =Vect k (X) and K(X) = Z e K(X) = Z Vectk (X) for any k such that2k dim X.Tensor product denes a ring structure on K(X). We dene (V )(W )=(V W ) for VW 2 Vect(X). Since (V 1 V 2 ) (W )=(V 1 W V 2 W ),this extends from F (Vect(X)) to dene a ring structure on K(X) inwhichthe trivial line bundle 1 functions as a multiplicative identity. e K(X) is anideal of K(X).K(X) isaZ-module. It is convenient tochange the coecient group anddene:K(X C)=K(X) Z Cto permit complex coecients. K(X C) is the free C-vector space generatedby Vect(X) subject to the relations V W = V + W . By usingcomplex coecients, we eliminate torsion which makes calculations muchsimpler. The Chern character is a morphism:ch:Vect(X) ! H even (X C) = M qH 2q (X C):We dene ch using characteristic classes in the second section if X is asmooth manifold it is possible to extend this denition using topologicalmethods to more general topological settings. The identities:ch(V W )=ch(V )+ch(W ) and ch(V W )=ch(V ) ch(W )imply that we can extend:ch: K(X) ! H even (X C)to be a ring homomorphism. We tensor this Z-linear map with C to getch: K(X C) ! H even (X C):Lemma 3.8.3 (Chern isomorphism). ch: K(X C) ! H even (X C) isa ring isomorphism.If e Heven(X C) = L q>o H2q (X C) is the reduced even dimensional cohomology,then ch: e K(X C) ! e Heven(X C) is a ring isomorphism. Forthis reason, e K(X) is often refered to as reduced K-theory. We emphasizethat in this isomorphism we are ignoring torsion and that torsion is crucial

218 3.8. The Chern Isomorphismto understanding K-theory in general. Fortunately, the index is Z-valuedand we can ignore torsion in K-theory for an understanding of the indextheorem.We now return to studying the relation between K(X) andK(X). Weshall let [X Y ] denote the set of homotopy classes of maps from X toY . We shall always assume that X and Y are equipped with base pointsand that all maps are basepoint preserving. We x 2k dim X and letf: X ! S 0 (2k). Since f(x) is self-adjoint, and f(x) 2 = I, the eigenvalues off(x) are 1. Since Tr f(x) =0,each eigenvalue appears with multiplicityk. We let (f) be the bundles over X which are sub-bundles of X 1 kso that the ber of (f) at x is just the 1 eigenspace of f(x). If wedene (f)(x) = 1 (1 f(x)) then these are projections of constant rank2k with range (f). If f and f 1 are homotopic maps, then they determineisomorphic vector bundles. Thus the assignment f 7! + (f) 2 Vect k (X)denes a map + :[X S 0 (2k)] ! Vect k (X).Lemma 3.8.4. The natural map [X S 0 (2k)] ! Vect k (X) is bijective for2k dim X.Proof: Given V 2 Vect k (X) we choose W 2 Vect k (X) so V W =1 2k .We choose ber metrics on V and on W and make this sum orthogonal.By applying the Gram-Schmidt process to the given global frame, we canassume that there is an orthonormal global frame and consequently thatV W = 1 2k is an orthogonal direct sum. We let + (x) be orthogonalprojection on the ber V x of 1 2k and f(x) = 2 + (x) ; I. Then it isimmediate that f: X ! S 0 (2k) and + (f) = V . This proves the map issubjective. The injectivity comes from the same sorts of considerations aswere used to prove Lemma 3.8.1 and is therefore omitted.If f: X ! GL 0 (k C), we can extend the denition to let (f)(x) bethe span of the generalized eigenvectors of f corresponding to eigenvalueswith positive/negative real part. Since there are no purely imaginaryeigenvalues, (f) has constant rank and gives a vector bundle over X.In a similar fashion, we can classify Vect k X =[X U(k)]. Since U(k) isa deformation retract of GL(k C), we identify [X U(k)] = [X GL(k C)].If g: X ! GL(k C), we use g as a clutching function to dene a bundle overX. Over D (X), we take thebundles D (X) C k . D + (X) \ D ; (X) =X. On the overlap, we identify (x z) + =(x z 0 ) ; if P z g(x) =z 0 . P If we lets + and s ; be the usual frames for C k over D then z i + g ijs + j = z i ; s; iso that we identify the frames using the identitys ; = gs + :We denote this bundle by V g . Homotopic maps dene isomorphic bundlesso we have a map [X GL(k C)] ! Vect k X. Conversely, given a vectorbundle V over X we can always choose local trivializations for V over

And Bott Periodicity 219D (X) since these spaces are contractible. The transition function s ; =gs + gives a map g: X ! GL(k C). It is convenient to assume X hasa base point x 0 and to choose s ; = s + at x 0 . Thus g(x 0 ) = I. Thisshows the map [X GL(k C)] ! Vect k X is surjective. If we had chosendierent trivializations ~s + = h + s + and ~s ; = h ; s ; then we would haveobtained a new clutching function ~g = h ; g(h + ) ;1 . Since h are denedon contractible sets, they are null homotopic so ~g is homotopic to g. Thisproves:Lemma 3.8.5. The map [X GL(k C)] = [X U(k)] ! Vect k (X), givenby associating to a map g the bundle dened by the clutching function g,is bijective.It is always somewhat confusing to try to work directly with this denition.It is always a temptation to confuse the roles of g and of its inverseas well as the transposes involved. There is another denition which avoidsthis diculty and which willbevery useful in computing specic examples.If g: X ! GL(k C), we shall let g(x)z denote matrix multiplication. Weshall regard C k as consisting of column vectors and let g act as a matrixfrom the left. This is, of course, the opposite convention from that usedpreviously.Let 2 [; 2 ]bethe suspension parameter. We dene:2: [X GL(k C)] ! [X GL 0 (2k C)]: [X GL 0 (k C)] ! [X GL(k C)]by!(sin )Ig(x )=k (cos )g (x)(cos )g(x) ;(sin )I kf(x ) = (cos )f(x) ; i(sin )I k :We check that has the desired ranges as follows. If g: X ! GL(k C),then it is immediate that g is self-adjoint. We compute:!(g) 2 = (sin2 )I k + (cos 2 )g g 00 (sin 2 )I k +(cos 2 )gg :!This is non-singular since g is invertible. Therefore g is invertible. If = =2, then g = I 0 is independent of x. If g is unitary, g 20 ;IS 0 (2k). If f 2 GL 0 (k C), then f has no purely imaginary eigenvalues sof is non-singular.

220 3.8. The Chern IsomorphismLemma 3.8.6. Let g: X ! GL(k C) and construct the bundle + (g)over X. Then this bundle is dened by the clutching function g.Proof: We may replace g by a homotopic map without changing theisomorphism class of the bundle + (g). Consequently, we may assumewithout loss of generality that g: X ! U(k) so g g = gg = I k . Consequently(g) 2 = I 2k and + (g) = 1 2 (I 2k +g). If z 2 C k , then: z + (g)(x ) =01 z + (sin )z:2 (cos )g(x)zThis projection from C k to + (g) is non-singular away from the southpole S and can be used to give a trivialization of + (g) onX ; S.From this description, it is clear that + (g) is spanned by vectors ofthe form 1 2 (1 + (sin ))z(cos )g(x)z + (g) consists of all vectors of the formon the second factor + (S):away from the south pole. At the south pole, ab! 0b 0w. Consequently, projectionis non-singular away from thenorth pole N and gives a trivialization of + (g)onX;N. We restrict tothe equator X and compute the composite of these two maps to determinethe clutching function: z07! 1 2 z 7! 1 0:g(x)z 2 g(x)zThe function g(x)=2 ishomotopic to g which completes the proof.This is a very concrete description of the bundle dened by the clutchingfunction g. In the examples we shall be considering, it will come equippedwith a natural connection which willmake computing characteristic classesmuch easier.We now compute the double suspension. Fix f: X ! S 0 (2k) andg: X !GL(k C). 2 f(x )= 2 g(x )sin cos f(cos )f(x) ; i sin gcos f(cos )f + i sin g; sin = cos sin ; i sin cos cos g (x)cos cos g(x) ; cos cos ; i sin !!

And Bott Periodicity 221We let U(1) bethe direct limit of the inclusions U(k) ! U(k +1) andS 0 (1) be the direct limit of the inclusions S 0 (2k) ! S 0 (2k +2)! .Then we have identied:eK(X) =[X S 0 (1)] and e K(X) =[X S0 (1)] = [S U(1)]We can now state:Theorem 3.8.7 (Bott periodicity). The map 2 : e K(X) =[X S0 (1)] ! e K( 2 X)=[ 2 X S 0 (1)]induces a ring isomorphism. Similarly, the map 2 : e K(X) =[X U(1)] ! e K( 2 X)=[ 2 X U(1)]is a ring isomorphism.We note that [X U(1)] inherits a natural additive structure from thegroup structure on U(1) by letting g g 0 be the direct sum of these twomaps. This group structure is compatible with the additive structure oneK(X) since the clutching function of the direct sum is the direct sum ofthe clutching functions. Similarly,we can put a ring structure on [X U(1)]using tensor products to be compatible with the ring structure on K(X). eThe Chern character identies K(X e C) with H e even (X C). We mayidentify 2 X with a certain quotient of X S 2 . Bott periodicity in thiscontext becomes the assertion K(X S 2 ) = K(X) K(S 2 ) which is theKunneth formula in cohomology.We now consider the case of a sphere X = S n . The unitary group U(2)decomposes as U(2) = U(1)SU(2) = S 1 S 3 topologically so 1 (U(2)) =Z, 2 (U(2)) = 0 and 3 (U(2)) = Z. This implies K(S e 1 ) = K(S e 3 ) = 0and K(S e 2 )= K(S e 4 )=Z. Using Bott periodicity, we know more generallythat:Lemma 3.8.8 (Bott periodicity). If n is odd, theneK(S n ) ' n;1 (U(1)) = 0:If n is even, theneK(S n ) ' n;1 (U(1)) = Z:It is useful to construct explicit generators of these groups. Let x =(x 0 ...x n ) 2 S n , let y = (x x n+1 ) 2 S n+1 , and let z = (y x n+2 ) 2S n+2 . If f: S n ! GL 0 (k C) and g: X ! GL(k C) we extend these to

222 3.8. The Chern Isomorphismbe homogeneous of degree 1 with values in the k k matrices. Then wecompute:f(y) =f(x) ; ix n+1! 2 xf(z) = n+2 f (x)+ix n+1f(x) ; ix n+1 ;x n+2!g(y)= x n+1 g (x)g(x) ;x n+1! 2 g(z)= x n+1 ; ix n+2 g (x):g(x) ;x n+1 ; ix n+2If we suppose that f is self-adjoint, then we can express: 2 f(z) =x n+21 00 ;1! I k + x n+10 i;i 0! I k + 0 11 0! f(x):We can now construct generators for n;1 U(1) and e K(S n ) using Cliffordalgebras. Let g(x 0 x 1 )=x 0 ; ix 1 generate 1 (S 1 )=Z thenforg(x 0 x 1 x 2 )=x 0 e 0 + x 1 e 1 + x 2 e 2!e 0 = 0 1 e1 01 =!!0 i e;i 02 = 1 0 :0 ;1The fe j g satisfy the relations e j e k +e k e j =2 jk and e 0 e 1 e 2 = ;iI 2 . + (g)is a line bundle over S 2 which generates e K(S 2 ). We compute: 2 g(x 0 x 1 x 2 x 3 )=x 0 e 0 + x 1 e 1 + x 2 e 2 ; ix 3 I 2 :If we introduce z 1 = x 0 + ix 1 and z 2 = x 2 + ix 3 then! 2 g(z 1 z 2 )= z 2 z 1:z 1 ;z 2Consequently 2 g generates 3 (U(1)). We suspend once to construct: 3 g(x 0 x 1 x 2 x 3 x 4 )= x 0 e 0 e 0 + x 1 e 0 e 1 + x 2 e 0 e 2 + x 3 e 1 I + x 4 e 2 I:The bundle + ( 3 g) is a 2-plane bundle over S 4 which generates e K(S 4).We express 3 g = x 0 e 4 0 + + x 4 e 4 4 then these matrices satisfy the commutationrelations e 4 j e4 j + e4 k e4 j =2 jk and e 4 0 e4 1 e4 2 e4 3 e4 4 = ;I.

And Bott Periodicity 223We proceed inductively to dene matrices e 2kj for 0 j 2k so thate i e j + e j e i = 2 ij and e 2k0 ...e 2k2k P= (;i)k I. These are matrics of shape2 k 2 k such that 2k;1 (g)(x) = x j e 2kj . In Lemma 2.1.5 we computedthat: Zchk( + 2k;1 g)=i k 2 ;k Tr(e 2kS 2k 0 ...e 2k2k) =1:These bundles generate e K(S 2k ) and 2k (g) generates 2k+1 (U(1)). Wesummarize these calculations as follows:Lemma 3.8.9. Let fe 0 ...e 2k g be a collection of self-adjoint matrices ofshape 2 k 2 k such thate i e j +e j e i =2 ij and such that e 0 ...e 2k =(;i) k I.We dene e(x) =x 0 e 0 + +x 2k e 2k for x 2 S 2k . Let + (e) be the bundle of+1 eigenvectors of e over S 2k . Then + (e) generates K(S e 2k ) ' Z. e(y) =e(x) ; ix 2k+1 generates R 2k+1 (U(1)) ' Z.an isomorphism andS 2k chk( + e)=1.RS 2k chk(V ): e K(S 2k ) ! Z isProof: We note that R S 2k chk(V ) is the index of the spin complex withcoecients in V since all the Pontrjagin forms of T (S 2k )vanish except forp 0 . Thus this integral, called the topological charge, is always an integer.We have checked that the integral is 1 on a generator and hence the mapis surjective. Since e K(S 2k )=Z, it must be bijective.It was extremely convenient to have the bundle with clutching function 2k;2 g so concretely given so that we could apply Lemma 2.1.5 to computethe topological charge. This will also be important in the next chapter.

3.9. The Atiyah-Singer Index Theorem.In this section, we shall discuss the Atiyah-Singer theorem for a generalelliptic complex by interpreting the index as a map in K-theory. Let Mbe smooth, compact, and without boundary. For the moment we makeno assumptions regarding the parity of the dimension m. We do not assumeM is orientable. Let P : C 1 (V 1 ) ! C 1 (V 2 ) be an elliptic complexwith leading symbol p(x ): S(T M) ! HOM(V 1 V 2 ). We let(T M)bethe berwise suspension of the unit sphere bundle S(T M). We identify(T M)=S(T M 1). We generalize the construction of section 3.8 todene p:(T M) ! END(V 1 V 2 ) by:!(sin )Ip(x )=V1 (cos )p (x ):(cos )p(x ) ; (sin )I V2This is a self-adjoint invertible endomorphism. We let (p) be the subbundleof V 1 V 2 over (T M) corresponding to the span of the eigenvectorsof p with positive/negative eigenvalues. If we have given connectionson V 1 and V 2 , we can project these connections to dene natural connectionson (p). The clutching function of + (p) is p in a sense weexplain as follows:We form the disk bundles D (M) over M corresponding to the northernand southern hemispheres of the ber spheres of (T M). Lemma3.8.6 generalizes immediately to let us identify + (p) with the bundleV +1 [ V ;2 over the disjoint union D + (M) [ D ; (M) attached using theclutching function p over their common boundary S(T M). If dim V 1 = k,then + (p) 2 Vect k ((T M)). Conversely, we suppose given a bundleV 2 Vect k ((T M)). Let N: M ! (T M) and S: M ! (T M) bethe natural sections mapping M to the northern and southern poles of theber spheres N(x) = (x 0 1) and S(x) = (x 0 ;1) in S(T M 1). Nand S are the centers of the disk bundles D (M). We let N (V ) = V 1and S (V )=V 2 be the induced vector bundles over M. D (M) deformationretracts to M fNg and M fSg. Thus V restricted to D (M)is cannonically isomorphic to the pull back of V 1 and V 2 . On the intersectionS(T M) = D + (M) \ D ; (M) we have a clutching or glueingfunction relating the two decompositions of V . This gives rise to a mapp: S(T M) ! HOM(V 1 V 2 ) which is non-singular. The same argument asthat given in the proof of Lemma 3.8.5 shows that V is completely determinedby the isomorphism class of V 1 and of V 2 together with the homotopyclass of the map p: S(T M) ! HOM(V 1 V 2 ).Given an order we can recover the leading symbol p by extending pfrom S(T M) to T M to be homogeneous of order . We use this todene an elliptic operator P : C 1 (V 1 ) ! C 1 (V 2 ) with leading symbol p .If Q is another operator with the same leading symbol, then we dene

The Atiyah-Singer Theorem 225P (t) = tP +(1; t)Q . This is a 1-parameter family of such operatorswith the same leading symbol. Consequently by Lemma 1.4.4 index(P )=index(Q ). Similarly, if we replace p by a homotopic symbol, then theindex is unchanged. Finally, suppose we give two orders of homgeneity 1 > 2 . We can choose a self-adjoint pseudo-dierential operator R onC 1 (V 2 ) with leading symbol jj 1; 2I V2 . Then we can let P 1 = RP 2 soindex(P 1 ) = index(R)+index(P 2 ). Since R is self-adjoint, its index is zeroso index(P 1 ) = index(P 2 ). This shows that the index only depends onthe homotopy class of the clutching map over S(T M) and is independentof the order of homogeneity andthe extension and the particular operatorchosen. Consequently, we can regard index: Vect((T M)) ! Z so thatindex( + (p)) = index(P )ifP is an elliptic operator. (We can always \rollup" an elliptic complex to give a 2-term elliptic complex in computing theindex so it suces to consider this case).It is clear from our denition that (p q) =(p) (q) and therefore + ((p q)) = + ((p)) + ((q)). Since index(P Q) = index(P )+index(Q) we conclude:index(V W ) = index(V ) + index(W )for VW 2 Vect((T M)):This permits us to extend index: K((T M)) ! Z to be Z-linear. Wetensor with the complex numbers to extend index: K((T M) C) ! C,so that:Lemma 3.9.1. There is a natural map index: K((T M) C) ! C whichis linear so that index(P ) = index( + (p)) if P : C 1 V 1 ! C 1 V 2 is anelliptic complex over M with symbol p.There is a natural projection map :(T M) ! M. This gives a naturalmap : K(M C) ! K((T M) C). If N denotes the north pole section,then N =1 M so N = 1 and consequently is injective. This permitsus to regard K(M C) as a subspace of K((T M) C). If V = V 1 ,thenthe clutching function dening V is just the identity map. Consequently,the corresponding elliptic operator P can be taken to be a self-adjointoperator on C 1 (V ) which has index zero. This proves:Lemma 3.9.2. If V 2 K((T M) C) can be written as V 1 for V 1 2K(M C) then index(V )=0. Thus index: K((T M) C)=K(M C) ! C.These two lemmas show that all the information contained in an ellipticcomplex from the point of view of computing its index is contained in thecorresponding description in K-theory. The Chern character gives an isomorphismof K(X C)totheeven dimensional cohomology. We will exploitthis isomorphism to give a formula for the index in terms of cohomology.In addition to the additive structure on K(X C), there is also a ringstructure. This ring structure also has its analogue with respect to elliptic

226 3.9. The Atiyah-Singer Theoremoperators as we have discussed previously. The multiplicative nature of thefour classical elliptic complexes played a fundamental role in determiningthe normalizing constants in the formula for their index.We give (T M) the simplectic orientation. If x = (x 1 ...x m ) is asystem of local coordinates on M, let =( 1 ... m )betheber coordinateson T M. Let u be the additional ber coordinate on T M 1. Weorient T M 1 using the form:! 2m+1 = dx 1 ^ d 1 ^ ^dx m ^ d m ^ du:Let ~ N be the outward normal on S(T M1) and dene ! 2m on S(T M1)by:! 2m+1 = ~ N ^ ! 2m = ! 2m ^ ~N:This gives the orientation of Stokes theorem.Lemma 3.9.3.(a) Let P : C 1 (V 1 ) ! C 1 (V 2 ) be an elliptic complex over M 1 and letQ: C 1 (W 1 ) ! C 1 (W 2 ) be an elliptic complex over M 2 . We assume Pand Q are partial dierential operators of the same order and we let M =M 1 M 2 and dene over MR =(P 1+1 Q) (P 1 ; 1 Q ): C 1 (V 1 W 1 V 2 W 2 )! C 1 (V 2 W 1 V 1 W 2 ):Then R is elliptic and index(R) = index(P ) index(Q).(b) The four classical elliptic complexes discussed earlier can be decomposedin this fashion over product manifolds.(c) Let p and q be arbitrary elliptic symbols over M 1 and M 2 and dener =p 1 ;1 q1 q p 1Let i 2 H even (M i C) for i =1 2, then:Z(T M) 1 ^ 2 ^ ch( + (r))=Z(T M 1 )! 1 ^ ch( + (p)) over M = M 1 M 2 :Z(T M 2 ) 2 ^ ch( + (q)):(d) Let q be an elliptic symbol over M 1 and let p be a self-adjoint ellipticsymbol over M 2 . Over M = M 1 M 2 dene the self-adjoint elliptic symbolr by:!r = 1 p q 1:q 1 ;1 p

The Atiyah-Singer Theorem 227Let 1 2 e (M 1 ) and 2 2 o (M 2 ) be closed dierential forms. GiveS(T M), (T M 1 ) and S(T M 2 ) the orientations induced from the simplecticorientations. Then:ZS(T M) 1 ^ 2 ^ ch( + r)=Z(T M 1 ) 1 ^ ch( + (q)) ZS(T M 2 ) 1 ^ ch( + (p)):Remark: We will use (d) to discuss the eta invariant in Chapter 4 weinclude this integral at this point since the proof is similar to that of (c).Proof: We let (p q r) bethe symbols of the operators involved. Then:r =p 1 ;1 q1 q p 1!r =p 1 1 q ;1 q p 1!so that:!r r = p p 1+1 q q 00 pp 1+1 qq !rr = pp 1+1 q q 00 p p 1+1 qq :r r and rr are positive self-adjoint matrices if ( 1 2 ) 6= (0 0). Thisveries the ellipticity. We note that if (P Q) are pseudo-dierential, R willstill be formally elliptic, but the symbol will not in general be smooth at 1 =0or 2 =0and hence R will not be a pseudo-dierential operator inthat case. We compute:!R R = P P 1+1 Q Q 00 PP 1+1 QQ !RR = PP 1+1 Q Q 00 P P 1+1 QQ N(R R)=N(P P ) N(Q Q) N(PP ) N(QQ )N(RR )=N(PP ) N(Q Q) N(P P ) N(QQ )index(R)=fdim N(P P ) ; dim N(PP )g= index(P ) index(Q)which completes the proof of (a).fdim N(Q Q) ; dim N(QQ )g

228 3.9. The Atiyah-Singer TheoremWe verify (b) on the symbol level. First consider the de Rham complexand decompose:(T M 1 )= e 1 o 1 (T M 2 )= e 2 o 2 (T M)=( e 1 e 2 o 1 o 2) ( o 1 e 2 e 1 o 2):Under this decomposition: L ((d + ) M )( 1 2 )= c(1 ) 1 ;1 c( 2 )1 c( 2 ) c( 1 ) 1c() denotes Cliord multiplication. This veries the de Rham complexdecomposes properly.The signature complex is more complicated. We decompose (T M 1 )= e1 o1to decompose the signature complex into two complexes:(d + ): C 1 ( +e1) ! C 1 ( ;o1)(d + ): C 1 ( +o1 ) ! C 1 ( ;e1 ):Under this decomposition, the signature complex of M decomposes intofour complexes. If, for example, we consider the complex:(d + ): C 1 ( +e1 +e2 ;o1 ;o2) !!C 1 ( ;o1 +e2 +e1 ;o2 )then the same argument as that given for the de Rham complex applies toshow the symbol is:If we consider the complex:c( 1 ) 1 ;1 c( 2 )1 c( 2 ) c( 1 ) 1(d + ): C 1 ( +o1 +e2 ;e1 ;o2 )then we conclude the symbol is!:! C 1 ( ;e1 +e2 +o1 ;o2)c( 1 ) 1 1 c( 2 );1 ( 2 ) c( 1 ) 1!:

The Atiyah-Singer Theorem 229which isn't right. We can adjust the sign problem for 2 either by changingone of the identications with + (M) orbychanging the sign of Q (whichwon't aect the index). The remaining two cases are similar. The spin andDolbeault complexes are also similar. If we take coecients in an auxilarybundle V , the symbols involved are unchanged and the same argumentshold. This proves (b).The proof of (c) is more complicated. We assume without loss of generalitythatp and q are homogeneous of degree 1. Let ( 1 2 u) parametrizethe bers of T M 1. At 1 = (1= p 2 0 ... 0), 2 = (1= p 2 0 ... 0),u =0the orientation is given by:; dx 1 1 ^ d 1 1 ^ dx 1 2 ^ d 1 2 ^^ dx 1 m 1 ^ d 1 m 1 ^ dx 2 1 ^ dx 2 2 ^ d 2 2 ^ ^dx 2 m 2 ^ d 2 m 2 ^ du:We have omitted d2 1 (and changed the sign) since it points outward at thispoint of the sphere to get the orientation on (T M).In matrix form we have on(V 1 W 1 )(V 2 W 2 )(V 2 W 1 )(V 1 W 2 )that:0 !! 1u 1 0p 1 1 q 0 1;1 q p 1r =B@p 1 ;1 q 1 q p 1!!;u 1 00 1This is not a very convenient form to work with. We dene 1 = 1 00 ;1 2 = 1 00 ;1!!and = 0 pp 0and = 0 qq 0!!:CAon V 1 V 2 = Von W 1 W 2 = Wand compute that r = u 1 2 + I W + 1 . We replace p and qby homotopic symbols so that:p p = j 1 j 2 I V1 pp = j 1 j 2 I V2 q q = j 2 j 2 I W1 qq = j 2 j 2 I W2 :Since f 1 2 I W 1 g all anti-commute and are self-adjoint,(r) 2 =(u 2 + j 1 j 2 + j 2 j 2 )I on V W:We parametrize (T M)by S(T M 1 ) 0 2 (T M 2 ) in the form:; 1 cos 2 sin u sin :

230 3.9. The Atiyah-Singer TheoremWe compute the orientation by studying = 4 , 1 = (1 0 ... 0), 2 =(1 0 ... 0), u =0. Since d 1 1 = ; sin d, the orientation is given by:dx 1 1 ^ d ^ dx 1 2 ^ d 1 2 ^^ dx 1 m 1 ^ d 1 m 1 ^ dx 2 1 ^ dx 2 2 ^ d 2 2 ^ ^dx 2 m 2 ^ d 2 m 2 ^ du:If we identify S(T M 1 ) 0 2 with D+ (T M 1 ) then this orientation is:! 1 2m 1 ^ ! 2 2m 2so the orientations are compatible.In this parametrization, we compute r( 1 2 u)=(sin) 1 (u 2 +( 2 )) + (cos )( 1 ) I W . Since q = u 2 + ( 2 ) satises (q) 2 =j 2 j 2 + u 2 2 on W , we may decompose W = + (q) ; (q). Then:r = f(sin ) 1 + (cos )( 1 )gI =p( 1 ) Ir = fsin(;) 1 +(cos)( 1 )gI =p( 1 ;) Ion V + (q)on V ; (q):Consequently: + (r) = + (p)( 1 ) + (q) + (p)( 1 ;) ; (q)over (D + M 1 ) (T M 2 ):If we replace ; by in the second factor, wemay replace + (p)( 1 ;) ; (q) by + (p)( 1 ) ; (q). Since we have changed the orientation,we must change the sign. Therefore:Z(T M)= 1 ^ 2 ^ ch( + (r))ZD + M 1 1 ^ ch( + (p)) ;ZZ(T M 2 )D ; M 1 1 ^ ch( + (p)) Z 2 ^ ch( + (q))(T M 2 ) 2 ^ ch( ; (q)):ch( + (q)) + ch( ; (q)) = ch(V 2 )doesnotinvolve the ber coordinatesof (T M 2 ) and thusZ(T M 2 ) 2 ^ ch(V 2 )=0:We may therefore replace ;ch( ; (q)) by ch( + (q)) in evaluating theintegral over D ; (M 1 ) (T M 2 )tocomplete the proof of (c).

The Atiyah-Singer Theorem 231We prove (d) in a similar fashion. We suppose without loss of generalitythat p and q are homogeneous of degree 1. We parameterize S(T M) =S(T M 1 ) [0 2 ] S(T M 2 )inthe form ( 1 cos 2 sin ). Thenr = sin 00 ; sin ! p( 2 )+Again we decompose V 2 = + (p) ; (p) so thatwhere; r = ; (sin ) 1 + (cos ) 1 ( 1 ) r = sin(;) 1 + (cos ) 1 ( 1 )! 1 = 1 00 ;10 (cos )q ( 1 )(cos )q( 1 ) 0! 1 = 0 q:q 0on V + (p)on V ; (p)The remainder of the argument is exactly as before (with the appropriatechange in notation from (c)) and is therefore omitted.We now check a specic example to verify some normalizing constants:Lemma 3.9.4. Let M = S 1 be the unit circle. Dene P : C 1 (M) !C 1 (M) by:P (e in )=nei(n;1)for n 0ne in for n 0then P is an elliptic pseudo-dierential operator with index(P )=1. Furthermore:Z(T S 1 )ch( + (p)) = ;1:Proof: Let P 0 = ;i@=@ and let P 1 = f;@ 2 =@ 2 g 1=2 . P 0 is a dierentialoperator while P 1 is a pseudo-dierential operator by the results of section1.10. It is immediate that: L (P 0 )= P 0 (e in )=ne in L (P 1 )=jj P 1 (e in )=jnje in :!:We dene:Q 0 = 1 2 e;i (P 0 + P 1 ) L Q 0 =e;i 00 0Q 0 (e in )=nei(n;1)n 00 n 0

232 3.9. The Atiyah-Singer TheoremandQ 1 = 1 2 (P 0 ; P 1 ) L Q 1 = 0 0 0Q 1 (e in )=n 0 n 0ne in n 0:Consequently, P = Q 0 + Q 1 is a pseudo-dierential operator ande;i>0 L P = p = :te ;i ;tt ;t!e +i!if 0if 0:Since p does not depend on for 0, we may restrict to the region 0 in computing the integral. (We must smooth out the symbol to besmooth where = 0 but suppress these details to avoid undue technicalcomplications).It is convenient to introduce the parameters:u = t v = cos w = sin for u 2 + v 2 + w 2 =1then this parametrizes the region of (T S 1 ) where 0ina1-1fashionexcept where =0. Since du ^ dv ^ dw = ;d^ d ^ dt, S 2 inherits thereversed orientation from its natural one. Lete 0 = 1 00 ;1! e 1 = 0 11 0! e 2 =!0 i;i 0so p(u v w)=ue 0 + ve 1 + we 2 . Then by Lemma 2.1.5 we have;which completes the proof.ZS 2 ch( + (p)) = (;1) i2 Tr(e 0 e 1 e 2 )=;1We can now state the Atiyah-Singer index theorem:

The Atiyah-Singer Theorem 233Theorem 3.9.5 (The index theorem). Let P : C 1 (V 1 ) ! C 1 (V 2 )be an elliptic pseudo-dierential operator. Let Td r(M) =Td(TMC) bethe Todd class of the complexication of the real tangent bundle. Then:index(P )=(;1) dim M Z (T M)Td r(M) ^ ch( + (p)):We remark that the additional factor of (;1) dim M could have been avoidedif we changed the orientation of (T M).We begin the proof by reducing to the case dim M = m even and Morientable. Suppose rst that m is odd. We take Q: C 1 (S 1 ) ! C 1 (S 1 )to be the operator dened in Lemma 3.9.4 with index +1. We then formthe operator R with index(R) = index(P ) index(Q) = index(P ) denedin Lemma 3.9.3. Although R is not a pseudo-dierential operator, it canbe uniformly approximated by pseudo-dierential operators in the naturalFredholm topology (once the order of Q is adjusted suitably). (This processdoes not work when discussing the twisted eta invariant and will involveus in additional technical complications in the next chapter). Therefore:Z(;1) m+1 Td r(M S 1 ) ^ ch( + (r))(T (MS 1 ))=(;1) m Z (T M)Td r(M) ^ ch( + (p)) (;1)=(;1) m Z (T M)Td r(M) ^ ch( + (p)):Z(T S 1 )ch( + (q))To show the last integral gives index(P ) it suces to show the top integralgives index(R) and therefore reduces the proof of Theorem 3.9.3 to the casedim M = m even.If M is not orientable, we let M 0 be the orientable double cover of M.It is clear the formula on the right hand side of the equation multipliesby two. More careful attention to the methods of the heat equation forpseudo-dierential operators gives a local formula for the index even inthis case as the left hand side is also multiplied by two under this doublecover.This reduces the proof of Theorem 3.9.3 to the case dim M = m evenand M orientable. We x an orientation on M henceforth.Lemma 3.9.6. Let P : C 1 ( + ) ! C 1 ( ; ) be the operator of the signaturecomplex. Let ! = chm=2( + (p)) 2 H m ((T M) C). Then if ! Mis the orientation class of M(a) ! M ^ ! gives the orientation of (T M).

234 3.9. The Atiyah-Singer Theorem(b) If S m is a ber sphere of (T M), then R S m ! =2 m=2 .Proof: Let (x 1 ...x m )beanoriented local coordinate system on M sothat the fdx j g are orthonormal at x 0 2 M. If =(... m ) are the dualber coordinates for T M), then:p() = X ji j (c(dx j )) = X ji j (ext(dx j ) ; int( j ))gives the symbol of (d + ) c() denotes Cliord multiplication as denedpreviously. We let e j = ic(dx j ) these are self-adjoint matrices such thate j e k + e k e j =2 jk . The orientation class is dened by:e 0 = i m=2 c(dx 1 )...c(dx m )=(;i) m=2 e 1 ...e m :The bundles are dened as the 1 eigenspaces of e 0 . Consequently,p(t) =te 0 + X j e j :Therefore by Lemma 2.1.5 when S m is given its natural orientation,ZS m chm=2 + (p) =i m=2 2 ;m=2 Tr(e 0 e 1 ...e m )= i m=2 2 ;m=2 Tr(e 0 i m=2 e 0 )=(;1) m=2 2 ;m=2 Tr(I) =(;1) m=2 2 ;m=2 2 m=(;1) m=2 2 m=2 :However, S m is in fact given the orientation induced from the orientationon (T M) and on M. At the point (x 0 ... 0 1) in T M R the naturalorientations are:of X:of (T M):dx 1 ^ ^dx m dx 1 ^ d 1 ^ ^dx m ^ d m=(;1) m=2 dx 1 ^ ^dx m ^ d 1 ^ ^d mof S m : (;1) m=2 d 1 ^ ^d m :Thus with the induced orientation, the integral becomes 2 m=2lemma is proved.Consequently, ! provides a cohomology extension and:and theLemma 3.9.7. Let :(T M) ! M where M is orientable and evendimensional. Then(a) : H (M C) ! H ((T M) C) is injective.

The Atiyah-Singer Theorem 235(b) If ! is as dened in Lemma 3.9.6, then we can express any 2H ((T M) C) uniquely as = 1 + 2 ^ ! for i 2 H (M C).Since is injective, we shall drop it henceforth and regard H (M C)as being a subspace of H ((T M) C).The Chern character gives an isomorphism K(X C) ' H e (X C). Whenwe interpret Lemma 3.9.7 in K-theory, we conclude that we can decomposeK((T M) C) =K(M C) K(M C) + (p) ch(V ) generatesH e (M C)asV ranges over K(M C). Therefore K((T M) C)=K(M C)is generated as an additive module by the twisted signature complex withcoecients in bundles over M. + (p V )=V + (p)ifp V is the symbolof the signature complex with coecients in V .In Lemma 3.9.2, we interpreted the index as a map in K-theory. Sinceit is linear, it suces to compute on the generators given by the signaturecomplex with coecients in V . This proves:Lemma 3.9.8. Assume M is orientable and of even dimension m. Let P Vbe the operator of the signature complex with coecients in V . The bundlesf + (p) V g V 2Vect (M) generate K((T M) C)=K(M C) additively.It suces to prove Theorem 3.9.5 in general by checking it on the specialcase of the operators P V .We will integrate out the ber coordinates to reduce the integral of Theorem3.9.5 from (T M) to M. We proceed as follows. Let W be anoriented real vector bundle of ber dimension k + 1 over M equippedwith a Riemannian inner product. Let S(W ) be the unit sphere bundleof W . Let : S(W ) ! M be the natural projection map. We dene a mapI : C 1 ((S(W ))) ! C 1 ((M)) which is a C 1 ((M)) module homomorphismand which commutes with integration|i.e., if 2 C 1 ((S(W )))and 2 C 1 ((M)), we require the map 7! I () to be linear so thatI ( ^ ) = ^I() and R S(W ) = R M I ().We construct I as follows. Choose a local orthonormal frame for W todene ber coordinates u = (u 0 ...u k ) on W . This gives a local representationof S(W ) = V S k over the coordinate patch U P on M. If 2 C 1 ((S(W ))) has support over V ,we can decompose = ^ for 2 C 1 ((U)) and 2 C 1 ((S k )). We permit the to havecoecients which depend upon x 2 U. This expression is, of course, notunique. Then I () is necessarily dened by:I ()(x) = X ZS k (x):It is clear this is independent of the particular way wehave decomposed .If we can show I is independent of the frame chosen, then this will deneI in general using a partition of unity argument.

236 3.9. The Atiyah-Singer TheoremLet u 0 i = a ij(x)u j be a change of ber coordinates. Then we compute:du 0 i = a ij (x) du j + da ij (x)u j :Clearly if a is a constant matrix, we are just reparamatrizing S k so theintegral is unchanged. We x x 0 and suppose a(x 0 )=I. Then over x 0 ,du I = dv I +XjIj

The Atiyah-Singer Theorem 237It is clear Todd(m) is uniquely determined by Lemma 3.9.9. Both theindex of an elliptic operator and the formula of Lemma 3.9.9 are multiplicativewith respect to products by Lemma 3.9.3 so Todd(m) is a multiplicativecharacteristic form. We may therefore drop the dependence uponthe dimension m and simply refer to Todd. We complete the proof of theAtiyah-Singer index theorem by identifying this characteristic form withthe real Todd polynomial of T (M).We work with the Dolbeault complex instead of with the signature complexsince the representations involved are simpler. Let m be even, then(;1) m = 1. Let M be a holomorphic manifold with the natural orientation.We orient the bers of T M using the natural orientation which arisesfrom the complex structure on the bers. If are the ber coordinates,this gives the orientation:dx 1 ^ dy 1 ^ ^dx n ^ dy n ^ d 1 ^ ^d mwhere m =2n:This gives the total space T M an orientation which is (;1) n times thesimplectic orientation. Let S m denote a ber sphere of (T M)withthisorientation and let q be the symbol of the Dolbeault complex. Then:index( @)=ZZ Todd ^ ch( + q) =(;1) n Todd ^I (ch( + q)):(T M) MWe dene the complex characteristic formS =(;1) m=2 I (ch( + (q)))then the Riemann-Roch formula implies that:S ^ Todd(m) =Todd(T c M):It is convenient to extend the denition of the characteristic form S toarbitrary complex vector bundles V . Let W = (V ) and let q(v): V !END(W ) be dened by Lemma 3.5.2(a) to be the symbol of @ + 00ifV = T c (M). We let ext: V ! HOM(WW) be exterior multiplication.This is complex linear and we let int be the dual of ext int(v) = int(v)for 2 C. ext is invariantly dened while int requires the choice of a bermatric. We let q = ext(v) ; int(v) (where we have deleted the factor of i=2which appears in Lemma 3.5.2 for the sake of simplicity).We regard q as a section to the bundle HOM(VHOM(WW)). Fix aRiemannian connection on V and covariantly dierentiate q to computerq 2 C 1 (T M HOM(VHOM(WW))). Since the connection is Riemannian,rq =0this is not true in general for non-Riemannian connections.If V is trivial with at connection, the bundles (q) have curvature d d as computed in Lemma 2.1.5. If V is not at, the curvature of

238 3.9. The Atiyah-Singer TheoremV enters into this expression. The connection and ber metric on V denea natural metric on T V . We use the splitting dened by the connectionto decompose T V into horizontal and vertical components. These componentsare orthogonal with respect to the natural metric on V . Over V , qbecomes a section to the bundle HOM(VV ). We let r V denote covariantdierentiation over V , then r V q has only vertical components in T V andhas no horizontal components.The calculation performed in Lemma 2.1.5 shows that in this more generalsetting that the curvatures of r are given by: = + (r V + ^r V + + W ):If we choose a frame for V and W which is covariant constant at a pointx 0 , then r V + = d + has only vertical components while W has onlyhorizontal components. If V is the curvature of V , then W =( V ).Instead of computing S on the form level, wework with the correspondinginvariant polynomial.Lemma 3.9.10. Let A be an n n complex skew-adjoint matrix. LetB =(A) acting on (C n )=C 2n . Dene:S (A) = X (;1) n i2 1!ZS 2n Trf( + d + d + + B) g:If x j = i j =2 are the normalized eigenvectors of A, thenS (A) = Y je x j; 1x j:Proof: If V = V 1 V 2 and if A = A 1 A 2 , then the Dolbeault complexdecomposes as a tensor product by Lemma 3.9.3. The calculations ofLemma 3.9.3 using the decomposition of the bundles shows S (A) is amultiplicative characteristic class. To compute the generating function, itsuces to consider the case n =1. !If n =1,A = so that B = 0 0 ,ifwe decompose W =0 00 01 =1 V . If x + iy give the usual coordinates on V = C, then:q(x y) =x 0 11 0!+ y0 ;ii 0by Lemma 3.5.2 which gives the symbol of the Dolbeault complex. Therefore:q(x y u)=x 0 11 0!+ y! !0 ;i+ u 1 0 = xei 0 0 ;10 + ye 1 + ue 2 :!

We compute:e 0 e 1 e 2 = i= X j>0The Atiyah-Singer Theorem 239!1 00 1and + d + d + = i 2 + dvol :Since n =1,(;1) n X = ;1 and:S (A)= ; i j i 1ZTr((2 2+ dvol + + B) j )j!j>0; i i j1 ZTr( + B) j;1 dvol :2 2 (j ; 1)!We calculate that: + B = 1 2!1+u x ; iyx + iy 1 ; u0 00 ( + B) j;1 =2 1;j j;1 (1 ; u) j;1 0 0 1!!= 1 2!0 0 (1 ; u)where \" indicates a term we are not interested in. We use this identityand re-index the sum X to express:1 i j Z 1S (A) =4 2 j! 2;j (1 ; u) j dvol :j0S 2We introduce the integrating factor of e ;r2Zso that:R 3 u 2k e ;r2 dx dy du = Z=ZRZ 10=(2k +1)=2u 2k e ;u2 dur 2k+2 e ;r2 dr Z 10to compute:ZS 2 u 2k dvolr 2k e ;r2 dr u 2k dvol=4=(2k +1):S 2The terms of odd order integrate to zero so:ZS X (1 ; u) j jdvol = 4 2=2Z 102k12k +1 =4 Z 1(1 + t) j +(1; t) j dt=2 (1 + t)j+1 ; (1 ; t) j+1j +1=4 2 jj +1 :100ZS 2 u 2k dvolX jt 2k dt2k

240 3.9. The Atiyah-Singer TheoremWe substitute this to conclude:S (A) = X j0If we introduce x = i=2 thenS (x) = X j0 i j12 (j + 1)! :x j(j + 1)! = ex ; 1xwhich gives the generating function for S . This completes the proofofthelemma.We can now compute Todd. The generating function of Todd(T c ) isx=(1 ; e ;x ) so that Todd = S ;1 Todd(T c ) will have generating function:x1 ; e ;x xe x ; 1 =x1 ; e ;x ;x1 ; e xwhich is, of course, the generating function for the real Todd class. Thiscompletes the proof. We have gone into some detail to illustrate that it isnot particularly dicult to evaluate the integrals which arise in applyingthe index theorem. If we had dealt with the signature complex instead ofthe Dolbeault complex, the integrals to be evaluated would have been overS 4 instead of S 2 but the computation would have been similar.

CHAPTER 4GENERALIZEDINDEX THEOREMSAND SPECIAL TOPICSIntroductionThis chapter is less detailed than the previous three as several lengthycalculations are omitted in the interests of brevity. In sections 4.1 through4.6, we sketch the interrelations between the Atiyah-Patodi-Singer twistedindex theorem, the Atiyah-Patodi-Singer index theorem for manifolds withboundary, and the Lefschetz xed point formulas.In section 4.1, we discuss the absolute and relative boundary conditionsfor the de Rham complex if the boundary is non-empty. We discussPoincare duality and the Hodge decomposition theorem in this context.The spin, signature, and Dolbeault complexes do not admit such localboundary conditions and the index theorem in this context looks quite different.In section 4.2, we prove the Gauss-Bonnet theorem for manifoldswith boundary and identify the invariants arising from the heat equationwith these integrands.In section 4.3, we introduce the eta invariant as a measure of spectralasymmetry and establish the regularity ats =0. We discuss without proofthe Atiyah-Patodi-Singer index theorem for manifolds with boundary. Theeta invariant enters as a new non-local ingredient whichwas missing in theGauss-Bonnet theorem. In section 4.4, we review secondary characteristicclasses and sketch the proof of the Atiyah-Patodi-Singer twisted index theoremwith coecients in a locally at bundle. We discuss in some detailexplicit examples on a 3-dimensional lens space.In section 4.5, we turn to the Lefschetz xed point formulas. We treatthe case of isolated xed points in complete detail in regard to the fourclassical elliptic complexes. We also return to the 3-dimensional examplesdiscussed in section 4.4 to relate the Lefschetz xed point formulas to thetwisted index theorem using results of Donnelly. We discuss in some detailthe Lefschetz xed point formulas for the de Rham complex if the xedpoint set is higher dimensional. There are similar results for both thespin and signature complexes which we have omitted for reasons of space.In section 4.6 we use these formulas for the eta invariant to compute theK-theory of spherical space forms.

242 Chapter 4In section 4.7, we turn to a completely new topic. In a lecture at M.I.T.,Singer posed the question:RSuppose P (G) is a scalar valued invariant of the metric so that P (M) =MP (G)dvol is independent of the metric. Then is there a universalconstant c so P (M) =c(M)?The answer to this question (and to related questions involving form valuedinvariants) is yes. This leads immediately to information regarding thehigher order terms in the expansion of the heat equation. In section 4.8,we use the functorial properties of the invariants to compute a n (x P ) foran arbitrary second order elliptic parital dierential operator with leadingsymbol given by the metric tensor for n = 0 2 4. We list (withoutproof) the corresponding formula if n = 6. This leads to Patodi's formulafor a n (x m p ) discussed in Theorem 4.8.18. In section 4.9 we discusssome results of Ikeda to give examples of spherical space forms which areisospectral but not dieomorphic. We use the eta invariant to show theseexamples are not even equivariant cobordant.The historical development of these ideas is quite complicated and inparticular the material on Lefschetz xed point formulas is due to a numberof authors. We have included a brief historical survey at the beginning ofsection 4.6 to discuss this material.

4.1. The de Rham Complex for Manifolds with Boundary.In section 1.9, we derived a formula for the index of a strongly ellipticboundary value problem. The de Rham complex admits suitable boundaryconditions leading to the relative and absolute cohomolgy groups. It turnsout that the other 3 classical elliptic complexes do not admit even a weakercondition of ellipticity.Let (d + ): C 1 ((M)) ! C 1 ((M)) be the de Rham complex. Inthe third chapter we assumed the dimension m of M to be even, but weplace no such restriction on the parity here. M is assumed to be compactwith smooth boundary dM. Near the boundary, we introduce coordinates(y r) where y =(y 1 ...y m;1 ) give local coordinates for dM and such thatM = f x : r(x) 0 g. We further normalize the coordinates by assumingthe curves x(r) =(y 0 r) are unit speed geodesics perpendicular to dM forr 2 [0).Near dM, we decompose any dierential form 2 (M) as = 1 + 2 ^ drwhere i 2 (dM)are tangential dierential forms. We use this decomposition to dene:() = 1 ; 2 ^ dr: is self-adjoint and 2 =1. We dene the absolute and relative boundaryconditionsB a () = 2 and B r () = 1 :We let B denote either B a or B r , then B:(M)j dM ! (dM). We notethat B a can be identied with orthogonal projection on the ;1 eigenspaceof while B r can be identied with orthogonal projection on the +1eigenspace of . There is a natural inclusion map i: dM ! M and B r () =i () is just the pull-back of . The boundary condition B r does not dependon the Riemannian metric chosen, while the boundary condition B adoes depend on the metric.Lemma 4.1.1. Let B = B a or B r , then (d + B) is self-adjoint andelliptic with respect to the cone C ; R + ; R ; .Proof: We choose a local orthonormal frame fe 0 ...e m;1 g for T Mnear dM so that e 0 = dr. Letp j = ie jact on (M) by Cliord multiplication. The p j are self-adjoint and satisfythe commutation relation:p j p k + p k p j =2 jk :

244 4.1. The de Rham ComplexThe symbol of (d + ) is given by:As in Lemma 1.9.5, we dene: m;1X(y )=ip 0j=1p(x ) =zp 0 + j p j ;We dene the new matrices:so that:m;1Xj=1 j p j : () 6= (0 0) 2 T (dM)fC;R + ;R ; g:q 0 = p 0 and q j = ip 0 p j for 1 j m ; 1(y )=;iq 0 +m;1Xj=1 j q j :The fq j g are self-adjoint and satisfy the commutation relations q j q k +q k q j =2 jk . Consequently:(y ) 2 =(jj 2 ; 2 )I:We have (jj 2 ; 2 ) 2 C ; R ; ; 0 so we can choose 2 =(jj 2 ; 2 ) withRe() > 0. Then is diagonalizable with eigenvalues and V () isthespan of the eigenvectors of corresponding to the eigenvalue . (We setV =(M) =(T M) to agree with the notation of section 9.1).We dened ( 1 + 2 ^ dr) = 1 ; 2 ^ dr. Since Cliord multiplicationby e j for 1 j m ; 1 preserves (dM), the corresponding p j commutewith . Since Cliord multiplication by dr interchanges the factors of(dM) (dM) ^ dr, anti-commutes with p 0 . This implies anticommuteswith all the q j and consequently anti-commutes with . Thusthe only common eigenvectors must belong to the eigenvalue 0. Since 0 isnot an eigenvalue,V () \ V () =f0g:Since B a and B r are just orthogonal projection on the 1 eigenspaces of, N(B a ) and N(B r ) are just the 1 eigenspaces of . ThusB: V () ! (dM)is injective. Since dim(V ()) = dim (dM) = 2 m;1 , this must be anisomorphism which proves the ellipticity. Since p 0 anti-commutes with ,p 0 : V () ! V ()

For Manifolds with Boundary 245so (d + ) is self-adjoint with respect to either B a or B r by Lemma 1.9.5.By Lemma 1.9.1, there is a spectral resolution for the operator (d + ) Bin the form f g 1 =1 where (d + ) = and B = 0. The 2 C 1 ((M)) and j j!1. We set=(d+) 2 = d +d and dene:N((d + ) B )=f 2 C 1 ((M)) : B =(d + ) =0gN((d + ) B ) j = f 2 C 1 ( j (M)) : B =(d + ) =0gN( B )=f 2 C 1 ((M)) : B = B(d + ) = =0gN( B ) j = f 2 C 1 ( j (M)) : B = B(d + ) = =0g:Lemma 4.1.2. Let B denote either the relative or the absolute boundaryconditions. Then(a) N((d + ) B )=N( B ).(b) N((d + ) B ) j = N( B ) j .(c) N((d + ) B )= L j N((d + ) B) j :Proof: Let B = B(d + ) = = 0. Since (d + ) B is self-adjointwith respect to the boundary condition B, we can compute that =(d + ) (d + ) =0so (d + ) =0. This shows N( B ) N((d + ) B )and N( B ) j N((d + ) B ) j . The reverse inclusions are immediate whichproves (a) and (b). It is also clear that N((d+) B ) j N((d+) B ) for eachj. Conversely, let 2 N((d + ) B ). We decompose = 0 + + m intohomogeneous pieces. Then = P j j =0implies j =0for each j.Therefore we must check B j = B(d + ) j = 0 since then j 2 N( B ) j =N((d + ) B ) j which will complete the proof.Since B preserves the grading, B j =0. Suppose B = B r is the relativeboundary conditions so B( 1 + 2 ^dr) = 1 j dM . Then Bd = dB so B(d+) = dB() +B() = B() = 0. Since B preserves the homogeneity,this implies B j = 0 for each j. We observed Bd = dB =0soBd j =0for each j as well. This completes the proof in this case. If B = B a is theabsolute boundary condition, use a similar argument based on the identityB = B.We illustrate this for m = 1 by considering M = [0 1]. We decompose = f 0 + f 1 dx to decompose C 1 ((M)) = C 1 (M) C 1 (M) dx. It isimmediate that:(d + )(f 0 f 1 )=(;f 0 1 f0 0)so (d+) = 0 implies is constant. B a corresponds to Dirichlet boundaryconditions on f 1 while B r corresponds to Dirichlet boundary conditions onf 0 . Therefore:H 0 a([0 1] C)=CH 0 r ([0 1] C)=0H 1 a([0 1] C)=0H 1 r ([0 1] C)=C:

246 4.1. The de Rham ComplexA priori, the dimensions of the vector spaces Ha(M j C) and Hr j (M C)depend on the metric. It is possible, however, to get a more invariantdenition which shows in fact they are independent of the metric. Lemma1.9.1 shows these spaces are nite dimensional.Let d: C 1 ( j ) ! C 1 ( j+1 ) be the de Rham complex. The relativeboundary conditions are independent of the metric and are d-invariant.Let Cr 1 ( p ) = f 2 C 1 p : B r = 0 g. There is a chain complexd: Cr 1 ( p ) ! Cr 1 ( p+1 ) ! . We dene Hr p (M C) =(ker d= image d) pon Cr1 ( p ) to be the cohomology of this chain complex. The de Rham theoremfor manifolds without boundary generalizes to identify these groupswith the relative simplicial cohomology H p (MdM C). If 2 N((d+) Br ) pthen d = B r =0so j 2 H p (MdM C). The Hodge decomposition theoremdiscussed in section 1.5 for manifolds without boundary generalizesto identify Hr p (M C)=N( Br ) p . If we use absolute boundary conditionsand the operator ,thenwe can dene Ha(M p C) = N( Ba ) p = H p (M C).We summarize these results as follows:Lemma 4.1.3. (Hodge decomposition theorem). There are naturalisomorphisms between the harmonic cohomology with absolute andrelative boundary conditions and the simplicial cohomology groups of M:andH p a (M C) = N((d + ) Ba ) j ' H p (M C)H p r (M C) = N((d + ) Br ) j ' H p (MdM C):If M is oriented, we let be the Hodge operator : p ! m;p . Since interchanges the decomposition (T dM)(T dM)^dr,itanti-commuteswith and therefore B a () = 0 if and only if B r () = 0. Since d = 0if and only if = 0 and similarly = 0 if and only if d = 0, weconclude:Lemma 4.1.4. Let M be oriented and let be the Hodge operator. Then induces a map, called Poincare duality,: H p (M C) ' Ha(M p C) ! ' Hrm;p (MC) ' H m;p (MdM C):We dene the Euler-Poincare characteristics by:(M) = X (;1) p dim H p (M C)(dM) = X (;1) p dim H p (dM C)(MdM) = X (;1) p dim H p (MdM C):The long exact sequence in cohomology:H p (dM C) H p (M C) H p (MdM C) H p;1 (dM C)

shows that:For Manifolds with Boundary 247(M) =(dM)+(MdM):If m is even, then (dM) = 0 as dM is an odd dimensional manifoldwithout boundary so (M) =(MdM). If m is odd and if M is orientable,then (M) = ;(MdM) by Poincare duality. (M) is the index of thede Rham complex with absolute boundary conditions (MdM) is theindex of the de Rham complex with relative boundary conditions. ByLemma 1.9.3, there is a local formula for the Euler-Poincare characteristic.Since (M) =;(MdM) if m is odd and M is orientable, by passing tothe double cover if necessary we see (M) =;(MdM) in general if m isodd. This proves:Lemma 4.1.5.(a) If m is even, (M) =(MdM) and (dM) =0.(b) If m is odd, (M) =;(MdM) = 1 2 (dM).In contrast to the situation of manifolds without boundary, if we passto the category of manifolds with boundary, there exist non-zero indexproblems in all dimensions m.In the next subsection, we will discuss the Gauss-Bonnet formula formanifolds with boundary. We conclude this subsection with a brief discussionof the more general ellipticity conditions considered by Atiyah andBott. Let Q: C 1 (V 1 ) ! C 1 (V 2 ) be an elliptic dierential operator of orderd > 0 on the interior|i.e., if q(x ) is the leading symbol of Q, thenq(x ): V 1 ! V 2 is an isomorphism for 6= 0. Let W 1 = V 1 1 d dMbethe bundle of Cauchy data. We assume dim W 1 is even and let W 0 1 be abundle over dM of dimension 1 2 (dim W 1). Let B: C 1 (W 1 ) ! C 1 (W 0 1) bea tangential pseudo-dierential operator. We consider the ODEq(y 0D r )f =0lim f(r) =0r!1and let V + ()() be the bundle of Cauchy data of solutions to this equation.We say that (Q B) iselliptic with respect to the cone f0g if for all 6= 0,the map: g (B)(y ): V + ()() ! W 0 1is an isomorphism (i.e., we can nd a unique solution to the ODE such that g (B)(y )f = f 0 is pre-assigned in W 0 1). V + () is a sub-bundle of W 1and is the span of the generalized eigenvectors corresponding to eigenvalueswith positive real parts for a suitable endomorphism () just as in the rstorder case. g is the graded leading symbol as discussed in section 1.9.This is a much weaker condition than the one we have been consideringsince the only complex value involved is =0. We study the pair(Q B): C 1 (V 1 ) ! C 1 (V 2 ) C 1 (W 0 1):

248 4.1. The de Rham ComplexUnder the assumption of elliptic with respect to the cone f0g, this operatoris Fredholm in a suitable sense with closed range, nite dimensional nullspaceand cokernel. We let index(Q B) bethe index of this problem. TheAtiyah-Bott theorem gives a formula for the index of this problem.There exist elliptic complexes which do not admit boundary conditionssatisfying even this weaker notion of ellipticity. Let q(x ) bea rst ordersymbol and expandAs in Lemma 1.9.5 we deneq(y 0z)=q 0 z + = iq ;10m;1Xj=1m;1Xj=1q j j q j j :the ellipticity condition on the interior shows has no purely imaginaryeigenvalues for 6= 0.We let V ()() be the sub-bundle of V correspondingto the span of the generalized eigenvectors of corresponding to eigenvalueswith positive/negative real part. Then (Q B) is elliptic if and only if g (B)(): V + ()() ! W 0 1is an isomorphism for all 6= 0.Let S(T (dM)) = f 2 T (dM) : jj 2 = 1 g be the unit sphere bundleover dM. V () dene sub-bundles of V over S(T (dM)). The existenceof an elliptic boundary condition implies these sub-bundles are trivial overthe ber spheres. We study the case in which q q = jj 2 I. In this case,q ;10 = q0. If we set p j = iq ;10 q j, then these are self-adjoint and satisfyp j p k + p k p j = 2 jk . If m is even, then the ber spheres have dimensionm ; 2 which will be even. The bundles V () were discussed in Lemma2.1.5 and in particular are non-trivial ifTr(p 1 ...p m;1 ) 6= 0:For the spin, signature, and Dolbeault complexes, the symbol is given byCliord multiplication and p 1 ...p m;1 is multiplication by the orientationform (modulo some normalizing factor of i). Since the bundles involvedwere dened by the action of the orientation form being 1, this proves:Lemma 4.1.6. Let Q: C 1 (V 1 ) ! C 1 (V 2 ) denote either the signature,the spin, or the Dolbeault complex. Then there does not exist a boundarycondition B so that (Q B) is elliptic with respect to the cone f0g.The diculty comes, of course, in not permitting the target bundle W 0to depend upon the variable . In the rst order case, there is a natural

For Manifolds with Boundary 249pseudo-dierential operator B() with leading symbol given by projectionon V + ()(). This operator corresponds to global (as opposed to local)boundary conditions and leads to a well posed boundry value problem forthe other three classical elliptic complexes. Because the boundary valueproblem is non-local, there is an additional non-local term which arises inthe index theorem for these complexes. This is the eta invariant we willdiscuss later.

4.2. The Gauss-Bonnet TheoremFor Manifolds with Boundary.Let B denote either the absolute or relative boundary conditions forthe operator (d + ) discussed previously. We let (M) B be either (M)or (MdM) be the index of the de Rham complex with these boundaryconditions. Let evenB and oddB be the Laplacian on even/odd formswith the boundary conditions B = B(d + ) = 0. Let a n (x d + ) =a n (x even ) ; a n (x odd ) be the invariants of the heat equation denedin the interior of M which were discussed in Lemma 1.7.4. On the boundarydM, let a n (y d + B) =a n (y evenB ) ; a n(y oddB ) be the invariantsof the heat equation dened in Lemma 1.9.2. Then Lemma 1.9.3 implies:(M) B =Trfexp(;t evenB )g;Trexpf(;t oddB )g1X Z t (n;m)=2 a n (x d + )dvol(x)n=0+1Xn=0Mt (n;m+1)=2 ZdMa n (y d + B)dvol(y):The interior invariants a n (x d+) do not depend on the boundary conditionso we can apply Lemma 2.4.8 to conclude:a n (x d + )=0a m (x d + )=E mif n

The Gauss-Bonnet Theorem 251Lemma 4.2.1. If B denotes either absolute or relative boundary conditions,then a n (y d + B) denes an element of P b mn.Proof: By Lemma 1.9.2, a n (y d + B) is given by a local formula inthe jets of the metric which is invariant. Either by examining the analyticproof of Lemma 1.9.2 in a way similar to that used to prove Lemma 1.7.5and 2.4.1 or by using dimensional analysis as was done in the proof ofLemma 2.4.4, we can show that a n must be homogeneous of order n andpolynomial in the jets of the metric.Our normalizations impose some additional relations on the g ij= variables.By hypothesis, the curves (y 0 r) are unit speed geodesics perpendicularto dM at r =0. This is equivalent to assuming:r N N =0 and g jm (y 0) = jmwhere N = @=@r is the inward unit normal. The computation of theChristoel symbols of section 2.3 shows this is equivalent to assuming:; mmj = 1 2 (g mj=m + g mj=m ; g mm=j )=0:If we take j = m, this implies g mm=m = 0. Since g mm (y 0) 1, we concludeg mm 1 so g mm= 0. Thus ; mmj = g mj=m = 0. As g mj (y 0) = mj we conclude g mj mj and therefore g jm= = 0, 1 j m.We can further normalize the coordinate system Y on dM by assumingg jk=l (YG)(y 0 ) = 0 for 1 jkl m ; 1. We eliminate all these variablesfrom the algebra dening P the remaining variables are algebraicallyindependent.The only 1-jets of the metric which are left are the fg jk=m g variables for1 jk m ; 1. The rst step in Chapter 2 was to choose a coordinatesystem in which all the 1-jets of the metric vanish this proved to be thecritical obstruction to studying non-Kaehler holomorphic manifolds. Itturns out that the fg jk=m g variables cannot be normalized to zero. Theyare tensorial and giveessentially the components of the second fundamentalform or shape operator.Let fe 1 e 2 g be vector elds on M which are tangent to dM along dM.We dene the shape operator:S(e 1 e 2 )=(r e1 e 2 N)along dM. It is clear this expression is tensorial in e 1 . We compute:(r e1 e 2 N) ; (r e2 e 1 N)=([e 1 e 2 ]N):Since e 1 and e 2 are tangent to dM along dM, [e 1 e 2 ] is tangent to dMalong dM and thus ([e 1 e 2 ]N) = 0 along dM. This implies S(e 1 e 2 ) =

252 4.2. The Gauss-Bonnet TheoremS(e 2 e 1 ) is tensorial in e 2 . The shape operator denes a bilinear map fromT (dM) T (dM) ! R. We compute(r @=@yj @=@y k N)=; jkm = 1 2 (g jm=k + g km=j ; g jk=m )=; 1 2 g jk=m:We can construct a number of invariants as follows: let fe j g be a localorthonormal frame for T (M) such that e m = N = @=@r. Dene:andre j =X1km! jk e k for ! jk 2 T M and ! jk + ! kj =0 jk = d! jk ;X1m! j ^ ! k :The ! jm variables are tensorial as ! jm = P m;1k=1 S(e je k ) e k . We dene:Q km = c kmX"(i1 ...i m;1 ) i1 i 2^ ^ i2k;1 i 2k^ ! i2k+1 m ^ ^! im;1 m 2 m;1forc km =(;1) k p k!2 k+p 1 3 (2p ; 2k ; 1)h mwhere p =2i:The sum dening Q k is taken over all permutations of m ; 1 indices anddenes an m ; 1 form over M. If m is even, we dene:E m =(;1)pf2 m p p!gX"(i1 ...i m ) i1 i 2^ ^ im;1 i mas the Euler form discussed in Chapter 2.Q km and E m are the SO-invariant forms on M. E m is dened on all ofM while Q km is only dened near the boundary. Chern noted that if mis even,E m = ;d XkQ km:This can also be interpreted in terms of the transgression of Chapter 2. Letr 1 and r 2 be two Riemannian connections on TM. We dened an m ; 1form TE m (r 1 r 2 ) so thatdT E m (r 1 r 2 )=E m (r 1 ) ; E m (r 2 ):Near dM, we split T (M) =T (dM)1 as the orthogonal complementoftheunit normal. We project the Levi-Civita connection on this decomposition,

For Manifolds with Boundary 253and let r 2 be the projected connection. r 2 is just the sum of the Levi-Civita connection of T (dM) and the trivial connection on 1 and is at inthe normal direction. As r 2 is a direct sum connection, E P m (r 2 ) = 0.r 1 ;r 2 is essentially just the shape operator. TE m = ; Q km anddT E m = E m (r 1 )=E m . It is an easy exercise to work out the Q km usingthe methods of section 2 and thereby compute the normalizing constantsgiven by Chern.The Chern-Gauss-Bonnet theorem for manifolds with boundary in theoriented category becomes:(M) =ZME m +ZdMXkQ km :In the unoriented category, we regard E m dvol(x) as a measure on MRandQkm dvol(y) as a measure on dM. If m is odd, of course, (M) =1R(dM) = 12 2 dM E m;1 so there is no diculty with the Chern-Gauss-Bonnet theorem in this case.We derive the Chern-Gauss-Bonnet theorem for manifolds with boundaryfrom the theorem for manifolds without boundary. Suppose m is evenand that the metric is product near the boundary. Let M be the doubleof M then (M) = R M E m = 2 R M E m = 2(M) ; (dM) = 2(M)so (M) = R M E m. If the metric is not product near the boundary, letM 0 = dM [;1 0] [ M be the manifold M with a collar sewed on. LetG 0 be the restriction of the metric on M to the boundary and let G 0 0 bethe product metric on the collar dM [;1 0]. Using a partition of unity,extend the original metric on M to a new metric which agrees with G 0 0near dM f;1g which is the boundary of M 0 . Then:(M) =(M 0 )==ZZM 1E m =ME m +ZZMdME m ;XkZdM[;10]Q km XdkQ kmby Stoke's theorem since the Q k vanish identically near dM f;1g thereis no contribution from this component of the boundary of the collar (wechange the sign since the orientation of dM as the boundary of M and asthe boundary of dM [;1 0] are opposite).We now study the invariants of the heat equation. We impose no restrictionson the dimension m. If M = S 1 M 1 and if is the usual periodicparameter on S 1 , then there is a natural involution on (T M) given byinterchanging with d ^ for 2 (M 1 ). This involution preserves theboundary conditions and the associated Laplacians, but changes the parity

254 4.2. The Gauss-Bonnet Theoremof the factors involved. This shows a n (y d + B) =0for such a productmetric. We dene:r: P b mn !P b m;1nto be the dual of the map M 1 ! S 1 M 1 . Then algebraically: 0 if deg1 (gr(g ij= )=ij= ) 6= 0 if deg 1 (g ij= )=0where \" is simply a renumbering to shift all the indices down one. (Atthisstage, it is inconvenient to have used the last index to indicate the normaldirection so that the rst index must be used to denote the at indexdenoting the normal direction by the last index is suciently cannonicalthat we have not attempted to adopt a dierent convention despite theconict with the notation of Chapter 2). This proves:Lemma 4.2.2. Let B denote either the relative or absolute boundaryconditions. Then a n (y d + B) 2 P b mn. Furthermore, r(a n ) = 0 wherer: P b mn !P b m;1n is the restriction map.We can now begin to identify a n (y d + B) using the same techniquesof invariance theory applied in the second chapter.Lemma 4.2.3.. Let P 2P b mn. Suppose that r(P )=0. Then:(a) P =0if n 0 is even for 1 j m ; 1. This yields the inequalities:2m ; 2 Xjm;1From this it follows that:deg j (A) and r deg m (A):2m ; 2+r X jdeg j (A) =2r +2k + X j j + r =2r +2k + n:

Since j j2 we concludeFor Manifolds with Boundary 2552k X j j = n ; r:We combine these inequalities to conclude 2m;2+r 2n+r so n m;1.This shows P = 0 if n < m ; 1 which proves (a). If n = m ; 1, all theseinequalities must be equalities. j j =2, deg m (A) =r, and deg j (A) =2for 1 j m ; 1. Since the index m only appears in A in the g ij=mvariables where 1 i j m ; 1, this completes the proof of (b).Lemma 2.5.1 only used invariance under the group SO(2). Since P isinvariant under the action of SO(m ; 1), we apply the argument used inthe proof of Theorem 2.4.7 to choose a monomial A of P of the form:A = g 11=22 ...g 2k;12k;1=kk g k+1k+1=m ...g m;1m;1=mwhere if k =0the terms of the rst kind do not appear and if k = m ; 1,the terms of the second kind do not appear. Since m ; 1 = 2k + r, it isclear that r m ; 1mod2. We denote such a monomial by A k . SinceP 6= 0implies c(A k P) 6= 0 for some k, we conclude the dimension of them+1space of such P is at most the cardinality of fA k g =2 . This proves:Lemma 4.2.4. Let r: P b mm;1 ! P b m;1m;1 be the restriction map de-m+1ned earlier. Then dim N(r) 2 .We can now show:Lemma 4.2.5. Let P 2P b mm;1 with r(P )=0. Let i: dM ! M be theinclusion map and i : m;1 (T M) ! m;1 (T (dM)) be the dual map.Let m;1 be the Hodge operator on the boundary so m;1 : m;1 (T (dM))! 0 (T (dM)). Let Q km = m;1 (i Q km ). Then the f Q km g form a basisfor N(r) so we can express P as a linear combination of the Q km .Proof: It is clear r( Qkm ) = 0 and that these elements are linearly independent.We have [(m +1)=2] such elements so by Lemma 4.2.4 they mustbe a basis for the kernel of r. (If we reverse the orientation we change boththe sign of and Q so Q is a scalar invariant.)P We note that if m is odd, then Q m;1m = c "(i 1 ...i m;1 ) i1 i 2^^ im;1 i m;1is not the Euler form on the boundary since we are using theLevi-Civita connection on M and not the Levi-Civita connection on dM.However, E m;1 can be expressed in terms of the Qkm in this situation.Before proceeding to discuss the heat equation, we need a uniquenesstheorem:

256 4.2. The Gauss-Bonnet TheoremP Lemma 4.2.6. Let P = k a kQ km be a linear combination of thefQ km g. Suppose R that P 6= 0. Then there exists a manifold M and ametric G sodMP (G)(y) 6= 0.Proof: By assumption not all the a k =0. Choose k maximal so a k 6=0.Let n = m ; 2k and let M = S 2k D m;2k with the standard metric. (Ifm ; 2k = 1, we let M = D m and choose a metric which is product nearS m;1 .) We let indices 1 i 2k index a frame for T (S 2k ) and indices2k +1 u m index a frame for T (D m;2k ). Since the metric is product, iu = ! iu =0in this situation. Therefore Q jm =0if j k by assumption, we concludedM P (G) = a ksuces to show this integral is non-zero. This is immediate if n = 1 asQ m;1m = E m;1 and m ; 1 is even.Let n 2. We have Q km (G) =E 2k (G 1 )Q 0m;2k (G 2 ), since the metricis product. SinceZE 2k = (S 2k )=26= 0S 2kR we must only showdD n Q 0n is non-zero for all n>1. Let be a systemof local coordinates on the unit sphere S n;1 and let r be the usual radialparameter. If ds 2 e is the Euclidean metric and ds 2 is the spherical metric,thends 2 e = r 2 ds 2 + dr dr:From the description of the shape operator given previously we concludethat S = ;ds 2 . Let fe 1 ...e n;1 g be a local oriented orthonormal framefor T (S n ), then ! in = ;e i and therefore Q 0n = c dvol n;1 where c is anon-zero constant. This completes the proof.We combine these results in the following Theorem:Theorem 4.2.7. (Gauss-Bonnet formula for manifolds withboundary).(a) Let the dimension m be even and let B denote either the relative orthe absolute boundary conditions. LetQ km = c kmX"(i1 ...i m;1 ) i1 i 2^ ^ i2k;1 i 2k^ ! i2k+1 m ^ ^! im;1 mfor c km = (;1) k =( m=2 k! 2 k+m=2 1 3 (m ; 2k ; 1)). Let Qkm =(Q km jdM) 2P b mm;1 . Then:(i) a n (x d + ) =0for n

For Manifolds with Boundary 257(b) Let the dimension m be odd and let B r be the relative and B a theabsolute boundary conditions. Then:(i) a n (x d + ) = 0 for all n and a n (y d + B r ) = a n (y d + B a ) = 0for n

4.3. The Regularity at s =0of the Eta Invariant.In this section, we consider the eta invariant dened in section 1.10.This section will be devoted to proving eta is regular at s =0. In the nextsection we will use this result to discuss the twisted index theorem usingcoecients in a locally at bundle. This invariant appears as a boundarycorrection term in the index theorem for manifolds with boundary.We shall assume P : C 1 (V ) ! C 1 (V ) is a self-adjoint elliptic pseudodierentialoperator of order d>0. We dene(s P )= X i >0( i ) ;s ; X i 0.(a) P (P 2 ) v is a self-adjoint elliptic pseudo-dierential operator for any vand if 2v +1> 0, R(P )=R(P (P 2 ) v ).(b) R(P Q) =R(P )+R(Q).(c) There is a local formula R a(x P ) in the jets of the symbol of P up toorder d so that R(P )=Ma(x P ) j dvol(x)j.(d) If P t is a smooth 1-parameter family of such operators, then R(P t ) isindependent of the parameter t.(e) If P is positive denite, then R(P )=0.(f ) R(;P )=;R(P ).Proof: We have the formal identity: (s P (P 2 ) v ) = ((2v +1)s P ).Since we normalized the residue by multiplying by the order of the operator,(a) holds. The fact that P (P 2 ) v is again a pseudo-dierential operatorfollows from the work of Seeley. (b) is an immediate consequence of thedenition. (c) and (d) were proved in Lemma 1.10.2 for dierential operators.The extension to pseudo-dierential operators again follows Seeley'swork. If P is positive denite, then the zeta function and the eta functioncoincide. Since zeta is regular at the origin, (e) follows by Lemma 1.10.1.(f) is immediate from the denition.We note this lemma continues to be true if we assume the weaker conditiondet(p(x ) ; it) 6= 0for (t) 6= (0 0) 2 T M R.We use Lemma 4.3.1 to interpret R(P ) as a map in K-theory. LetS(T M) be the unit sphere bundle in T M. Let V be a smooth vectorbundle over M equipped with a ber inner product. Let p: S(T M) !

The Regularity at s =0 259END(V ) be self-adjoint and elliptic we assume p(x ) = p (x ) anddet p(x ) 6= 0 for (x ) 2 S(T M). We x the order of homogeneityd>0 and let p d (x ) be the extension of p to T M which is homogeneousof degree d. (In general, we must smooth out the extension near = 0to obtain a C 1 extension, but we suppress such details in the interests ofnotational clarity.)Lemma 4.3.2. Let p: S(T M) ! END(V ) be self-adjoint and elliptic.Let d > 0 and let P : C 1 (V ) ! C 1 (V ) have leading symbol p d . ThenR(P ) depends only on p and not on the order d nor the particular operatorP .Proof: Let P 0 have order d with the same leading symbol p d . Weformtheelliptic family P t = tP 0 +(1; t)P . By Lemma 4.3.1, R(P t ) is independentof t so R(P 0 ) = R(P ). Given two dierent orders, let (1 + 2v)d = d 0 .Let Q = P (P 2 ) v then R(Q) = R(P ) by Lemma 4.3.1(a). The leadingsymbol of Q is p(p 2 ) v . p 2 is positive denite and elliptic. We constructthe homotopy of symbols q t (x ) = p(x )(tp 2 (x ) +(1; t)jj 2 ). Thisshows the symbol of Q restricted to S(T M) is homotopic to the symbolof P restricted to S(T M) where the homotopy remains within the classof self-adjoint elliptic symbols. Lemma 4.3.1 completes the proof.We let r(p) =R(P ) for such an operator P . Lemma 4.3.1(d) shows r(p)is a homotopy invariant ofp. Let (p) be the subspaces of V spanned bythe eigenvectors of p(x ) corresponding to positive/negative eigenvalues.These bundles have constant rank and dene smooth vector bundles overS(T M) so + ; = V . In section 3.9, an essential step in provingthe Atayah-Singer index theorem was to interpret the index as a map inK-theory. To show R(P ) = 0, we must rst interpret it as a map in K-theory. The natural space in which to work is K(S(T M) Q) and notK((T M) Q).Lemma 4.3.3. Let G be an abelian group and let R(P ) 2 G be denedfor any self-adjoint elliptic pseudodierential operator. Assume R satisesproperties (a), (b), (d), (e) and (f ) [but not necessarily (c)] of Lemma 4.3.1.Then there exists a Z-linear map r: K(S(T M)) ! G so that:(a) R(P )=r( + (p)),(b) If : S(T M) ! M is the natural projection, then r( V ) = 0 for allV 2 K(M) so thatr: K(S(T M))=K(M) ! G:Remark: We shall apply this lemma in several contexts later, so state it insomewhat greater generality than is needed here. If G = R, we can extendr to a Q linear mapr: K(S(T M) Q) ! R:

260 4.3. The Regularity at s =0Proof: Let p: S(T M) ! END(V ) be self-adjoint and elliptic. We de-ne the bundles (p) and let (x ) denote orthogonal projection on (p)(x ). We let p 0 = + ; ; and dene p t = tp +(1; t)p 0 as ahomotopy joining p and p 0 . It is clear that the p t are self-adjoint. Fix(x ) andlet f i v i g be a spectral resolution of the matrix p(x ). Then:Consequentlyp(x )p 0 (x ) Xci v i Xci v i= X i c i v i= X sign( i )c i v i :p t Xci v i= X (t i +(1; t) sign( i ))c i v i :Since t i + (1 ; t)sign( i ) 6= 0 for t 2 [0 1], the family p t is elliptic.We therefore assume henceforth that p(x ) 2 = I on S(T M) and =1(1 p).2We let k be large and choose V 2 Vect k (S(T M)). Choose W 2Vect k (S(T M)) so V W ' 1 2k . We may choose the metric on 1 2k sothis direct sum is orthogonal and we dene to be orthogonal projectionon V and W . We let p = + ; ; so + = V and ; = W . We dener(V )=r(p). We must show this is well dened on Vect k . W is unique upto isomorphism but the isomorphism V W = 1 2k is non-canonical. Letu: V ! V and v: W ! W be isomorphisms where we regard V and W asorthogonal complements of 1 2k (perhaps with another trivialization). Wemust show r(p) =r(p). For t 2 [0 1] we letV (t) = span ; t v (1 ; t) u(v) v2V V W V W =1 2k 1 2k =1 4k :This is a smooth 1-parameter family of bundles connecting V 0to0 Vin 1 4k . This gives a smooth 1-parameter family of symbols p(t) connectingp(;1 2k )to(;1 2k )p. Thus r(p) =r(p;1 2k )=r(p(t)) = r(;1 2k p) =r(p) so this is in fact a well dened map r:Vect k (S(T M)) ! G. If V isthe trivial bundle, then W is the trivial bundle so p decomposes as thedirect sum of two self-adjoint matrices. The rst is positive denite andthe second negative denite so r(p) = 0 by Lemma 4.3.1(f). It is clearthat r(V 1 V 2 )=r(V 1 )+r(V 2 )by Lemma 4.3.1(b) and consequently sincer(1) = 0 we conclude r extends to an additive map from e K(S(T M)) ! G.We extend r to be zero on trivial bundles and thus r: K(S(T M)) ! G.Suppose V = V 0 for V 0 2 Vect k (M). We choose W 0 2 Vect k (M)so V 0 W 0 ' 1 2k . Then p = p + p ; for p + : S(T M) ! END(V 0 V 0 )

Of the Eta Invariant 261and p ; : S(T M) ! END(W 0 W 0 ). By Lemma 4.3.1, we conclude r(p) =r(p + )+r(p ; ). Since p + is positive denite, r(p + )=0. Since p ; is negativedenite, r(p ; )=0. Thus r(p) =0and r( V 0 )=0.This establishes the existence of the map r: K(S(T M)) ! G with thedesired properties. We must now check r(p) = r( + ) for general p. Thisfollows by denition if V is a trivial bundle. For more general V , werst choose W so V W = 1 over M. We let q = p 1 on V W ,then r(q) = r(p) +r(1) = r(p). However, q acts on a trivial bundle sor(q) = r( + (q)) = r( + (p) W ) = r( + (p)) + r( W ) = r( + (p))which completes the proof.Of course, the bundles (p) just measure the innitesimal spectralasymmetry of P so it is not surprising that the bundles they represent inK-theory are related to the eta invariant. This construction is completelyanalogous to the construction given in section 3.9 which interpreted theindex as a map in K-theory. We will return to this construction again indiscussing the twisted index with coecients in a locally at bundle.Such operators arise naturally from considering boundary value problems.Let M be a compact oriented Riemannian manifold of dimensionm = 2k ; 1 and let N = M [0 1) we let n 2 [0 1) denote the normalparameter. Let(d + ) + : C 1 ( + (T N)) ! C 1 ( ; (T N))be the operator of the signature complex. The leading symbol is given byCliord multiplication. We can use c(dn), where c is Cliord multiplication,to identify these two bundles over N. We express:(d + ) + = c(dn)(@=@n + A):The operator A is a tangential dierential operator on C 1 ( + (T N)) itis called the tangential operator of the signature complex. Since we havec(dn) c(dn) = ;1, the symbol of A is ;ic(dn)c() for 2 T (M). It isimmediate that the leading symbol is self-adjoint and elliptic since A isnatural this implies A is a self-adjoint elliptic partial dierential operatoron M.Let fe 1 ...e m g be a local oriented orthonormal frame for T M. Dene:! m = i k e 1 e m and ! m+1 = i k (;dn) e 1 e m = ;dn mas the local orientations of M and N. ! m is a central element of CLIF(M)! 2 m = ! 2 m+1 =1. If 2 (M) dene () = c(! m+1 ). This gives an

262 4.3. The Regularity at s =0isomorphism :(M) ! (N). We compute:( + ) ;1 f;c(dn)c()g + =( + ) ;1 f;c(dn)c() ; c(dn)c()c(;dn)c(! m )g=( + ) ;1 f;c(dn)c(! m )c(! m )c()+c(! m )c()g=( + ) ;1 f(c(! m+1 )+1)c(! m )c()g= c(! m )c():If we use + to regard A as an operator on C 1 ((M)) then this shows thatA is given by the diagram:C 1 ((M)) r ! C 1 (T (M (M)) c ! C 1 ((M)) ! ! C 1 ((M))where ! = ! m . This commutes with the operator (d + ) A = !(d + ).Both ! and (d+) reverse the paritysowe could decompose A = A even A odd acting on P smooth forms of even and odd degree. We letp = L (A) =ic(! ) = j jf j . The ff j g are self-adjoint matrices satisfying thecommutation relationWe calculate:f j f k + f k f j =2 jk :f 1 f m = i m c(! m e 1 e m )=i m;k c(! m !) =i m;k c(1)so that if we integrate over a ber sphere with the natural (not simplectic)orientation,ZS m;1 ch( + (p)) = i k;1 2 1;k Tr(f 1 f m )= i k;1 2 1;k i m;k 2 m = i k;1 2 1;k i k;1 2 2k;1=(;1) k;1 2 k :In particular, this is non-zero, so this cohomology class provides a cohomologyextension to the ber.As a H (M Q) module, we can decompose H (S(M) Q) =H (M Q)xH (M Q) where x = ch( + (p)). If we twist the operator A by takingcoecients in an auxilary bundle V , then we generate xH (M Q). Thesame argument as that given in the proof of Lemma 3.9.8 permits us tointerpret this in K-theory:

Of the Eta Invariant 263Lemma 4.3.4. Let M be a compact oriented Riemannian manifold ofdimension m =2k ; 1. Let A be the tangential operator of the signaturecomplex on M [0 1). If fe j g is an oriented local orthonormal basis forT (M), let ! = i k e 1 e m be the orientation form acting by Cliordmultiplication on the exterior algebra. A = !(d + ) on C 1 ((M)). If thesymbol of A is p,ZS m;1 ch( + (p)) = 2 k (;1) k;1 :The natural map K(M Q) ! K(S(T (M) Q) is injective and the groupK(S(T M) Q)=K(M Q) is generated by the bundles f + (A V )g as V runsover K(M).Remark: This operator can also be represented in terms of the Hodgeoperator. On C 1 ( 2p )forexample it is given by i k (;1) p+1 (d ; d). Weare using the entire tangential operator (and not just the part acting oneven or odd forms). This will produce certain factors of 2 diering fromthe formulas of Atiyah-Patodi-Singer. If M admits a SPIN c structure, onecan replace A by the tangential operator of the SPIN c complex in this casethe corresponding integrand just becomes (;1) k;1 .We can use this representation to prove:Lemma 4.3.5. Let dim M be odd and let P : C 1 (V ) ! C 1 (V ) be a selfadjointelliptic pseudo-dierential operator of order d > 0. Then R(P ) =0|i.e., (s P ) is regular at s =0.Proof: We rst suppose M is orientable. By Lemma 4.3.4, it suces toprove R(A V ) = 0 since r is dened in K-theory and would then vanishon the generators. However, by Lemma 4.3.1(c), the residue is given by alocal formula. The same analysis as that done for the heat equation showsthis formula must be homogeneous of order m in the jets of the metricand of the connection on V . Therefore, it must be expressible in terms ofPontrjagin forms of TM and Chern forms of V by Theorem 2.6.1. As mis odd, this local formula vanishes and R(A V )=0. If M is not orientable,we pass to the oriented double cover. If P : C 1 (V ) ! C 1 (V ) over M, welet P 0 : C 1 (V 0 ) ! C 1 (V 0 ) be the lift to the oriented double cover. ThenR(P ) = 1 2 R(P 0 ). But R(P 0 ) = 0 since the double cover is oriented andthus R(P )=0. This completes the proof.This result is due to Atayah, Patodi, and Singer. The trick used insection 3.9 to change the parity of the dimension by taking products witha problem over the circle does not go through without change as we shallsee. Before considering the even dimensional case, we must rst prove aproduct formula.

264 4.3. The Regularity at s =0Lemma 4.3.6. Let M 1 and M 2 be smooth manifolds. Let P : C 1 (V 1 ) !C 1 (V 2 ) be an elliptic complex over M 1 . Let Q: C 1 (V ) ! C 1 (V ) be aself-adjoint elliptic operator over M 2 . We assume P and Q are dierentialoperators of the same order and form:R = Q P P ;Q!then (s R) = index(P ) (s Q).on C 1 (V 1 V V 2 V )Proof: This lemma gives the relationship between the index and the etainvariant whichwe will exploit henceforth. We perform a formal computation.Let f g 1 =1 be a spectral resolution of the operator Q on C 1 (V ).We let=P P and decompose C 1 (V 1 )=N()+R(). We letf j j gbe a spectral resolution of restricted to N() ? =R(). The j are positivereal numbers f j P j = p j g form a spectral resolution of 0 = PP on N( 0 ) ? = R( 0 )=R(P ).We decompose L 2 (V 1 V ) = N()L 2 (V ) R()L 2 (V )andL 2 (V 2 V ) = N( 0 )L 2 (V ) R( 0 )L 2 (V ). In R()L 2 (V ) R( 0 )L 2 (V )we study the two-dimensional subspace that is spanned by the elements:f j P j = p j g. The direct sum of these subspaces as j varyis R() L 2 (V ) R( 0 ) L 2 (V ). Each subspace is invariant under theoperator R. If we decompose R relative to this basis, it is represented bythe 2 2 matrix:p! jp :j ; This matrix has two eigenvalues with opposite signs: p 2 + j . Since > 0 these eigenvalues are distinct and cancel in the sum dening eta.Therefore the only contribution to eta comes from N() L 2 (V ) andN( 0 ) L 2 (V ). On the rst subspace, R is 1 Q. Each eigenvalue of Q isrepeated dim N() times so the contribution to eta is dim N()(s Q).On the second subspace, R is 1 ;Q and the contribution to eta is; dim N()(s Q). When we sum all these contributions, we concludethat:(s R) = dim N()(s Q) ; dim N( 0 )(s Q) =index(P )(s Q):Although this formal cancellation makes sense even if P and Q are notdierential operators, R will not be a pseudo-dierential operator if P andQ are pseudo-dierential operators in general. This did not matter whenwe studied the index since the index was constant under approximations.The eta invariant is a more delicate invariant, however, so we cannot usethe same trick. Since the index of any dierential operator on the circle is

Of the Eta Invariant 265zero, we cannot use Lemmas 4.3.5 and 4.3.6 directly to conclude R(P )=0if m is even.We recall the construction of the operator on C 1 (S 1 ) having index 1.We x a 2 R as a real constant and x a positive order d 2 Z. We dene:Q 0 = ;i@=@Q 1 (a) =(Q 2d0 + a 2 ) 1=2dQ 2 (a) = 1 2 (e;i (Q 0 + Q 1 )+Q 0 ; Q 1 )Q(a) =Q 2 (a) (Q 1 (a)) d;1 :The same argument as that given in the proof of Lemma 3.9.4 shows theseare pseudo-dierential operators on C 1 (S 1 ). Q 2 (a) andQ(a) areellipticfamilies. If a = 0, then Q 2 (0) agrees with the operator of Lemma 3.9.4so index Q 2 (0) = 1. Since the index is continuous under perturbation,index Q 2 (a) = 1 for all values of a. Q 1 (a) is self-adjoint so its index iszero. Consequently index Q(a) =indexQ 2 (a)+indexQ 1 (a) =1for all a.We let P : C 1 (V ) ! C 1 (V ) be an elliptic self-adjoint partial dierentialoperator of order d > 0 over a manifold M. On M S 1 we dene theoperators:Q 0 = ;i@=@Q 1 =(Q 2d0 + P 2 ) 1=2dQ 2 = 1 2 (e;i (Q 0 + Q 1 )+Q 0 ; Q 1 ) Q = Q 2 Q d;11on C 1 (V ). These are pseudo-dierential operators over M S 1 sinceQ 2d0 + P 2 has positive denite leading order symbol. We dene R by:R = P QQ ;P!: C 1 (V V ) ! C 1 (V V ):This is a pseudo-dierential operator of order d which is self-adjoint. Wecompute:!R 2 = P 2 + Q Q 00 P 2 + QQ so! L (R 2 )(z) = p()2 + q q(z) 00 p() 2 + qq (z)for 2 T M and z 2 T S 1 . Suppose R is not elliptic so L (R)(z)v =0for some vector v 2 V V . Then f L (R)(z)g 2 v = 0. We decomposev = v 1 v 2 . We conclude:(p() 2 + q q(z))v 1 =(p() 2 + qq (z))v 2 =0:

266 4.3. The Regularity at s =0Using the condition of self-adjointness this implies:p()v 1 = q(z)v 1 = p()v 2 = q (z)v 2 =0:We suppose v 6= 0 so not both v 1 and v 2 are zero. Since P is elliptic,this implies = 0. However, for = 0, the operators Q 1 and Q 2 agreewith the operators of Lemma 3.9.4 and are elliptic. Therefore q(z)v 1 =q (z)v 2 =0implies z =0. Therefore the operator R is elliptic.Lemma 4.3.6 generalizes in this situation to become:Lemma 4.3.7. Let P , Q, R be dened as above, then (s R) =(s P ).Proof: We let f g be a spectral resolution of P on C 1 (V )over M.This gives an orthogonal direct sum decomposition:L 2 (V ) over M S 1 = M L 2 (S 1 ) :Each of these spaces is invariant under both P and R. If R denotes therestriction of R to this subspace, then(s R) = X (s R ):On L 2 ( ), P is just multiplication by the real eigenvalue . If wereplace P by we replace Q by Q( ), so R becomes:R = Q( )Q ( ); :!We now apply the argument given to prove Lemma 4.3.6 to conclude:(s R ) = sign( )j j ;s index(Q( )):Since index Q( )=1,this shows (s R) = sign( )j j ;s and completesthe proof.We can now generalize Lemma 4.3.5 to all dimensions:Theorem 4.3.8. Let P : C 1 (V ) ! C 1 (V ) be a self-adjoint ellipticpseudo-dierential operator of order d > 0. Then R(P )=0|i.e., (s P )is regular at s =0.Proof: This result follows from Lemma 4.3.5 if dim M = m is odd. Ifdim M = m is even and if P is a dierential operator, then we form thepseudo-dierential operator P over M S 1 with (s R) =(s P ). Then(s R) is regular at s =0implies (s P ) is regular at s =0. This proves

Of the Eta Invariant 267Theorem 4.3.8 for dierential operators. Of course, if P is only pseudodierential,then R need not be pseudo-dierential so this construction doesnot work. We complete the proof of Theorem 4.3.8 by showing the partialdierential operators of even order generate K(S(T M) Q)=K(M Q) ifm is even. (We already know the operators of odd order generate if m isodd and if M is oriented).Consider the involution 7! ; of the tangent space. This gives a naturalZ 2 action on S(T M). Let : S(T M) ! S(T M)=Z 2 = RP (T M)be the natural projection on the quotient projective bundle. Since m iseven, m ; 1 is odd and denes an isomorphism between the cohomologyof two bers : H (RP m;1 Q) =Q Q ! H (S m;1 Q) =Q Q:The Kunneth formula and an appropriate Meyer-Vietoris sequence implythat : H (RP (T M) Q) ! H (S(T M) Q)is an isomorphism in cohomology for the total spaces. We now use theChern isomorphism between cohomology and K-theory to conclude thereis an isomorphism in K-theory : K(RP (T M) Q) ' K(S(T M) Q):Let S 0 (k) be the set of all k k self-adjoint matrices A such thatA 2 = 1 and Tr(A) = 0. We noted in section 3.8 that if k is large,eK(X) = [X S 0 (2k)]. Thus K(S(T e M) Q) = K(RP e (T M) Q) is generatedby maps p: S(T M) ! S 0 (2k) such that p(x ) = p(x ;). Wecan approximate any even map by an even polynomial using the Stone-Weierstrass theorem. Thus we may Xsupposep has the form:p(x ) =jjnjj evenp (x) S(T M) where the p : M ! S 0 (2k) and where n is large. As is even, we canreplace by jj fn;jjg=2 and still have a polynomial with the samevalues on S(T M). We may therefore assume that p is a homogeneouseven polynomial this is the symbol of a partial dierential operator whichcompletes the proof.If m is odd and if M is orientable, we constructed specic examples ofoperators generating K(S(T M) Q)=K(M Q) usingthe tangential operatorof the signature complex with coecients in an arbitrary coecientbundle. If m is even, it is possible to construct explicit second order opratorsgenerating this K-theory group. One can then prove directly that

268 4.3. The Regularity at s =0eta is regular at s = 0 for these operators as they are all \natural" in acertain suitable sense. This approach gives more information by explcitlyexhibiting the generators as the consturction is quite long and technicalwe have chosen to give an alternate argument based on K-theory and referto (Gilkey, Theresidue of the global eta function at the origin) for details.We have given a global proof of Theorem 4.3.8. In fact such a treatmentis necessary since in general the local formulas giving the residue at s =0are non-zero. Let P : C 1 (V ) ! C 1 (V ) be self-adjoint and elliptic. Iff g is a spectral resolution of P ,we dene:(s P x)= X sign( )j j ;s ( )(x)so that:(s P )=ZM(s P x)dvol(x):Thus Res s=0 (s P x)=a(P x) isgiven by a local formula.We present the following example (Gilkey, The residue of the local etafunction at the origin) to show this local formula need not vanish identicallyin general.Example 4.3.9. Lete 1 = 1 00 ;1!! e 2 = 0 1 e1 03 =!0 i;i 0be Cliord matrices acting on C 2 . Let T m be the m-dimensional toruswith periodic parameters 0 x j 2 for 1 j m.(a) Let m = 3 and let b(x) be a real scalar. Let P = i P j e j@=@x j + b(x)I.Then P is self-adjoint and elliptic and a(x P )=cb where c 6= 0is someuniversal constant.(b) Let m = 2 and let b 1 and b 2 be imaginary scalar functions. Let P =e 1 @ 2 =@x 2 1 + e 2 @ 2 =@x 2 2 +2e 3 @ 2 =@x 1 @x 2 + b 1 e 1 + b 2 e 2 . Then P is self-adjointand elliptic and a(x P ) = c 0 (@b 1 =@x 2 ; @b 2 =@x 1 ) where c 0 6= 0 is someuniversal constant.(c) By twisting this example with a non-trivial index problem and usingLemma 4.3.6, we can construct examples on T m so that a(x P ) does notvanish identically for any dimension m 2.The value of eta at the origin plays a central role in the Atiyah-Patodi-Singer index theorem for manifolds with boundary. In section 2, we discussedthe transgression briey. Let r i be two connections on T (N). Wedened TL k (r 1 r 2 ) so that dT L k (r 1 r 2 ) = L k (r 1 ) ; L k (r 2 ) this isa secondary characteristic class. Let r 1 be the Levi-Civita connection ofN and near M = dN let r 2 be the product connection arising from the

Of the Eta Invariant 269product metric. As L k (r 2 ) = 0 we see dT L k (r 1 r 2 ) = 0. This is theanalogous term which appeared in the Gauss-Bonnet theorem it can becomputed in terms of the rst and second fundamental forms. For example,if dim N =4L 1 (R) = ;124 2 Tr(R ^ R) TL 1(R !) =;1Tr(R ^ !)24 8 2 where R is the curvature 2-form and ! is the second fundamental form.Theorem 4.3.10 (Atiyah-Patodi-Singer index theorem formanifolds with boundary). Let N be a 4k dimensional oriented compactRiemannian manifold with boundary M. Then:signature(N) =ZNL k ;ZMTL k ; 1 2 (0A)where A is the tangential operator of the signature complex discussed inLemma 4.3.4.In fact, the eta invariant more generally is the boundary correction termin the index theorem for manifolds with boundary. Let N be a compactRiemannian manifold with boundary M and let P : C 1 (V 1 ) ! C 1 (V 2 ) bean elliptic rst order dierential complex over N. We take ametric whichis product near the boundary and identify a neighborhood of M in N withM [0 1). We suppose P decomposes in the form P = (dn)(@=@n + A)on this collared neighborhood where A is a self-adjoint elliptic rst orderoperator over M. This is in fact the case for the signature, Dolbeault,or spin compexes. Let B be the spectral projection on the non-negativeeigenvalues of A. Then:Theorem 4.3.11. (The Atiyah-Patodi-Singer Index Theorem) .Adopt the notation above. P with boundary condition B is an ellipticproblem andindex(P B) =ZNfa n (x P P ) ; a n (x P P )g; 1 2 f(0A)+dimN(A)g:In this expression, n = dim N and the invariants a n are the invariants ofthe heat equation discussed previously.Remark: If M is empty then this is nothing but the formula for index(P )discussed previously. If the symbol does not decompose in this productstructure near the boundary of N, there are corresponding local boundarycorrection terms similar to the ones discussed previously. If one takes Pto be the operator of the signature complex, then Theorem 4.3.10 can bederived from this more general result by suitably interpreting index(P B)=signature(N) ; 1 dim N(A).2

4.4. The Eta Invariant with CoecientsIn a Locally Flat Bundle.The eta invariant plays a crucial role in the index theorem for manifoldswith boundary. It is also possible to study the eta invariant with coecientsin a locally at bundle to get a generalization of the Atiyah-Singer theorem.Let : 1 (M) ! U(k) be a unitary representation of the fundamentalgroup. Let M be the universal cover of M and let m ! gm for m 2 Mand g 2 1 (M) be the acion of the deck group. We dene the bundle V over M by the identication:V = M C k mod (m z) =(gm(g)z):The transition functions of V are locally constant andV inherits a naturalunitary structure and connection r with zero curvature. The holonomyof r is just the representation . Conversely, given a unitary bundle withlocally constant transition functions, we can construct the connection rto be unitary with zero curvature and recover as the holonomy of theconnection. We assume is unitary to work with self-adjoint operators,but all the constructions can be generalized to arbitrary representations inGL(k C).Let P : C 1 (V ) ! C 1 (V ). Since the transition functions of V are locallyconstant, we can dene P on C 1 (V V ) uniquely using a partition ofunity if P is a dierential operator. If P is only pseudo-dierential, P iswell dened modulo innitely smoothing terms. We dene:~(P )= 1 f(P )+dimN(P )g mod Z2ind( P )=~(P ) ; k~(P ) mod Z:Lemma 4.4.1. ind( P ) is a homotopy invariant of P . If we x , wecan interpret this as a mapind( ): K(S(T M))=K(M) ! R mod Zsuch that ind( P ) = ind( + ( L (P ))).Proof: We noted previously that ~ was well dened in R mod Z. Let P (t)be a smooth 1-parameter family of such operators and let P 0 (t) = d dt P (t).In Theorem 1.10.2, we proved:ddt ~(P t)=ZMa(x P (t)P 0 (t)) dvol(x)was given by a local formula. Let P k = P 1 k acting on V 1 k . Thiscorresponds to the trivial representation of 1 (M) inU(k). The operators

The Eta Invariant 271P and P k are locally isomorphic modulo 1 smoothing terms which don'taect the local invariant. Thus a(x P k (t)P 0 k (t)) ; a(x P (t)P 0 (t)) = 0.This implies d dt ind( P t) = 0 and completes the proof of homotopy invariance.If the leading symbol of P is denite, then the value (0P) is givenby a local formula so ind( ) = 0 in this case, as the two local formulascancel. This veries properties (d) and (e) of Lemma 4.3.1 properties (a),(b) and (f) are immediate. We therefore apply Lemma 4.3.3 to regardind( ): K(S(T M))=K(M) ! R mod Zwhich completes the proof.In sections 4.5 and 4.6 we will adopt a slightly dierent notation for thisinvariant. Let G be a group and let R(G) be the group representation ringgenerated by the unitary representations of G. Let R 0 (G) be the ideal ofrepresentations of virtual dimension 0. We extend ~: R(G) ! R mod Z tobe a Z-linear map. We let ind( P ) denote the restriction to R 0 (G). If is a representation of dimension j, then ind( P ) = ind( ; j 1P) =(P ) ; j(P ). It is convenient to use both notations and the contextdetermines whether we are thinking of virtual representations of dimension0 or the projection of an actual representation to R 0 (G). ind( P )isnottopological in , as the following example shows. We will discuss K-theoryinvariants arising from the dependence in Lemma 4.6.5.Example 4.4.2: Let M = S 1 be the circle with periodic parameter 0 2. Let g() =e i be the generator of 1 (M) ' Z. We let " belong to Rand dene: " (g) =e 2i"as a unitary representation of 1 (M). The locally at bundle V is topologicallytrivial since any complex bundle over S 1 is trivial. If we dene alocally at section to S 1 C by:~s() =e i"then the holonomy dened by ~s gives the representation " since~s(2) =e 2i" ~s(0):We let P = ;i@=@ on C 1 (S 1 ). ThenP " = P " = e +i" Pe ;i" = P ; ":The spectrum of P is fng n2Z so the spectrum of P " is fn ; "g n2Z . Therefore:X(s P " )= sign(n ; ")jn ; "j ;s :n;"6=0

272 4.4. The Eta InvariantWe dierentiate with respect to " to get:d Xd" (s P ")=s jn ; "j ;s;1 :n;"6=0We evaluate at s = 0. Since the sum dening this shifted zeta functionranges over all integers, the pole at s =0has residue 2 so we conclude:dd" ~(P ")=2 1 2 =1 and ind( "P)=Z "01 d" = ":We note that if we replace " by " + j for j 2 Z, then the representation isunchanged and the spectrum of the operator P " is unchanged. Thus reductionmod Z is essential in making ind( " P)well dened in this context.If V is locally at, then the curvature of r is zero so ch(V )=0. Thisimplies V is a torsion element in K-theory so V 1 n ' 1 kn for someinteger n. We illustrate this withExample 4.4.3: Let M = RP 3 = S 3 =Z 2 = SO(3) be real projective spacein dimension 3 so 1 (M) = Z 2 . Let : Z 2 ! U(1) be the non-trivialrepresentation with (g) =;1 where g is the generator. Let L = S 3 Cand identify (x z) = (;x ;z) to dene a line bundle L over RP 3 withholonomy . We show L is non-trivial. Suppose the contrary, then L is trivial over RP 2 as well. This shows there is a map f: S 2 ! S 1 with;f(;x) = f(x). If we restrict f to the upper hemisphere of S 2 , thenf: D+ 2 ! S 1 satises f(x) = ;f(;x) on the boundary. Therefore f hasodd degree. Since f extends to D + , f must have zero degree. Thiscontradiction establishes no such f exists and L is non-trivial.The bundle L L is S 3 C modulo the relation (x z) = (;x ;z).We let g(x): S 3 ! SU(2) be the identity map:!g(x) = x 0 + ix 1 ; x 2 + ix 3x 2 + ix 3 x 0 ; ix 1then g(;x) =;g(x). Thus g descends to give a global frame on L L sothis bundle is topologically trivial and L represents a Z 2 torsion class inK(RP 3 ). Since L is a line bundle and is not topologically trivial, L ; 1is a non-zero element of e K(RP3 ). This construction generalizes to deneL over RP n . We use the map g: S n ! U(2 k ) dened by Cliord algebrasso g(x) =;g(;x) to show 2 k L ; 2 k =0in e K(RPn ) where k =[n=2] werefer to Lemma 3.8.9 for details.We suppose henceforth in this section that the bundle V is topologicallytrivial and let ~s be a global frame for V . (In section 4.9 we will study the

With Coefficients in a Locally Flat Bundle 273more general case). We can take P k = P 1 relative to the frame ~s sothat both P k and P are dened on the same bundle with the same leadingsymbol. We form the 1-parameter family tP k +(1; t)P = P (t ~s ) anddene:whereind( P ~s )=Z 10a(x P ~s )=ddtZM~(P (t ~s )) dt = a(x P ~s )dvol(x)Z 10a(x P (t ~s)P 0 (t ~s )) dt:The choice of a global frame permits us to lift ind from R mod Z toR. If we take the operator and representation of example 4.4.2, thenind( " P~s )=" and thus in particular the lift depends on the global framechosen (or equivalently on the particular presentation of the representation on a trivial bundle). This also permits us to construct non-trivial realvalued invariants even on simply connected manifolds by choosing suitableinequivalent global trivializations of V .Lemma 4.4.4.R (a) ind( P ~s )=Ma(x P ~s )dvol(x) is given by a local formula whichdepends on the connection 1-form of r relative to the global frame ~s andon the symbol of the operator P .(b) If ~s t is a smooth 1-parameter family of global sections, then ind( P ~s t )is independent of the parameter t.Proof: The rst assertion follows from the denition of a(x P ~s )givenabove and from the results of the rst chapter. This shows ind( P ~s t )varies continuously with t. Since its mod Z reduction is ind( P ), thismod Z reduction is constant. This implies ind( P ~s t ) itself is constant.We can use ind( P ~s ) to detect inequivalent trivializations of a bundleand thereby study the homotopy [MU(k)] even if M is simply connected.This is related to spectral ow.The secondary characteristic classes are cohomological invariants of therepresentation . They are normally R mod Z classes, but can be liftedto R and expressed in terms of local invariants if the bundle V is givena xed trivialization. We rst recall the denition of the Chern character.Let W be a smooth vector bundle with connection r. Relative to somelocal frame, we let ! be the connection 1-form and = ; d! ; ! ^ ! bei kthe curvature. TheCherncharacter is given by chk(r) = 2 1k!Tr( k ).This is a closed 2k form independent of the frame ~s chosen. If r i are twoconnections for i = 0 1, we form r t = tr 1 +(1; t)r 0 . If = ! 1 ; ! 0 ,then transforms like a tensor. If t is the curvature of the connectionr t , then:chk(r 1 ) ; chk(r 0 )=Z 10ddt ch k(r t ) dt = d(Tchk(r 1 r 0 ))

274 4.4. The Eta Invariantwhere the transgression Tchk is dened by: iTchk(r 1 r 0 )=2 k1(k ; 1)! Tr Z 10 k;1tdt :We refer to the second chapter for further details on this construction.We apply this construction to the case in which both r 1 and r 0 havezero curvature. We choose a local frame so ! 0 = 0. Then ! 1 = and 1 = d ; ^ =0. Consequently:so that:! t = t and t = td ; t 2 ^ =(t ; t 2 ) ^ Tchk(r 1 r 0 )= i k12 (k ; 1)!Z 1We integrate by parts to evaluate this coecient:Z 10Therefore(t ; t 2 ) k;1 dt =Z 10t k;1 (1 ; t) k;1 dt = k ; 1k(k ; 1)!=k (k +1)(2k ; 2)(k ; 1)! (k ; 1)!= :(2k ; 1)!Tchk(r 1 r 0 )=0(t ; t 2 ) k;1 dt Tr( 2k;1 ):Z 10Z 10t 2k;2 dt i k (k ; 1)!2 (2k ; 1)! Tr(2k;1 ):t k (1 ; t) k;2 dtWe illustrate the use of secondary characteristic classes by giving anotherversion of the Atiyah-Singer index theorem. Let Q: C 1 (1 k ) ! C 1 (1 k ) bean elliptic complex. Let ~s be global frames on + (q) over D (T M)sothat ~s ; = q t (x )~s + on D + (T M)\D ; (T M)=S(T M). (The clutchingfunction is q to agree with the notation adopted in the third section weexpress ~s ; = q t ~s + as we think of q being a matrix acting on column vectorsof C k . The action on the frame is therefore the transpose action). Wechoose connections r on + (q) so r (~s )=0on D (T M). Then:index(Q) =(;1) m Z (T M)Todd(M) ^ ch(r ; )=(;1) m Z D + (M)Todd(M) ^ ch(r ; ):

With Coefficients in a Locally Flat Bundle 275However, on D + we have + =0so we can replace ch(r ; ) by ch(r ; ) ;ch(r + ) without changing the value of the integral. ch(r ; ) ; ch(r + ) =d Tch(r ; r + ) so an application of Stokes theorem (together with a carefulconsideration of the orientations involved) yields:index(Q) =(;1) m Z S(T M)Todd(M) ^ Tch(r ; r + ):Both connections have zero curvature near the equator S(T M). The bersover D + are glued to the bers over D ; using the clutching function q.With the notational conventions we have established, if f + is a smoothsection relative to the frame ~s + then the corresponding representation isqf + relative to the frame ~s ; . Therefore r ; (f + ) = q ;1 dq f + + df + andconsequentlyr ; ;r + = = q ;1 dq:(In obtaining the Maurer-Cartan form one must be careful which conventionone uses|right versus left|and we confess to having used both conventionsin the course of this book.) We use this to compute:forTchk(r ; r + )= X c k Tr((q ;1 dq) 2k;1 )c k = i k (k ; 1)!2 (2k ; 1)! :P Let = g ;1 dg be the Maurer-Cartan form and let Tch = c k Tr( 2k;1 ).kThis denes an element of the odd cohomology of GL( C) such thatTch(r ; r + )=q (Tch). We summarize these computations as follows:Lemma 4.4.5. Let = g ;1 dg be the Maurer-Cartan form on the generallinear group. Dene:Tch = X k i k (k ; 1)!2 (2k ; 1)! Tr(2k;1 )as an element of the odd cohomology. If Q: C 1 (1 : ) ! C(1 : ) is an ellipticcomplex dened on the trivial bundle, let q be the symbol so q: S(T M) !GL( C). Then index(Q) =(;1) m RS(T M) Todd(M) ^ q (Tch).We compute explicitly the rst few terms in the expansion:Tch =i ;1Tr()+2 24 2 Tr(3 )+ ;i960 3 Tr(5 )+:In this version, the Atiyah-Singer theorem generalizes to the case of dM 6= as the Atiyah-Bott theorem. We shall discuss this in section 4.5.We can now state the Atiyah-Patodi-Singer twisted index theorem:

276 4.4. The Eta InvariantTheorem 4.4.6. Let P : C 1 (V ) ! C 1 (V ) be an elliptic self-adjointpseudo-dierential operator of order d > 0. Let : 1 (M) ! U(k) be aunitary representation of the fundamental group and assume the associatedbundle V is toplogically trivial. Let ~s be a global frame for V and letr 0 (~s ) 0 dene the connection r 0 . Let r be the connection dened bythe representation and let = r (~s ) so that:Tch(r r 0 )= X k i2 k (k ; 1)!(2k ; 1)! Tr(2k;1 ):Then:ind( P ~s )=(;1) m Z S(T M)Todd(M) ^ ch( + p) ^ Tch(r r 0 ):S(T M) is given the orientation induced by the simplectic orientation dx 1^d 1 ^ ^dx m ^ d m on T M where we use the outward pointing normalso N ^ ! 2m;1 = ! 2m :We postpone the proof of Theorem 4.4.6 for the moment to return to theexamples considered previously. In example 4.4.2, we let = e 2i" on thegenerator of 1 S 1 ' Z. We let \1" denote the usual trivialization of thebundle S 1 C so that:r (1) = ;i"d and Tch(r r 0 )="d=2:The unit sphere bundle decomposes S(T M) = S 1 f1g [S 1 f;1g.The symbol of the operator P is multiplication by the dual variable soch( + p) = 1 on S 1 f1g and ch( + p) = 0 on S 1 f;1g. The inducedorientation on S 1 f1g is ;d. Since (;1) m = ;1, we compute:;ZS(T M)Todd(M) ^ ch( + p) ^ Tch(r r 0 )=Z 20"d=2 = ":This also gives an example in which Tch 1 is non-trivial.In example 4.4.3, we tookthenon-trivial representation of 1 (RP 3 )=Z 2 to dene a non-trivial complex line bundle V such that V V =1 2 .The appropriate generalization of this example is to 3-dimensional lensspaces and provides another application of Theorem 4.4.6:Example 4.4.7: Let m = 3 and let n and q be relatively prime positiveintegers. Let = e 2i=n be a primitive n th root of unity and let =diag( q ) generate a cyclic subgroup ; of U(2) of order n. If s () = s ,then f s g 0s

With Coefficients in a Locally Flat Bundle 277quotient manifold. As 1 (S 3 ) = 0, we conclude 1 (L(n q)) = ;. Let V s bethe line bundle corresponding to the representation s . It is dened fromS 3 C by the equivalence relation (z 1 z 2 w)=(z 1 q z 2 s w).Exactly the same arguments (working mod n rather than mod 2) used toshow V 1 is non-trivial over RP 3 show V s is non-trivial for 0

278 4.4. The Eta Invariantso that Tr( 3 )=;12 dvol. Therefore:Zs 3 Tch 2 (r 1 r ;1 r 0 r 0 )= 1224 2 volume(S 3 )=1:We can study other values of (s q) by composing with the map (z 1 z 2 ) 7!(z1z s sq02 )=j(zs 1z sq02 j. This gives a map homotopic to the original map anddoesn't change the integral. This is an s 2 q 0 -to-one holomorophic map sothe corresponding integral becomes s 2 q 0 . If instead of integrating over S 3we integrate over L(n q), we must divide the integral by n so that:ZL(nq)Tch 2 (r s r ;s r 0 r 0 )= s2 q 0n :Let P be the tangential operator of the signature complex. By Lemma4.3.2, we have:Zs 2 ch( + (P )) = ;4:S 3 is parallelizable. The orientation of S(T S 3 )isdx 1 ^ dx 2 ^ d 2 ^ dx 3 ^d 3 = ;dx 1 ^ dx 2 ^ dx 3 ^ d 2 ^ d 3 so S(T S 3 ) = ;S 3 S 2 given theusual orientation. ThusZ(;1) 3 Tch 2 (r s r ;s r 0 r 0 ) ^ ch( + P )S(T S 3 )=(;1)(;1)(;1) 4 s 2 q 0 =n = ;4s 2 q 0 =n:Consequently by Theorem 4.4.6, we conclude, since Todd(S 3 )=1,ind( s + ;s 0 + 0 P)=;4s 2 q 0 =nusing the given framing.The operator P splits into an operator on even and odd forms with equaleta invariants and a corresponding calculation shows that ind( s + ;s 0 + 0 P even ) = ;2s 2 q 0 =n. There is an orientation preserving isometryT : L(p q) ! L(p q) dened by T (z 1 z 2 ) = (z 1 z 2 ). It is clear T interchangesthe roles of s and ;s so that as R mod Z-valued invariants,ind( s P even )=ind( ;s P even ) so that:ind( s P) = 2 ind( s P even ) = ind( s P even )+ind( ;s P even )= ;2s 2 q 0 =n:This gives a formula in R mod Z for the index of a representation whichneed not be topologically trivial. We will return to this formula to discussgeneralizations in Lemma 4.6.3 when discussing ind( ) for generalspherical space forms in section 4.6.

With Coefficients in a Locally Flat Bundle 279We sketch the proof of Theorem 4.4.6. We shall omit many of the detailsin the interests of brevity. We refer to the papers of Atiyah, Patodi, andSinger for complete details. An elementary proof is contained in (Gilkey,The eta invariant and the secondary characteristic classes of locally atbundles). Dene:ind 1 ( P s)=(;1) m Z S(T M)Todd(M) ^ ch( + p) ^ Tch(r r 0 )then Lemmas 4.3.6 and 4.3.7 generalize immediately to:Lemma 4.4.8.(a) Let M 1 and M 2 be smooth manifolds. Let P : C 1 (V 1 ) ! C 1 (V 2 ) bean elliptic complex over M 1 . Let Q: C 1 (V ) ! C 1 (V ) be a self-adjointelliptic operator over M 2 . We assume P and ! Q are dierential operators ofthe same order and form R = Q P over M = M 1 M 2 . Let beP ;Qa representation of 1 (M 2 ). Decompose 1 (M) = 1 (M 1 ) 1 (M 2 ) andextend to act trivially on 1 (M 1 ). Then we can identify V over M withthe pull-back of V over M 2 . Let s be a global trivialization of V then:ind( R ~s ) = index(P )ind( Q ~s )ind 1 ( R ~s ) = index(P )ind 1 ( Q ~s ):(b) Let M 1 = S 1 be the circle and let (R Q) be as dened in Lemma 4.3.7,then:ind( R ~s ) = ind( P ~s )ind 1 ( R ~s )=ind 1 ( P ~s ):Proof: The assertions about ind follow directly from Lemmas 4.3.6 and4.3.7. The assertions about ind 1 follow from Lemma 3.9.3(d) and from theAtiyah-Singer index theorem.Lemma 4.4.8(c) lets us reduce the proof of Theorem 4.4.6 to the casedim M odd. Since both ind and ind 1 are given by local formulas, we mayassume without loss of generality that M is also orientable. Using the samearguments as those given in subsection 4.3, we can interpret both ind andind 1 as maps in K-theory once the representation and the global frame~s are xed. Consequently, the same arguments as those given for the proofof Theorem 4.3.8 permit us to reduce the proof of Theorem 4.4.6 to thecase in which P = A V is the operator discussed in Lemma 4.3.4.ind( P ~s ) is given by a local formula. If we express everything withrespect to the global frame ~s, then P k = P 1 and P is functoriallyexpressible in terms of P and in terms of the connection 1-form ! = r ~sind( P ~s )=ZMa(x G !) dvol(x):

280 4.4. The Eta InvariantThe same arguments as those given in discussing the signature complexshow a(x G !) ishomogeneous of order n in the jets of the metric and ofthe connection 1-form. The local invariant changes sign if the orientationis reversed and thus a(x G !) dvol(x) should be regarded as an m-formnot as a measure. The additivity of with respect to direct sums showsa(x G ! 1 ! 2 ) = a(x G ! 1 )+a(x G ! 2 ) when we take the direct sumof representations. Arguments similar to those given in the second chapter(and which are worked out elsewhere) prove:Lemma 4.4.9. Let a(x G !) be an m-form valued invariant which isdened on Riemannian metrics and on 1-form valued tensors ! so thatd! ; ! ^ ! = 0. Suppose a is homogeneous of order m in the jets of themetric and the tensor ! and suppose that a(x G ! 1 ! 2 )=a(x G ! 1 )+a(x G ! 2 ). Then we can decompose:a(x G !) = X f (G) ^ Tch (!) = X f (G) ^ c Tr(! 2;1 )where f (G) are real characteristic forms of T (M) of order m +1; 2.We use the same argument as that given in the proof of the Atiyah-Singerindex theorem to show there must exist a local formula for ind( P s) whichhas the form:ind( P s)=ZS(T M)X4i+2j+2k=m+1Td 0 im(M) ^ chj( + p) ^ Tchk(r r 0 ):When the existence of such a formula is coupled with the product formulagiven in Lemma 4.4.8(a) and with the Atiyah-Singer index theorem, wededuce that the formula must actually have the form:ind( P s)=ZS(T M)Xj+2k=m+1;Todd(M) ^ ch( + p) j ^ c(k) Tch k(r r 0 )where c(k) is some universal constant which remains to be determined.If we take an Abelian representation, all the Tchk vanish for k>1. Wealready veried that the constant c(1) = 1 by checking the operator ofexample 4.4.2 on the circle. The fact that the other normalizing constantsare also 1 follows from a detailed consideration of the asymptotics of theheat equation which arise we refer to (Gilkey, The eta invariant and thesecondary characteristic classes of locally at bundles) for further detailsregarding this verication. Alternatively, the Atiyah-Patodi-Singer index

With Coefficients in a Locally Flat Bundle 281theorem for manifolds with boundary can be used to check these normalizingconstants we refer to (Atiyah, Patodi, Singer: Spectral asymmetryand Riemannian geometry I{III) for details.The Atiyah-Singer index theorem is a formula on K((T M)). TheAtiyah-Patodi-Singer twisted index theorem can be regarded as a formulaon K(S(T M)) = K 1 ((T M)). These two formulas are to be regardedas suspensions of each other and are linked by Bott periodicity inapurelyformal sense which we shall not make explicit.If V is not topologically trivial, there is no local formula for ind( P )in general. It is possible to calculate this using the Lefschetz xed pointformulas as we will discuss later.We conclude this section by discussing the generalization of Theorem4.4.6 to the case of manifolds with boundary. It is a fairly straightforwardcomputation using the methods of section 3.9 to show:Lemma 4.4.9. We adopt the notation of Theorem 4.4.6. Then for anyk 0,ind( P ~s )=(;1) m Z 2k (T M)Todd(M)^ch( + ( 2k (p)))^Tch(r r 0 ):Remark: This shows that we can stabilize by suspending as often as weplease. The index can be computed as an integral over 2k+1 (T M) whilethe twisted index is an integral over 2k (T M). These two formulas areat least formally speaking the suspensions of each other.It turns out that the formula of Theorem 4.4.6 does not generalizedirectly to the case of manifolds with boundary, while the formula ofLemma 4.4.9 with k = 1 does generalize. Let M be a compact manifoldwith boundary dM. We choose a Riemannian metric on M which is productnear dM. Let P : C 1 (V ) ! C 1 (V )bea rst order partial dierentialoperator with leading symbol p which is formally self-adjoint. We supposep(x ) 2 = jj 2 P 1 V so that p is dened by Cliord matrices. If we decomposep(x ) = j p j(x) j relative to a local orthonormal frame for T (M),then the p j are self-adjoint and satisfy the relations p j p k + p k p j =2 jk .Near the boundary we decompose T (M) =T (dM) 1into tangentialand normal directions. We let = P (z) for 2 T (dM) reect thisdecomposition. Decompose p(x ) =1jm;1 p j(x) j + p m z. Let t bea real parameter and dene: X (x t)=i p m p j (x) j ; it1jm;1as the endomorphism dened in Chapter 1. We suppose given a self-adjointendomorphism q of V which anti-commutes with . Let B denote theorthogonal projection 1 (1 + q) onthe+1 eigenspace of q. (P B) isaselfadjointelliptic boundary value problem. The results proved for manifolds2without boundary extend to this case to become:

282 4.4. The Eta InvariantTheorem 4.4.10. Let (P B) be an elliptic rst order boundary valueproblem. We assume P is self-adjoint and L (p) 2 = jj 2 I V . We assumePB = 1 (1+q) where 2 q anti-commutes with = ip m 1jm;1 p j j ; it .Let f g 1 =1 denote the spectrum of the operator P B . Dene:(s P B)= X sign( )j j ;s :(a) (s P B) is well dened and holomorphic for Re(s) 0.(b) (s P B) admits a meromorphic extension to C with isolated simplepoles at s = (n ; m)=2 for n = 0 1 2 ... . The residue of at such asimple pole is given by integrating a local formula a n (x P ) over M and alocal formula a n;1 (x P B) over dM.(c) The value s =0is a regular value.(d) If (P u B u ) is a smooth 1-parameter family of such operators, then thederivative ddu (0P uB u ) is given by a local formula.Remark: This theorem holds in much greater generality than we are statingit we restrict to operators and boundary conditions given by Cliord matricesto simplify the discussion. The reader should consult (Gilkey-Smith)for details on the general case.Such boundary conditions always exist if M is orientable and m is eventhey may not exist if m is odd. For m even and M orientable, we cantake q = p 1 ...p m;1 . If m is odd, the obstruction is Tr(p 1 ...p m ) asdiscussed earlier. This theorem permits us to dene ind( P B) if is aunitary representation of 1 (M) just as in the case where dM is empty.In R mod Z, it is a homotopy invariant of (P B). If the bundle V istopologically trivial, we can deneind( P B ~s )as a real-valued invariant which is given by a local formula integrated overM and dM.The boundary condition can be used to dene a homotopy of p to anelliptic symbol which doesn't depend upon the tangential ber coordinates.Homotopy 4.4.11: Let (P B) beas in Theorem 4.4.10 with symbols givenby Cliord matrices. Let u be an auxilary parameter and dene:(x t u) = cos ; 2 u (x t) + sin ; 2 u q for u 2 [;1 0]:It is immediate that (x t 0) = (x t)and(x t ;1) = ;q. As andq are self-adjoint and anti-commute, this is a homotopy through self-adjointmatrices with eigenvalues cos 2 ; 2 u (jj 2 + t 2 )+sin 2 ; 2 u 1=2.

With Coefficients in a Locally Flat Bundle 283Therefore (x t u) ; iz is a non-singular elliptic symbol for ( t z) 6=(0 0 0). We havep(x z t)=p(x z) ; it = ip m f(x t) ; izgso we dene the homotopyp(x z t u)=ip m f(x t u) ; izg for u 2 [;1 0]:We identiy a neighborhood of dM in M with dM [0). We sew on acollared neighborhood dM [;1 0] to dene f M. We use the homotopyjust dened to dene an elliptic symbol (p) B on (T f M) which doesnotdepend upon the tangential ber coordinates on the boundary dM f;1g. We use the collaring to construct a dieomorophism of M and f Mto regard (p) B on (T M) where this elliptic symbol is independent ofthe tangential ber coordinates on the boundary. We call this constructionHomotopy 4.4.11.We can now state the generalization of Theorem 4.4.6 to manifolds withboundary:Theorem 4.4.12. We adopt the notation of Theorem 4.4.10 and let(P B) be an elliptic self-adjoint rst order boundary value problem withthe symbols given by Cliord matrices. Let be a representation of 1 (M).Suppose V is topologically trivial and let ~s be a global frame. Thenind( P B ~s )=(;1) m Z 2k (T M)Todd(M) ^ ch( + ( 2k;1 ((p) B ))) ^ Tch(r r 0 )for any k 1. Here (p) B is the symbol on (T M) dened by Homotopy4.4.11 so that it is an elliptic symbol independent of the tangentialber variables on the boundary.Remark: This theorem is in fact true in much greater generality. It istrue under the much weaker assumption that P is a rst order formallyself-adjoint elliptic dierential operator and that (P B) isself-adjoint andstrongly elliptic in the sense discussed in Chapter 1. The relevant homotopyis more complicated to discuss and we refer to (Gilkey-Smith) for bothdetails on this generalization and also for the proof of this theorem.We conclude by stating the Atiyah-Bott formula in this framework. LetP : C 1 (V 1 ) ! C 1 (V 2 ) be a rst order elliptic operator and let B be anelliptic boundary value problem. The boundary value problem gives ahomotopy ofp through self-adjoint elliptic symbols to a symbol independentof the tangential ber variables. We call the new symbol fpg B . TheAtiyah-Bott formula isindex(P B) =(;1) m Z (T M)Todd(M) ^ ch( + ((p) B )):

4.5. Lefschetz Fixed Point Formulas.In section 1.8, we discussed the Lefschetz xed point formulas using heatequation methods. In this section, we shall derive the classical Lefschetzxed point formulas for a non-degenerate smooth map with isolated xedpoints for the four classical elliptic complexes. We shall also discuss thecase of higher dimensional xed point sets for the de Rham complex. Thecorresponding analysis for the signature and spin complexes is much moredicult and is beyond the scope of this book, and we refer to (Gilkey,Lefschetz xed point formulas and the heat equation) for further details.We will conclude by discussing the theorem of Donnelly relating Lefschetzxed point formulas to the eta invariant.Anumber of authors have worked on proving these formulas. Kotake in1969 discussed the case of isolated xed points. In 1975, Lee extended theresults of Seeley to yield the results of section 1.8 giving a heat equationapproach to the Lefschetz xed point formulas in general. We also derivedthese results independently not being aware of Lee's work. Donnelly in1976 derived some of the results concerning the existence of the asymptoticexpansion if the map concerned was an isometry. In 1978 he extended hisresults to manifolds with boundary.During the period 1970 to 1976, Patodi had been working on generalizinghis results concerning the index theorem to Lefschetz xed point formulas,but his illness and untimely death in December 1976 prevented him frompublishing the details of his work on the G-signature theorem. Donnellycompleted Patodi's work and joint papers by Patodi and Donnelly containthese results. In 1976, Kawasaki gave a proof of the G-signature theoremin his thesis on V -manifolds. We also derived all the results of this sectionindependently at the same time. In a sense, the Lefschetz xed pointformulas should have been derived by heat equation methods at the sametime as the index theorem was proved by heat equation methods in 1972and it remains a historical accident that this was not done. The problemwas long over-due for solution and it is not surprising that it was solvedsimultaneously by a number of people.We rst assume T : M ! M is an isometry.Lemma 4.5.1. If T is an isometry, then the xed point setof T consistsof the disjoint union of a nite number of totally geodesic submanifoldsN 1 ... . If N is one component of the xed point set, the normal bundle is the orthogonal complement ofT (N) in T (M)j N . is invariant underdT and det(I ; dT ) > 0 so T is non-degenerate.Proof: Let a>0 be the injectivity radius of M so that if dist(x y)

Lefschetz Fixed Point Formulas 285set. This shows the xed point set is totally geodesic. Fix T (x) = x anddecompose T (M) x = V 1 where V 1 = f v 2 T (M) x : dT (x)v = v g. SincedT (x) is orthogonal, both V 1 and are invariant subspaces and dT is anorthogonal matrix with no eigenvalue 1. Therefore det(I ; dT ) > 0. Let be a geodesic starting at x so 0 (0) 2 V 1 . Then T is a geodesic startingat x with T 0 (0) = dT (x) 0 (0) = 0 (0) so T = pointwise and exp x (V 1 )parametrizes the xed point set near x. This completes the proof.We let M be oriented and of even dimension m =2n. Let (d+): C 1 ( + )! C 1 ( ; ) be the operator of the signature complex. We letH (M C) =N( ) on C 1 ( ) so that signature(M) = dim H + ; dim H ; . If T is anorientation preserving isometry,thenT d = dT and T = T where \"denotes the Hodge operator. Therefore T induces maps T on H (M C).We dene:L(T ) signature =Tr(T + ) ; Tr(T ; ):Let A = dT 2 SO(m) at an isolated xed point. Denedefect(A signature) =fTr( + (A)) ; Tr( ; (A))g= det(I ; A)as the contribution from Lemma 1.8.3. We wish to calculate this characteristicpolynomial. If m =2,letfe 1 e 2 g be an oriented orthonormal basisfor T (M) =T (M) such thatAe 1 = (cos )e 1 + (sin )e 2 and Ae 2 = (cos )e 2 ; (sin )e 1 :The representation spaces are dened by: + = spanf1+ie 1 e 2 e 1 + ie 2 g and ; =spanf1 ; ie 1 e 2 e 1 ; ie 2 gso that:Tr( + (A)) = 1 + e ;i Tr( ; (A)) = 1 ; e ;i det(I ; A) =2; 2 cos and consequently:defect(A signature) = (;2i cos )=(2 ; 2 cos ) =;i cot(=2):More generally let m =2n and decompose A = dT into a product of mutuallyorthogonal and commuting rotations through angles j correspondingto complex eigenvalues j = e i jfor 1 j n. The multiplicative natureof the signature complex then yields the defect formula:defect(A signature) =nYj=1f;i cot( j =2)g =nYj=1 j +1 j ; 1 :

286 4.5. Lefschetz Fixed Point FormulasThis is well dened since the condition that the xed point be isolated isjust 0 < j < 2 or equivalently j 6=1.There are similar characteristic polynomials for the other classical ellipticcomplexes. For the de Rham complex, we noted in Chapter 1 thecorresponding contribution to be 1 = sign(det(I ; A)). If M is spin, letA: C 1 ( + ) ! C 1 ( ; ) be the spin complex. If T is an isometry whichcan be lifted to a spin isometry, we can dene L(T ) spin to be the Lefschetznumber of T relative to the spin complex. This lifts A from SO(m) toSPIN(m). We dene:defect(A spin) =fTr( + (A) ; Tr( ; (A))g= det(I ; A)and use Lemma 3.2.5 to calculate if m =2that:defect(A spin) =(e ;i=2 ; e i=2 )=(2 ; 2 cos )= ;i sin(=2)=(1 ; cos ) =; i 2 cosec(=2)=np p o ; =(2 ; ; )= p =( ; 1):Using the multiplicative nature we get a similar product formula in general.Finally, let M be a holomorphic manifold and let T : M ! M be aholomorphicPmap. Then T and @ commute. We let L(T ) Dolbeault =q (;1)q Tr(T on H 0q ) be the Lefschetz number of the Dolbeault complex.Just because T has isolated xed points does not imply that it isnon-degenerate the map z 7! z +1denes a map on the Riemann sphereS 2 which has a single isolated degenerate xed point at 1. We supposeA = L(T ) 2 U( m ) is in fact non-degenerate and dene2defect(A Dolbeault) =fTr( 0even (A)) ; Tr( 0odd (A))g det(I ; A real ):If m =2,it is easy to calculatedefect(A Dolbeault) =(1; e i )=(2 ; 2 cos ) =(1; )=(2 ; ; )= =( ; 1)with a similar multiplicative formula for m > 2. We combine these resultwith Lemma 1.8.3 to derive the classical Lefschetz xed point formulas:Theorem 4.5.2. Let T : M ! M be a non-degenerate smooth map withisolated xed points at F (T )=fx 1 ...x r g. Then(a) L(T ) de Rham = P j sign(det(I ; dT ))(x j).(b) Suppose T is an orientation preserving isometry. Dene:defect(A signature) = Y jf;i cot j g = Y j j +1 j ; 1

Lefschetz Fixed Point Formulas 287P then L(T ) signature = j defect(dT (x j) signature).(c) Suppose T is an isometry preserving a spin structure. Dene:Y defect(A spin) = ; i Y pj2 cosec( j=2) =jj j ; 1 then L(T ) spin = P j defect(dT (x j) spin).(d) Suppose T is holomorphic. Dene:defect(A Dolbeault) = Y j j j ; 1 then L(T ) Dolbeault= P j defect(dT (x j) Dolbeault).We proved (a) in Chapter 1. (b){(d) follow from the calculations we havejust given together with Lemma 1.8.3. In dening the defect, the rotationangles of A are f j g so Ae 1 = (cos )e 1 + (sin )e 2 and Ae 2 =(; sin )e 1 +(cos )e 2 . The corresponding complex eigenvalues are f j = e i jg where1 j m=2. The formula in (a) does not depend on the orientation. Theformula in (b) depends on the orientation, but not on a unitary structure.The formula in (c) depends on the particular lift of A to spin. The formulain (d) depends on the unitary structure. The formulas (b){(d) are allfor even dimensional manifolds while (a) does not depend on the parityof the dimension. There is an elliptic complex called the PIN c complexdened over non-orientable odd dimensional manifolds which also has aninteresting Lefschetz number relative to an orientation reversing isometry.Using Lemmas 1.8.1 and 1.8.2 it is possible to get a local formula for theLefschetz number concentrated on the xed pont set even if T has higherdimensional xed point sets. A careful analysis of this situation leads to theG-signature theorem in full generality. We refer to the appropriate papersof (Gilkey, Kawasaki, and Donnelly) for details regarding the signature andspin complexes. We shall discuss the case of the de Rham complex in somedetail.The interesting thing about the Dolbeault complex is that Theorem 4.5.2does not generalize to yield a corresponding formula in the case of higherdimensional xed point sets in terms of characteristic classes. So far, ithas not proven possible to identify the invariants of the heat equationwith generalized cohomology classes in this case. The Atiyah-Singer indextheorem in its full generality also does not yield such a formula. If oneassumes that T is an isometry of a Kaehler metric, then the desired resultfollows by passing rst to the SPIN c complex as was done in Chapter 3.However, in the general case, no heat equation proof of a suitable generalizationis yet known. We remark that Toledo and Tong do have a formula

288 4.5. Lefschetz Fixed Point Formulasfor L(T ) Dolbeault in this case in terms of characteristic classes, but theirmethod of proof is quite dierent.Before proceeding to discuss the de Rham complex in some detail, wepause to give another example:Example 4.5.3: Let T 2 be the 2-dimensional torus S 1 S 1 with usualperiodic parameters 0 x y 1. Let T (x y) = (;y x) be a rotationthrough 90 . Then(a) L(T ) de Rham =2.(b) L(T ) signature =2i.(c) L(T ) spin =2i= p 2.(d) L(T ) Dolbeault =1; i.Proof: We use Theorem 4.5.2 and notice that there are two xedpointsat (0 0) and (1=2 1=2). We could also compute directly using the indicatedaction on the cohomology groups. This shows that although thesignature of a manifold is always zero if m 2 (4), there do exist nontrivialL(T ) signature in these dimensions. Of course, there are many otherexamples.We now consider the Lefschetz xed point formula for the de Rham complexif the xed point set is higher dimensional. Let T : M ! M be smoothand non-degenerate. We assume for the sake of simplicity forthe momentthat the xed point set of T has only a single component N of dimensionn. The general case will be derived by summing over the components ofthe xed point set. Let be the sub-bundle of T (M)j N spanned by thegeneralized eigenvectors of dT corresponding to eigenvalues other than 1.We choose the metric on M so the decomposition T (M)j N = T (N) isorthogonal. We further normalize the choice of metric by assumingthatNis a totally goedesic submanifold.Let a k (x T p ) denote the invariants of the heat equation discussed inLemma 1.8.1 so that1X ZTr(T e ;tp ) t (k;n)=2 a k (x T p )dvol(x):k=0Let a k (x T ) de Rham = P (;1) p a k (x T p ), then Lemma 1.8.2 implies:L(T ) de Rham 1Xk=0t (k;n)=2 ZNNa k (x T ) de Rham dvol(x):Thus:ZN 0a k (x T ) de Rham dvol(x) =L(T ) de Rhamif k 6= dimNif k =dimN:

Lefschetz Fixed Point Formulas 289We choose coordinates x = (x 1 ...x n ) on N. We extend these coordinatesto a system of coordinates z = (x y) for M near N wherey =(y 1 ...y m;n ). We adopt the notational convention:indices 1 i j k mindex a frame for T (M)j N1 a b c n index a frame for T (N)n 0, jj > 1 with coecients c(g ij T i=j )j det(I ; dT )j ;1 where thec(g ij T i=j ) are smooth in the g ij and T i=j variables. The results of section1.8 imply a k (x T ) de Rham 2 T . If Z is a coordinate system, if G is ametric, and if T is the germ of a non-degenerate smooth map we canevaluate p(Z T G)(x) for p 2 T and x 2 N. p is said to be invariant ifp(Z T G)(x) =p(Z 0 TG)(x) for any two coordinate systems Z, Z 0 of thisform. We let T mn be the sub-algebra of T of all invariant polynomialsand we let T mnk be the sub-space of all invariant polynomials which arehomogeneous of degree k using the grading dened above. L It is not dicultto show there is a direct sum decomposition T mn = k T mnk as a gradedalgebra.Lemma 4.5.4. a k (x T ) de Rham 2T mnk .Proof: The polynomial dependence upon the jets involved together withthe form of the coecients follows from Lemma 1.8.1. Since a k (x T ) de Rhamdoes not depend upon the coordinate system chosen, it is invariant. Wecheck the homogeneity using dimensional analysis as per usual. If we replacethe metric by a new metric c 2 G, then a k becomes c ;k a k . We replacethe coordinate system Z by Z 0 = cZ to replace g ij= by c ;jj g ij= and T i=by c ;jj+1 T i= . This completes the proof. The only feature dierent fromthe analysis of section 2.4 is the transformation rule for the variables T i= .The invariants a k are multiplicative.Lemma 4.5.5. Let T 0 : M 0 ! M 0 be non-degenerate with xed submanifoldN 0 . Dene M = S 1 M 0 , N = S 1 N 0 , and T = I T 0 . Then

290 4.5. Lefschetz Fixed Point FormulasT : M ! M is non-degenerate with xed submanifold N. We give M theproduct metric. Then a k (x T ) de Rham =0.Proof: We may decompose p (M) = 0 (S 1 ) p (M 0 ) 1 (S 1 ) p;1 (M 0 ) ' p (M 0 ) p;1 (M 0 ):This decomposition is preserved by the map T . Under this decomposition,the Laplacian M p splits into 0 p 00 p. The natural bundle isomorphismidenties 00 p = 0 p;1. Since a k (x T M p )=a k (x T 0 p)+a k (x T 00 p) =a k (x T 0 p)+a k (x T 0 p;1), the alternating sum dening a k (x T ) de Rhamyields zero in this example which completes the proof.We normalize the coordinate system Z by requiring that g ij (x 0 ) = ij .There are further normalizations we shall discuss shortly. There is a naturalrestriction map r: T mnk !T m;1n;1k dened algebraically by: 0 if deg1 (g ij= ) 6= 0r(g ij= )=gij= 0 if deg 1 (g ij= )=0 0 if deg1 (T i= ) 6= 0r(T i= )=Ti= 0 if deg 1 (T i= )=0:In this expression, we let g 0 and T 0 denote the renumbered indices to referto a manifold of one lower dimension. (It is inconvenient to have used thelast m ; n indices for the normal bundle at this point, but again this is thecannonical convention which we have chosen not to change). In particularwe note that Lemma 4.5.5 implies r(a k (x T ) de Rham )=0. Theorem 2.4.7generalizes to this setting as:Lemma 4.5.6. Let p 2T mnk be such that r(p) =0.(a) If k is odd or if k

Lefschetz Fixed Point Formulas 291function h. We know h(dT ) = sign(det(I ; dT )) if n = m by Theorem1.8.4. The multiplicative nature of the de Rham complex with respect toproducts establishes this formula in general which proves (b). SinceL(T ) de Rham = X sign(det(I ; dT ))ZN E n dvol(x )by Theorem 1.8.2, (c) follows from the Gauss-Bonnet theorem already established.This completes the proof of Theorem 4.5.7.Before beginning the proof of Lemma 4.5.6, we must further normalizethe coordinates being considered. We have assumed that the metric chosenmakes N a totally geodesic submanifold. (In fact, this assumption isinessential, and the theorem is true in greater generality. The proof, however,is more complicated in this case). This implies that we can normalizethe coordinate system chosen so that g ij=k (x 0 )=0for 1 i j k m. (Ifthe submanifold is not totally geodesic, then the second fundamental formenters in exactly the same manner as it did for the Gauss-Bonnet theoremfor manifolds with boundary).By hypothesis, we decomposed T (M)j N = T (N) and decomposeddT = I dT . This implies that the Jacobian matrix T i=j satises:Consequently:T a=b = a=b T a=u = T u=a =0along N:T a=N =0for j j < 2 and T u=N =0for j j =0:Consequently, for the non-zero T i= variables:Xandeg a (T i= ) ord(T i= )=jj;1:Let p satisfy the hypothesis of Lemma 4.5.6. Let p 6= 0 and let A be amonomial of p. We decompose:A = f(dT ) A 1 A 2whereA1 = g i1 j 1 = 1...g ir j r = rA 2 = A k1 = 1...A ks = sfor j j2for j j2:Since r(p) =0, deg 1 A 6= 0. Since p is invariant under orientation reversingchanges of coordinates on N, deg 1 A must be even. Since p is invariantunder coordinate permutations on N, deg a A 2 for 1 a n. We nowcount indices:2n Xdeg a (A) =Xdeg a (A 1 )+X1an1an1andeg a (A 2 ) 2r +ord(A 1 ) + ord(A 2 ) 2 ord(A 1 )+2ord(A 2 )=2ord(A):

292 4.5. Lefschetz Fixed Point FormulasThis shows p =0if ord(A)

Lefschetz Fixed Point Formulas 293P . Since g commutes with the operator A, it commutes with the boundaryconditions. Let L(P B g) be the Lefschetz number of this problem.Theorems 4.3.11 and 1.8.3 generalize to this setting to become:Theorem 4.5.8 (Donnelly). Let N be a compact Riemannian manifoldwith boundary M. Let the metric on N be product near M. LetP : C 1 (V 1 ) ! C 1 (V 2 ) be a rst order elliptic dierential complex overN. Assume near M that P has the form P = p(dn)(@=@n + A) where Ais a self-adjoint elliptic L tangential operator on C 1 (V 1 ) over the boundaryM. Let L 2 (V 1 j M ) = E() be a spectral resolution of A and let B beorthonormal projection on the non-negative spectrum of A. (P B) is anelliptic boundary value problem. Assume given an isometry g: N ! Nwith isolated xed points x 1 ...x r in the interior of N. Assume given anaction g on V i so gP = Pg. Then gA = Ag as well. Dene:(s A g)= X sign()jj ;s Tr(g on E()):This converges for Re(s) 0 and extends to an entire function of s. LetL(P B g) denote the Lefschetz number of g on this elliptic complex andletdefect(P g)(x i )=f(Tr(g on V 1 ) ; Tr(g on V 2 ))= det(I ; dT )g(x i )then:L(P B g)= Xidefect(P G)(x i ) ;2 1 (0Ag)+Tr(g on N(A)) :Remark: Donnelly's theorem holds in greater generality as one does notneed to assume the xed points are isolated and we refer to (Donnelly, Theeta invariant of G-spaces) for details.We use this theorem to compute the eta invariant on the quotient manifoldM = M=G. Equivariant eigensections for A over M correspond to theeigensections of A over M. Then:~( A)= 1 2 f(0 A)+dimN( A)g= 1jGj 1 X 2 f(0Ag)+Tr(g on N(A))gg2G= 1jGjXg2G+ 1jGjf;L(P B g)g + 1jGjXg2Gg6=IXx2F (g)ZNdefect(P g)(x):(a n (x P P ) ; a n (x P P )) dx

294 4.5. Lefschetz Fixed Point FormulasThe rst sum over the group gives the equivariant index of P with thegiven boundary condition. This is an integer and vanishes in R mod Z.The second contribution arises from Theorem 4.3.11 for g = I. The nalcontribution arises from Theorem 4.5.8 for g 6= I and we sumover the xedpoints of g (which may bedierent for dierent group elements).If we suppose dim M is even, then dim N is odd so a n (x P P ) ;a n (x P P ) = 0. This gives a formula for ( A) over M in terms of thexed point data on N. By replacing V i by V i 1 k and letting G act by arepresentation of G in U(k), we obtain a formula for ( A ).If dim M is odd and dim N is even, the local interior formula forindex(P B) given by the heat equation need not vanish identically. Ifwe twist by a virtual representation , we alter the defect formulas bymultiplying by Tr((g)). The contribution from Theorem 4.3.11 is multipliedby Tr((1)) = dim . Consequently this term disappears if is arepresentation of virtual dimension 0. This proves:Theorem 4.5.9. Let N be a compact Riemannian manifold with boundaryM. Let the metric on N be product near M. Let P : C 1 (V 1 ) ! C 1 (V 2 )be a rst order elliptic dierential complex over N. Assume near M that Phas the form P = p(dn)(@=@+A) where A is a self-adjoint elliptic tangentialoperator on C 1 (V 1 ) over M. Let G be a nite group acting by isometrieson N. Assume for g 6= I that g has only isolated xed points in the interiorof N, and let F (g) denote the xed point set. Assume given an action onV i so gP = Pg and gA = Ag. Let M = M=G be the quotient manifoldand A the induced self-adjoint elliptic operator on C 1 ( V1 = V 1 =G) overM. Let 2 R(G) be a virtual representation. If dim M is odd, we assumedim =0. Then:for~( A )= 1jGjXg2Gg6=IXs2F (g)Tr((g)) defect(P g)(x)mod Zdefect(P g)(x)=f(Tr(g on V 1 ) ; Tr(g on V 2 ))= det(I ; dg)g(x):We shall use this result in the next section and also in section 4.9 todiscuss the eta invariant for sherical space forms.

4.6. The Eta Invariant and the K-Theory ofSpherical Space Forms.So far, we have used K-theory as a tool to prove theorems in analysis.K-theory and the Chern isomorphism have played an important role in ourdiscussion of both the index and the twisted index theorems as well as inthe regularity of eta at the origin. We nowreverse the process and will useanalysis as a tool to compute K-theory. We shall discuss the K-theory ofspherical space forms using the eta invariant to detect the relevant torsion.In section 4.5, Corollary 4.5.9, we discussed the equivariant computationof the eta invariant. We apply this to spherical space forms as follows. LetG be a nite group and let : G ! U(l) be a xed point free representation.We suppose det(I ;(g)) 6= 0forg 6= I. Such a representation is necessarilyfaithful the existence of such a representation places severe restrictions onthe group G. In particular, all the Sylow subgroups for odd primes must becyclic and the Sylow subgroup for the prime 2 is either cyclic or generalizedquaternionic. These groups have all been classied by (Wolf, Spaces ofConstant Curvature) and we refer to this work for further details on thesubject.(G) acts without xed points on the unit sphere S 2l;1 in C l . Let M =M() =S 2l;1 =(G). We suppose l>1 so, since S 2l;1 is simply connected, induces an isomorphism between G and 1 (M). M inherits a naturalorientation and Riemannian metric. It also inherits a natural Cauchy-Riemann and SPIN c structure. (M is not necessarily a spin manifold). Themetric has constant positive sectional curvature. Such a manifold is calleda spherical space form all odd dimensional compact manifolds withoutboundary admitting metrics of constant positive sectional curvature arisein this way. The only even dimensional spherical space forms are the sphereS 2l and the projective space RP 2l . We concentrate for the moment ontheodd dimensional case we will return to consider RP 2l later in this section.We have the geometrical argument:T (S 2l;1 ) 1=T (R 2l )j S 2l;1= S 2l;1 R 2l = S 2l;1 C lis the trivial complex bundle of dimension l. The dening representation is unitary and acts naturally on this bundle. If V is the locally atcomplex bundle over M() dened by the representation of 1 (M()) = G,then this argument shows(V ) real = T (M()) 1this is, of course, the Cauchy-Riemann structure refered to previously. Inparticular V admits a nowhere vanishing section so we can split V = V 1 1where V 1 is an orthogonal complement ofthe trivial bundle correspondingto the invariant normal section of T (R 2l )j S 2l;1 . ThereforeX(;1) (V )= X (;1) (V 1 1) = X (;1) f (V 1 ) ;1 (V 1 )g =0

296 4.6. The Eta InvariantinPK(M). This bundle corresponds to the virtual representation = (;1) () 2 R 0 (G). This proves:Lemma 4.6.1. Let be a xed point free representation P of a nite groupG in U(l) and let M() =S 2l;1 =(G). Let = (;1) () 2 R 0 (G).Then:T (M()) 1=(V ) real and V =0in e K(M()):The sphere bounds a disk D 2l in C l . The metric is not product nearthe boundary of course. By making a radial change of metric, we can puta metric on D 2l agreeing with the standard metric at the origin and witha product metric near the boundary so that the action of O(2l) continuesto be an action by isometries. The transition functions of V are unitaryso T (M()) inherits a natural SPIN c structure. Let = signature or Dolbeaultand let A be the tangential operator of the appropriate ellipticcomplex over the disk. (G) acts on D 2l and the action extends to an actionon both the signature and Dolbeault complexes. There is a single xedpoint at the origin of the disk. Let defect((g) ) denote the appropriateterm from the Lefschetz xed point formulas. Let f g denote the complexeigenvalues of (g), and let (g) r denote the corresponding element ofSO(2l). It follows from section 4.5 that:defect((g) signature)= Y defect((g) Dolbeault)= +1 ; 1 det((g))det((g) ; I) = Y ; 1 :We apply Corollary 4.5.9 to this situation to compute:4.6.2. Let : G ! U(l) be a xed point free representation of a nitegroup. Let M() =S 2l;1 =(G) be a spherical space form. Let = signatureor Dolbeault and let A be the tangential operator of the appropriateelliptic complex. Let 2 R 0 (G) be a virtual representation of dimension0. Then:~((A ) )= 1jGjXg2Gg6=ITr((g)) defect((g) ) in R mod Z.Remark: A priori, this identity is in R mod Z. It is not dicult to showthat this generalized Dedekind sum is always Q mod Z valued and thatjGj l ~ 2 Z so one has good control on the denominators involved.The perhaps somewhat surprising fact is that this invariant is polynomial.Suppose G = Z n is cyclic. Let x = (x 1 ...x l ) be a collection of

K-Theory of Spherical Space Forms 297indeterminates. Let Td and L be the Todd and Hirzebruch polynomialsdiscussed previously. We dene:Td 0 (~x) =1 L 0 (~x) =1Td 1 (~x) = 1 2Xxj L 1 (~x) =0Td 2 (~x) = 1 12 Xj

298 4.6. The Eta Invariantwhich is the formula obatined previously in section 4.4.Proof: We refer to (Gilkey, The eta invariant and the K-theory of odddimensional spherical space forms) for the proof it is a simple residuecalculation using the results of Hirzebruch-Zagier.This is a very computable invariant and can be calculated using a computerfrom either Lemma 4.6.2 or Lemma 4.6.3. Although Lemma 4.6.3 atrst sight only applies to cyclic groups, it is not dicult to use the Brauerinduction formula and some elementary results concerning these groups toobtain similar formulas for arbitrary nite groups admitting xed pointfree representations.Let M be a compact manifold without boundary with fundamental groupG. If is a unitary representation of G and if P is a self-adjoint ellipticdierential operator, we have dened the invariant ~(P ) as an R mod Zvalued invariant. (In fact this invariant can be dened for arbitrary representationsand for elliptic pseudo-dierential operators with leading symbolhaving no purely imaginary eigenvalues on S(T M) and most of what wewill say will go over to this more general case. As we are only interested innite groups it suces to work in this more restricted category).Let R(G) be the group representation ring of unitary virtual representationsof G and let R 0 (G) be the ideal of virtual representations of virtualdimension 0. The map 7! V denes a ring homomorphism from R(G) toK(M) andR 0 (G) to e K(M). We shall denote the images by Kat (M) andeK at (M) these are the rings generated by virtual bundles admitting locallyat structures, or equivalently by virtual bundles with constant transitionfunctions. Let P be a self-adjoint and elliptic dierential oprator. Themap 7! ~(A ) is additive with respect to direct sums and extends to amap R(G) ! R mod Z as already noted. We let ind( P ) be the mapform R 0 (G) to R mod Z. This involves a slight change of notation fromsection 4.4 if is a representation of G, thenind( ; dim() 1P)denotes the invariant previously dened by ind( P ). This invariant is constantunder deformations of P within this class the Atiyah-Patodi-Singerindex theorem for manifolds with boundary (Theorem 4.3.11) implies it isalso an equivariant cobordism invariant. We summarize its relevant properties:Lemma 4.6.4. Let M be a compact smooth manifold without boundary.Let P be a self-adjoint elliptic dierential operator over M and let 2R 0 ( 1 (M)).(a) Let P (a) be a smooth 1-parameter family of such operators, thenind( P (a)) is independent ofa in R mod Z. If G is nite, this is Q mod Zvalued.

K-Theory of Spherical Space Forms 299(b) Let P be a rst order operator. Suppose there exists a compact manifoldN with dN = M. Suppose there is an elliptic complex Q: C 1 (V 1 ) !C 1 (V 2 ) over N so P is the tangential part of Q. Suppose the virtual bundleV can be extended as a locally at bundle over N. Then ind( P )=0.Proof: The rst assertion of (a) follows from Lemma 4.4.1. The locallyat bundle V is rationally trivial so by multipying by a suitable integerwe can actually assume V corresponds to a at structure on the dierenceof trivial bundles. The index is therefore given by a local formula. If welift to the universal cover, we multiply this local formula by jGj. On theuniversal cover, the index vanishes identically as 1 =0. Thus an integermultiple of the index is 0 in R=Z so the index is in Q=Z which proves(a). To prove (b) we take the operator Q with coecients in V . Thelocal formula of the heat equation is just multiplied by the scaling constantdim() = 0 since V is locally at over N. Therefore Theorem 4.3.11yields the identity index(Q B coe in V )=0; ind( P ). As the index isalways an integer, this proves (b). We will use (b) in section 4.9 to discussisospectral manifolds which are not dieomorphic.Examle 4.4.2 shows the index is not an invariant in K-theory. We getK-theory invariants as follows:Lemma 4.6.5. Let M be a compact manifold without boundary. Let Pbe an elliptic self-adjoint pseudo-dierential operator. Let 2 R 0 ( 1 (M))and dene the associative symmetric bilinear form on R 0 R 0 by:ind( 1 2 P) = ind( 1 2 P):This takes values in Q mod Z and extends to an associative symmetricbilinear form ind( P): e Kat (M) e Kat (M) ! Q mod Z. If we considerthe dependence upon P ,thenwe get a trilinear mapind: e Kat (M) e Kat (M) K(S(T M))=K(M) ! Q mod Z:Proof: The interpretation of the dependence in P as a map in K-theoryfollows from 4.3.3 and is therefore omitted. Any virtual bundle admittinga locally at structure has vanishing rational Chern character and mustbe a torsion class. Once we have proved the map extends to K-theory, itwill follow it must be Q mod Z valued. We suppose given representations 1 , ^ 1 , 2 and a bundle isomorphism V 1 = V^1 . Let j = dim( 1 ) andk = dim( 2 ). If we can showind(( 1 ; j) ( 2 ; k)P) = ind((^ 1 ; j) ( 2 ; k)P)then the form will extend to e Kat (M) e Kat (M) and the lemma will beproved.

300 4.6. The Eta InvariantWe calculate that:ind(( 1 ; j) ( 2 ; k)P)= ~(P 1 2)+j k ~(P ) ; j ~(P 2 ) ; k ~(P 1 )= ~f(P 1 ) 2 g;k ~(P 1 ) ; j f~(P 2 ) ; k ~(P )g= ind( 2 P 1 ) ; j ind( 2 P):By hypothesis the bundles dened by 1 and ^ 1 are isomorphic. Thusthe two operators P 1 and P^1 are homotopic since they have the sameleading symbol. Therefore ind( 2 P 1 ) = ind( 2 P^1 )which completes theproof. We remark that this bilinear form is also associative with respect tomultiplication by R(G) and K at (M).We will use this lemma to study the K-theory of spherical space forms.Lemma 4.6.6. Let be a xed point free representation of a nite groupG. Deneind ( 1 2 )= 1jGjXg2Gg6=Idet((g))Tr( 1 2 )(g)det((g) ; I) :Let = P (;1) () 2 R 0 (G). Let 1 2 R 0 (G) and suppose ind ( 1 2 )=0in Q mod Z for all 2 2 R 0 (G). Then 2 2 R(G).Proof: The virtual representation is given by the dening relation thatTr((g)) = det(I ; (g)). det() denes a 1-dimensional representationof G as this is an invertible element of R(G) we see that the hypothesisimplies1jGjXg2Gg6=ITr( 1 2 )= det(I ; (g)) 2 Zfor all 2 2 R 0 (G):If f and f ~ are any two class functions on G, we dene the symmetric innerproduct (f f)= ~ PjGj1g2G f(g) f(g). ~ The orthogonality relations showthatf is a virtual character if and only if (fTr()) 2 Z for all 2 R(Z). Wedene:f(g)=Tr( 1 (g))= det(I ; (g)) for g 6= If(g)=; X h2Gh6=ITr( 1 (h))= det(I ; (h)) if g = I:Then (f1) = 0 by denition. As (f 2 ) 2 Z by hypothesis for 2 2 R 0 (G)we see (f 2 ) 2 Z for all 2 2 R(G) so f is a virtual character. We letTr()(g) =f(g). The dening equation implies:Tr( )(g) =Tr( 1 (g)) for all g 2 G:

K-Theory of Spherical Space Forms 301This implies 1 = and completes the proof.We can now compute the K-theory of odd dimensional spherical spaceforms.Theorem 4.6.7. Let : G ! U(l) be a xed point free representation of anite group G. Let M() =S 2l;1 =(G) be a spherical space form. Supposel>1. Let = P (;1) () 2 R 0 (G). Then e Kat (M) =R 0 (G)=R(G)andind( A Dolbeault ): e Kat (M) e Kat (M) ! Q mod Zis a non-singular bilinear form.Remark: It is a well known topological fact that for such spaces e K = e Katso we are actually computing the reduced K-theory of odd dimensionalspherical space forms (which was rst computed by Atiyah). This particularproof gives much more information than just the isomorphism and wewill draw some corollaries of the proof.Proof: We have a surjective map R 0 (G) ! e Kat (M). By Lemma 4.6.1we have 7! 0 so R(G) is in the kernel of the natural map. Conversely,suppose V =0in e K. By Lemma 4.6.5 wehave ind( 1 A Dolbeault )=0forall 1 2 R 0 (G). Lemma 4.6.2 lets us identify this invariant with ind ( 1 ).Lemma 4.6.6 lets us conclude 2 R(G). This shows the kernel of thismap is precisely R(G)whichgives the desired isomorphism. Furthermore, 2 ker(ind ( )) if and only if 2 R(G) if and only if V = 0 so thebilinear form is non-singular on e K.It is possible to prove anumber of other results about the K-theory ringusing purely group theoretic methods the existence of such a non-singularassociative symmetric Q mod Z form is an essential ingredient.Corollary 4.6.8. Adopt the notation of Theorem 4.6.7.(a) e K(M) only depends on (G l) as a ring and not upon the particular chosen.(b) The index of nilpotency for this ring is at most l|i.e., if 2 R 0 (G)then Q 1l 2 R 0 (G) so the product of l virtual bundles of e K(M)always gives 0 in e K(M).(c) Let V 2 e K(M). Then V = 0 if and only if (V ) = 0 for all possiblecovering projections : L(n ~q ) ! M by lens spaces.There is, of course, a great deal known concerning these rings and wereferto (N. Mahammed, K-theorie des formes spheriques) for further details.If G = Z 2 ,thenthe resulting space is RP 2l;1 which is projective space.There are two inequivalent unitary irreducible representations 0 , 1 ofG. Let x = 1 ; 0 generate R 0 (Z 2 ) = Z the ring structure is given by

302 4.6. The Eta Invariantx 2 = ;2x. Let A be the tangential operator of the Dolbeault complex:ind(x A)= 1 2 Tr(x(;1)) det(;I l)= det((;I) l ; I l )= ;2 ;lusing Lemma 4.6.2. Therefore ind(x x A) = 2 ;l+1 which implies thateK(RP 2l;1 ) = Z=2 l;1 Z. Let L = V 1 be the tautological bundle overRP 2l;1 . It is S 2l;1 C modulo the relation(x z) = (;x ;z). UsingCliord matrices we can construct a map e: S 2l;1 ! U(2 l;1 ) such thate(;x) = ;e(x). This gives an equivariant trivialization of S 2l;1 C 2l;1which descends to give a trivialization of 2 l;1 L. This shows explicitlythat 2 l;1 (L ; 1) = 0 in K(RP 2l;1 ) the eta invariant is used to show nolower power suces. This proves:Corollary 4.6.9. Let M = RP 2l;1 then e K(M) ' Z=2l;1Z where thering structure is x 2 = ;2x.Let l =2and let G = Z n be cyclic. Lemma 4.6.3 showsind( s ; 0 A Dolbeault )= ;q0n s2 2 :Let x = 1 ; 0 so x 2 = 2 ; 0 ; 2( 1 ; 0 ) andind(x x A Dolbeault )= ;q0n 4 ; 22is a generator of Z[ 1 n ]modZ. As x2 =0in e K and as x R(G) =R0 (G) wesee:Corollary 4.6.10. Let M = L(n1q) be a lens space of dimension 3.eK(M) =Z n with trivial ring structure.We have computed the K-theory for the odd dimensional spherical spaceforms. e K(S 2l ) = Z and we gave a generator in terms of Cliord algebrasin Chapter 3. To complete the discussion, it suces to consider even dimensionalreal projective space M = RP 2l . As e Heven(M Q) = 0, e K ispure torsion by the Chern isomorphism. Again, it is known that the atand regular K-theory coincide. Let x = L ; 1 = V 1 ; V 0 . This is therestriction of an element of e K(RP2l+1) so 2 l x = 0 by Corollary 4.6.9. Itis immediate that x 2 = ;2x. We show e K(RP2l+1) = Z=2 l Z by giving asurjective map to a group of order 2 l .We construct an elliptic complex Q over the disk D 2l+1 . Let fe 0 ...e 2l gbe a collection of 2 l 2 l skew-adjoint matrices so e j e k +e k e j = ;2 jk . Up tounitary equivalence, the only invariant of such a collection is Tr(e 0 ...e 2l )=

K-Theory of Spherical Space Forms 303(2i) l . There are two inequivalent collections the other is obtained bytaking f;e 0 ... ;e 2l g. Let Q be the operator P j e j@=@x j acting to mapQ: C 1 (V 1 ) ! C 1 (V 2 ) where V i are trivial bundles of dimension 2 l overthe disk. Let g(x) = ;x be the antipodal map. Let g act by +1 on V 1and by ;1 on V 2 , then gQ = Qg so this is an equivariant action. Let Abe the tangential operator of this complex on S 2l and A the correspondingoperator on S 2l =Z 2 = M. A is a self-adjoint elliptic rst order operator onC 1 (1 2l ). If we replace Q by ;Q, the tangential operator is unchanged soA is invariantly dened independent of the choice of the fe j g. (In fact, Ais the tangential operator of the PIN c complex.)The dimension is even so we can apply Corollary 4.5.9 to conclude~( A)= 1 2 det(I 2l+1 ; (;I 2l+1 )) ;1 f2 l ; (;2 l )g =2 ;l;1 :If we interchange the roles of V 1 and V 2 , we change the sign of the etainvariant. This is equivalent to taking coecients in the bundle L = V 1 .Let x = L ; 1 then ind( 1 ; 0 A)=;2 ;l;1 ; 2 ;l;1 = ;2 ;l :The even dimensional rational cohomology of S(T M) is generated byH 0 and thus K(S(T M))=K(M) Q =0. Suppose 2 l;1 x =0in K(M). eThen there would exist a local formula for 2 l;1 ind( 1 ; 0 ) sowe couldlift this invariant from Q mod Z to Q. As this invariant is dened on thetorsion group K(S(T M))=K(M)itwould havetovanish. As 2 l;1 ind( 1 ; 0 A)=; 1 does not vanish, we conclude 2 2l;1 x is non-zero in K-theoryas desired. This proves:Corollary 4.6.11. K(RP e 2l ) ' Z=2 l Z. If x = L ; 1 is the generator,then x 2 = ;2x.We can squeeze a bit more out of this construction to compute theK-theory of the unit sphere bundle K(S(T M)) where M is a sphericalspace form. First suppose dim M =2l;1 isoddsoM = S 2l;1 =(G)where is a unitary representation. Then M has a Cauchy-Riemann structureand we can decompose T (M) =1V where V admits a complex structureT (M) 1 = (V ) real . Thus S(T M) has a non-vanishing section and theexact sequence 0 ! K(M) ! K(S(T M)) ! K(S(T M))=K(M) ! 0splits. The usual clutching function construction permits us to identifyK(S(T M))=K(M) with K(V ).As V admits a complex structure, theThom isomorphism identies K(V )=x K(M) where x is the Thom class.This gives the structure K(S(T M)) = K(M) xK(M). The bundle xover S(T M) can be taken to be + (p) where p is the symbol of the tangentialoperator of the Dolbeault complex. The index form can be regardedas a pairing e K(M) x e K(M) ! Q mod Z which is non-degenerate.It is more dicult to analyse the even dimensional case. We wish tocompute K(S(T RP 2l )). 2 l ind( 1 ; 0 ) is given by a local formula

304 4.6. The Eta Invariantsince 2 l x = 0. Thus this invariant must be zero as K(S(T M))=K(M) istorsion. We have constructed an operator A so ind( 1 ; 0 A)=;2 ;l andthus ind( 1 ; 0 ): K(S(T M))=K(M) ! Z[2 ;l ]=Z is surjective with thegiven range. This is a cyclic group of order 2 l so equivalentlyind( 1 ; 0 ): K(S(T M))=K(M) ! Z=2 l Z ! 0:Victor Snaith (private communication) has shown us that the existence ofsuch a sequence together with the Hirzebruch spectral sequence in K-theoryshows0 ! K(M) ! K(S(T M)) is exact and jK(S(T M))=K(M)j =2 lso that index( 1 ; 0 ) becomes part of a short exact sequence:0 ! K(M) ! K(S(T M)) ! Z=2 l Z ! 0:To compute the structure of K(S(T M)) = Z e K(S(T M)), we mustdetermine the group extension.Let x = L ; 1 = V 1 ; V 0 generate e K(M). Let y = + ( L A)) ;2 l;1 1, then this exact sequence together with the computation ind( 0 ; 1 A)=2 ;l shows e K(S(T M)) is generated by x and y. We know 2 l x =0 and that j e K(S(T M))j = 4 l to determine the additive structure ofthe group, we must nd the order of y. Consider the de Rham complex(d + ): C 1 ( even (D)) ! C 1 ( odd (D)) over the disk. The antipodal mapacts by +1 on e and by ;1 on o . Let A1 be the tangential operatorof this complex. We decompose (d + ) into 2 l operators each of whichis isomorphic to Q. This indeterminacy does not aect the tangentialoperator and thus A1 =2 l A. The symbol of A1 on even (D) is ;ic(dn)c() for 2 T (M). Letf even + odd g = even +c(dn) odd provide an isomorphism between (M)and even (Dj M ). We may regard A1 as an operator on C 1 ((M)) withsymbol a 1 given by:a 1 (x )( even + odd )= f ;1 ;ic(dn)c() gf even + odd g= f ;1 ;ic(dn)c()gf even + c(dn) odd g= ;1 f;ic(dn)c() even ; ic() odd g= ;ic()f even + odd gso that A1 = ;(d + ) on C 1 ((M)). Let "( odd ) = odd ; ic() oddprovide an isomorphism between odd (M) and + (a 1 ). This shows:2 l y = odd (M) ; 2 2l;1 1:

K-Theory of Spherical Space Forms 305Let (W ) = W ; dim(W ) 1 be the natural projection of K(M) oneK(M). We wish to compute ( odd (M)). As complex vector bundles wehave T (M) ; 1 =(2l +1) L so that we have the relation j (T M) j;1 (T 2l+1M)=j L j . This yields the identities:( j (T M)) + ( j;1 (T M)) = 0( j (T M)) + ( j;1 (T 2l +1M)) =j (L)if j is evenif j is odd.; Thus ( j (T M)) = ( j;2 (T 2l+1M)) +j (L) if j is odd. This leadsto the identity:n;( 2j+1 (T M)) = 2l+1 ;2j+1 + 2l+1 ;2j;1 + + 2l+1o1n ; ;( odd (T 2l+1M)) = l1 +(l ; 1) 2l+13 (L) + +; 2l+12l;1We complete this calculation by evaluating this coecient. Letf(t) = 1 ;(t +1) 2l+1 ; (t ; 1) 2l+12= t 2l; ; 2l+11 + t2l;2 2l+13f 0 (t) = 1 2 (2l +1); (t +1) 2l ; (t ; 1) 2l=2nt 2l;1 l ; 2l+11We evaluate at t =1to conclude:; 2l+1 1l + + t2 ; 2l+1+1 + t 2l;3 (l ; 1) ; 2l+13o (L): ; + + t 1 2l+1o:2l;1

306 4.6. The Eta Invariantind( ) gives a perfect pairing K(RP 2l ) K(X)=K(RP 2l ) ! Q mod Z.The generators of K(X) are f1xyg for x = L ; 1 and y = + (a).Remark: This gives the additive structure. We have x 2 = ;2x. We cancalculate x y geometrically. Let p(u) =i P v j e j . We regard p:1 v ! L vfor v = 2 l over RP 2 l. Let a L = a I on L v . Then pa L p = a L so (a) L = (a). Therefore:(L ; 1) + (a) ; 1 2l;1 = ;2 l;1 (L ; 1)+(L + (a) ; + (a))=2 l;1 (L ; 1) + ; (a) ; + (a)=2 l;1 (L ; 1) + + (a)+ ; (a) ; 2 + (a)=2 l;1 x ; 2yso that x y =2 l;1 x ; 2y. This gives at least part of the ring structurewe do not know a similar simple geometric argument tocompute y y.

4.7. Singer's Conjecture for the Euler Form.In this section, we will study a partial converse to the Gauss-Bonnettheorem as well as other index theorems. This will lead to informationregarding the higher order terms which appear in the heat equation. In alecture at M.I.T., I. M. Singer proposed the following question:Suppose that R P (G) is a scalar valued invariant of the metric such thatP (M) =MP (G)dvol is independent of the metric. Then is theresome universal constant c so that P (M) =c(M)?Put another way, the Gauss-Bonnet theorem gives a local formula for atopological invariant (the Euler characteristic). Is this the only theoremof its kind? The answer to this question is yes and the result is due toE. Miller who settled the conjecture using topological means. We alsosettled the question in the armative usinglocal geometry independentlyand we would like to present at least some the the ideas involved. If theinvariant is allowed to depend upon the orientation of the manifold, thenthe characteristic numbers also enter as we shall see later.We let P (g ij= ) be a polynomial invariant of the metric real analyticor smooth invariants can be handled similarly. We suppose P (M) =RMP (G)dvol is independent of the particular metric chosen on M.Lemma 4.7.1. Let P be a polynomial invariant of the metric tensor withcoecients which depend smoothly on the g ij variables. Suppose P (M) isindependent of the metric chosen. We decompose P = P P n for P n 2P mnhomogeneous of order n in the metric. Then P n (M) is independent of themetric chosen separately for each n P n (M) =0for n 6= m.Proof: This lets us reduce the questions involved to the homogeneouscase. If we replace the metric G by c 2 G then P n (c 2 G) = c ;n P n (G)by Lemma 2.4.4. Therefore R M P (c2 G)dvol(c 2 G) = P n cm;n RM P n(G).Since this is independent of the constant c,P m (M) =P (M) which completes the proof.RP n (M) = 0 for n m andIf Q is 1-form valued, we let P =divQ be scalar valued. It is clear thatMdiv Q(G)dvol(G) =0 so P (M) =0in this case. The following gives apartial converse:R Lemma 4.7.2. Let P 2 P mn for n 6= m satisfy P (M) =M P (G)dvolis independent of the metric G. Then there exists Q 2 P mn;11 so thatP = div Q.Proof: Since n 6= m, P (M) = 0. Let f(x) be a real valued functionon M and let G t be the metric e tf(x) G. If n = 0, then P is constant soP =0and the lemma is immediate. We assume n>0 henceforth. We let

308 4.7. Singer's ConjectureP (t) =P (e tf(x) G) and compute:ddt; P (e tf(x) G)dvol(e tf(x) G) = d ;P (e tf(x) G)e tmf(x)=2 dvol(G)dt= Q(G f)dvol(G):Q(fG) is a certain expression which is linear in the derivatives of thescaling function f. We letf i1 ...i pdenote the multiple covariant derivativesof f and let f :i1 ...i pdenote the symmetrized covariant derivatives of f. Wecan expressQ(fG) = X Q i1 ...i pf :i1 ...i pwhere the sum ranges over symmetric tensors of length less than n. Weformally integrate by parts to express:Q(fG) =divR(fG)+ X (;1) p Q i1 ...i p :i 1 ...i pf:By integrating over the structure group O(m) P we can ensure that thisprocess is invariantly dened. If S(G) = (;1) p Q i1 ...i p :i 1 ...i pthen theidentity:0=ZMQ(fG)dvol(G) =ZMS(G)f dvol(G)is true for every real valued function f. This implies S(G) =0soQ(fG) =div R(fG). We set f =1. Sincee tm=2 P (e t G)=e (m;n)t=2 P (G)div R(1G) which com-we conclude Q(1G)= m;n P (G) so P (G) = 22pletes the proof.m;nThere is a corresponding lemma for form valued invariants. The proof issomewhat more complicated and we refer to (Gilkey, Smooth invariants ofa Riemannian manifold) for further details:Lemma 4.7.3.(a) Let R P 2 P mnp . We assume n 6= p and dP = 0. If p = m, weassumeMP (G) is independent of G for every G on M. Then there existsQ 2P mn;1p;1 so that dQ = P .(b) Let R P 2 P mnp . We assume n 6= m ; p and P = 0. If p = 0 weassumeMP (G)dvol is independent of G for every G in M. Then thereexists Q 2P mn;1p+1 so that Q = P .Remark: (a) and (b) are in a sense dual if one works with SO(m)invarianceand not just O(m) invariance. We can use this Lemma together with theresults of Chapter 2 to answer the generalized Singer's conjecture:

For the Euler Form 309Theorem 4.7.4.R (a) Let P be a scalar valued invariant so that P (M) =MP (G)dvol isindependent of the particular metric chosen. Then we can decompose P =c E m + div Q where E m is the Euler form and where Q is a 1-form valuedinvariant. This implies P (M) =c(M).(b) Let P be a p-form R valued invariant so that dP = 0. If p = m, weassume P (M) =MP (G) is independent of the particular metric chosen.Then we can decompose P = R + dQ. Q is p ; 1 form valued and R is aPontrjagin form.P Proof: We decompose P = P j into terms which are homogeneous oforder j. Then each P j satises the hypothesis of Theorem 4.7.4 separatelyso wemay assume without loss of generalitythatP is homogeneous of ordern. Let P be as in (a). If n 6= m, then P =divQ be Lemma R 4.7.2. If n = m,we let P 1 = r(P ) 2 P m;1m . It is immediateM 1P 1 (G 1 )dvol(G 1 ) =1R2 S 1 M 1P (1 G 1 ) is independent of the metric G 1 so P 1 satises thehypothesis of (a) as well. Since m ; 1 6= m we conclude P 1 = div Q 1 forQ 1 2 P m;1m;11 . Since r is surjective, we can choose Q 2 P mm;11 sor(Q) =Q 1 . Therefore r(P ;div Q) =P 1 ;div Q 1 = 0 so by Theorem 2.4.7,P ;div Q = cE m for some constant c which completes the proof. (The factthat r : P mnp ! P m;1np ! 0 is, of course, a consequence of H. Weyl'stheorem so we are using the full force of this theorem at this point for therst time).If P is p-form valued, the proof is even easier. We decompose P into homogeneousterms and observe each term satises the hypothesis separately.If P is homogeneous of degree n 6= p then P = dQ be Lemma 4.7.3. If Pis homogeneous of degree n = p, then P is a Pontrjagin form by Lemma2.5.6 which completes the proof in this case.The situation in the complex catagory is not as satisfactory.Theorem 4.7.5. Let M be a holomorphic manifold R and let P be a scalarvalued invariant of the metric. Assume P (M) =MP (G)dvol is independentof the metric G. Then we can express P = R +divQ + E . Q is a1-form valued invariant and P = R 0 where R 0 is a Chern form. The additionalerror term E satises the conditions: r(E )=0and E vanishes forKaehler metric. Therefore P (M) is a characteristic number if M admits aKaehler metric.The additional error term arises because the axiomatic characterizationof the Chern forms given in Chapter 3 were only valid for Kaehler metrics.E in general involves the torsion of the holomorphic connection and to showdiv Q 0 = E for some Q 0 is an open problem. Using the work of E. Miller,it does follow that R E dvol = 0 but the situation is not yet completelyresolved.

310 4.7. Singer's ConjectureWe can use these results to obtain further information regarding thehigher terms in the heat expansion:Theorem 4.7.6.(a) Let a n (x d + ) denote the invariants of the heat equation for thede Rham complex. Then(i) a n (x d + ) =0if m or n is odd or if nm, then a n (x d + ) 6 0 in general. However,there does exist a 1-form valued invariant q mn so a n = div q mn andr(q mn )=0.(b) Let a sign n (x V ) be the invariants of the heat equation for the signaturecomplex with coecients in a bundle V . Then(i) a sign n (x V )=0for n

For the Euler Form 311Proof: We let Ae k be a monomial of P . We express A in the form:A = g i1 j 1 = 1...g ir j r = r:By Lemma 2.5.5, we could choose a monomial B of P with deg k (B) = 0for k > 2`(A) = 2r. Since r(Q) = 0, we conclude 2r m. On the otherhand, 2r P j j = m +1so we seem 2r m +1:Since m is even, we conclude 2r = m. This implies one of the j j = 3 whileall the other j j =2. We choose the notation so j 1 j = 3 and j j =2for > 1. By Lemma 2.5.1, we may choose A in the form:A = g ij=111 g i2 j 2 =k 2 l 2...g ir j r =k r l r:We rst suppose deg 1 (A) =3. This implies A appears in the expressionAe 1 in P so deg j (A) 2 is even for j > 1. We estimate 2m + 1 =4r +1= P j deg j(A) = 3+ P j>1 deg j(A) 3+2(m ; 1) = 2m +1toshow deg j (A) = 2 for j > 1. If m = 2, then A = g 22=111 which showsdim N(r) = 1 and Q is a multiple of dE 2 . We therefore assume m > 2.Since deg j (A) = 2 for j > 1, we can apply the arguments used to proveTheorem 2.4.7 to the indices j > 1 to show thatg 22=111 g 33=44 ...g m;1=mm e 1is a monomial of P .Next we suppose deg 1 (A) > 3. If deg 1 (A) is odd, then deg j (A) 2 iseven for j 2 implies 2m+1 = 4r+1 = P j deg j(A) 5+2(m;1) = 2m+3which is false. Therefore deg 1 (A) is even. We choose the notation in thiscase so Ae 2 appears in P and therefore deg 2 (A) is odd and deg j (A) 2forj > 2. This implies 2m +1=4r + 1 = deg j (A) 4+1+ P j>2 deg j(A) 5+2(m ; 2) = 2m +1. Since all the inequalities must be equalities weconcludedeg 1 (A) =4 deg 2 (A) =1 deg 3 (A) =2for j > 2:We apply the arguments of the second chapter to choose A of this formso that every index j > 2 which does not touch either the index 1 or theindex 2 touches itself. We choose A so the number of indices which touchthemselves in A is maximal. Suppose the index 2 touches some other indexthan the index 1. If the index 2 touches the index 3, then the index 3cannot touch itself in A. An argument using the fact deg 2 (A) = 1 andusing the arguments of the second chapter shows this would contradict themaximalityofA and thus the index 1 must touch the index 2 in A. We use

312 4.7. Singer's Conjecturenon-linear changes of coordinates to show g 12=111 cannot divide A. Usingnon-linear changes of coordinates to raplace g 44=12 by g 12=44 if necessary,we conclude A has the formA = g 33=111 g 12=44 g 55=66 ...g m;1m;1=mm :This shows that if m 4, then dim N(r: P mm+11 ! P m;1m+11 ) 2.Since these twoinvariants are linearly independent for m 4 this completesthe proof of the lemma.We apply the lemma and decomposea m+2 (x d + ) =c 1 (m)(E m ) kk + c 2 (m)div m :The c 1 (m) and c 2 (m) are universal constants defending only on the dimensionm. We consider a product manifold of the form M = S 2 M m;2 todene a map:r (2) : P mn0 ! M qnP m;2q0 :This restriction map does not preserve the grading since by throwing derivativesto the metric over S 2 ,we can lower the order of the invariantinvolved.Let x 1 2 S 2 and x 2 2 M m;2 . The multiplicative nature of the de Rhamcomplex implies:a n (x d + ) =Xj+k=ma j (x 1 d+ ) S 2 a k (x 2 d+ ) M m;2 :However, a j (x 1 d+ ) is a constant since S 2 is a homogeneous space. Therelations R S 2 a j (x 1 d+ ) =2 j2 implies thereforeso thata n (x d + ) = 12 a n;2(x 2 d+ )r (2) a m m+2 = 12 am;2 m :Since r (2) (E m ) kk =2 1 (E m;2) kk and r (2) div m =2 1 div m;2, we concludethat in fact the universal constants c 1 and c 2 do not depend upon m.(If m = 2, these two invariants are not linearly independent so we adjustc 1 (2) = c 1 and c 2 (2) = c 2 ). It is not dicult to use Theorem 4.8.16 whichwill be discussed in the next section to compute that if m =4,then:a 6 (x d + ) = 112 (E m) kk ; 1 6 div m:We omit the details of the verication. This proves:

For the Euler Form 313Theorem 4.7.8. Let m be even and let a m+2 (x d + ) denote the invariantof the Heat equation for the de Rham complex. Then:a m+2 = 112 (E m) kk ; 1 6 div m:Wegave a dierent proof of this result in (Gilkey,Curvature and the heatequation for the de Rham complex). This proof, due to Willis, is somewhatsimpler in that it uses Singer's conjecture to simplify the invariance theoryinvolved.There are many other results concerning the invariants which appear inthe heat equation for the de Rham complex. Gunther and Schimming havegiven various shue formulas which generalize the alternating sum denedpreviously. The combinatorial complexities are somewhat involved so weshall simply give an example of the formulas which can be derived.Theorem 4.7.9. Let m be odd. ThenX(;1) p (m ; p)a n (x m p )= 0 if n

4.8. Local Formulas for the InvariantsOf the Heat Equation.In this subsection, we will compute a n (x m p ) for n =0 1 2. In principle,the combinatorial formulas from the rst chapter could be used in thiscalculation. In practice, however, these formulas rapidly become much toocomplicated for practical use so we shall use instead some of the functorialproperties of the invariants involved.Let P : C 1 (V ) ! C 1 (V ) be a second order elliptic operator with leadingsymbol given by the metric tensor. This means we can express:P = ; Xijg ij @ 2 =@x i @x j + X kA k @=@x k + Bwhere the A k , B are endomorphisms of the bundle V and where the leadingterm is scalar. This category of dierential operators includes all theLaplacians we have been considering previously. Our rst task is to get amore invariant formulation for such an operator.Let r be a connection on V and let E 2 C 1 (END(V )) be an endomorphismof V . We use the Levi-Civita connection on M and the connectionr on V to extend r to tensors of all orders. We dene the dierentialoperator P r by the diagram:P r : C 1 (V ) r ! C 1 (T M V ) r ! C 1 (T M T M V ) ;g1;;;! C 1 (V ):Relative to a local orthonormal frame for T M,we can expressP r (f) =;f iiso this is the trace of second covariant dierentiation. We deneand our rst result is:P (rE)=P r ; ELemma 4.8.1. Let P : C 1 (V ) ! C 1 (V ) be a second order operatorwith leading symbol given by the metric tensor. There exists a uniqueconnection on V and a unique endomorphism so P = P (rE).Proof: We let indices i, j, k index a coordinate frame for T (M). Weshall not introduce indices to index a frame for V but shall always usematrix notation. The Christoel symbols ; ij k = ;; ik j of the Levi-Civitaconnection are given by:; ij k = 1 2 gkl (g il=j + g jl=i ; g ij=l )r @=@xi (@=@x j ) =; ij k (@=@x k )r @=@xi (dx j )=; ij k (dx k )

Local Formulas for the Invariants 315where we sum over repeated indices. We let ~s be a local frame for V andlet ! i be the components of the connection 1-form ! so that:r(f ~s )=dx i (@f=@x i + ! i (f)) ~s:With this notational convention, the curvature is given by: ij = ! j=i ; ! i=j + ! i ! j ; ! j ! i ( = d! + ! ^ !):We now compute:r 2 (f ~s )=dx j dx i (@ 2 f=@x i @x j + ! j @f=@x i +; jk i @f=@x kfrom which it follows that:+ ! i @f=@x j + ! i=j f + ! j ! i f +; jk i ! k f) ~sP r (f ~s )=;fg ij @f=@x i @x j +(2g ij ! i ; g jk ; jk i )@f=@x iWe use this identity to compute:+(g ij ! i=j + g ij ! i ! j ; g ij ; ij k ! k )fg~s(P ; P r )(f ~s )=;f(A i ; 2g ij ! j + g jk ; jk i )@f=@x i +()fg~swhere we have omitted the 0 th order terms. Therefore (P ; P r ) is a 0 thorder operator if and only ifA i ; 2g ij ! j + g jk ; jk i =0 or equivalently ! i = 1 2 (g ijA j + g ij g kl ; kl j ):This shows that the f! i g are uniquely determined by the condition that(P ; P r )isa0 th order operator and specicies the connection r uniquely.We dene:E = P r ; P so E = B ; g ij ! i=j ; g ij ! i ! j + g ij ! k ; ij k :We x this connection r and endomorphism E determined by P .summarize these formulas as follows:Corollary 4.8.2. If (rE) are determined by the second order operatorP = ;(g ij @ 2 f=@x i @x j + A i @j=@x i + Bf) ~s, then! i = 1 2 (g ijA j + g ij g kl ; kl j )E = B ; g ij ! i=j ; g ij ! i ! j + g ij ! k ; ij k :We digress briey to express the Laplacian p in this form. If r is theLevi-Civita connection acting on p-forms, it is clear that p ; P r is a rstWe

316 4.8. Local Formulas for the Invariantsorder operator whose leading symbol is linear in the 1-jets of the metric.Sinceitisinvariant, the leading symbol vanishes so p ; P r is a 0 th orderand the Levi-Civita connection is in fact the connection determined by theoperator p . We must now compute the curvature term. The operator(d + ) is dened by Cliord multiplication:(d + ): C 1 ((T M)) ! C 1 (T M (T M))Cliord multiplication;;;;;;;;;;;;;;;;! C 1 ((T M)):If we expand = f I dx I and r = dx i (@f I =@x i +; iI J f I ) dx J then it isimmediate that if \" denotes Cliord multiplication,(d + )(f I dx I )=(f I=i dx i dx I )+(f I ; iIJ dx i dx J )(f I dx I )=(f I=ij dx j dx i dx I )+(f I ; iIJ=j dx j dx i dx J )+where we have omitted terms involving the 1-jets of the metric. Similary,we compute:P r (f I dx I )=;g ij f I=ij dx I ; g ij ; iIJ=j f I dx J + :We now x a point x 0 of M and let X be a system of geodesic polarcoordinates centered at x 0 . Then ; iIJ=i =0and; iIJ=j = 1 2 R jiIJ gives thecurvature tensor at x 0 . Using the identities dx i dx j + dx j dx i = ;2 ijwe see f I=ij dx j dx i = ;f I=ii and consequently:(P r ;)(f I dx I )=; 1 2 R ijIJ f I dx i dx j dx J = X i

Of the Heat Equation 317order 0 in the jets of the symbol, a 0 ( P) = a 0 (P )= vol(M) is constant.We compute:X Ze ;tn2 t ;1=2 a 0 ( P) d = t ;1=2 (2)a 0 ( P):nMHowever, the Riemann sums approximating the integral show thatp =Z 1;1e ;tx2 dx = limt!0Xne ;(p tn) 2 ptfrom which it follows immediately that a 0 ( P) = p =(2) = (4) ;1=2 .More generally, by taking the direct sum of these operators acting onC 1 (S 1 C k ) we conclude a 0 ( P) = (4) ;1=2 dim V which completesthe proof if m =1.More generally, a 0 (x P ) is homogeneous of order 0 in the jets of thesymbol so a 0 (x P ) is a constant which only depends on the dimension ofthe manifold and the dimension of the vector bundle. Using the additivityof Lemma 1.7.5, we conclude a 0 (x P ) must have the form:a 0 (x P )=c(m) dim V:We now let M = S 1 S 1 , the at m-torus, and let = ; P @2 =@ 2 .The product formula of Lemma 1.7.5(b) implies that:a 0 (x ) = Y a 0 ( ;@ 2 =@ 2 )=(4) ;m=2which completes the proof of the lemma.The functorial properties of Lemma 1.7.5 were essential to the proof ofLemma 4.8.4. We will continue to exploit this functoriality in computinga 2 and a 4 . (In principal one could also compute a 6 in this way, but thecalculations become of formidable diculty and will be omitted).It is convenient to work with more tensorial objects than with the jetsof the symbol of P . We let dim V = k and dim M = m. We introduceformal variables f R i1 i 2 i 3 i 4 ... i1 i 2 ... E ... g for the covariant derivativesof the curvature tensor of the Levi-Civita connection, of the curvaturetensor of r, and of the covariant derivatives of the endomorphism E.We let S be the non-commutative algebra generated by these variables.Since there are relations among these variables, S isn't free. If S 2 Sand if e is a local orthonormal frame for T (M), we dene S(P )(e)(x) =S(G rE)(e)(x)(END(V )) by evaluation. We say S is invariant ifS(P )=S(P )(a) is independent of the orthonormal frame e chosen for T (M). Wedene:ord(R i1 i 2 i 4 i 4 i 5 ...i k)=k +2ord( i1 i 2 i 3 ...i k)=kord(E i1 ...i k)=k +2

318 4.8. Local Formulas for the Invariantsto be the degree of homogeneity in the jets of the symbol of P , and letS mnk be the nite dimensional subspace of S consisting of the invariantpolynomials which are homogeneous of order n.If we apply H. Weyl's theorem to this situation and apply the symmetriesinvolved, it is not dicult to show:Lemma 4.8.5.(a) S m2k is spanned by the two polynomials R ijij I, E.(b) S m4k is spanned by the eight polynomialsR ijijkk I R ijij R klkl I R ijik R ljlk I R ijkl R ijkl IE 2 E ii R ijij E ij ij :We omit the proof in the interests of brevity the corresponding spanningset for S m6k involves 46 polynomials.The spaces S mnk are related to the invariants a n (x P ) of the heat equationas follows:Lemma 4.8.6. Let (m n k) be given and let P satisfy the hypothesisof Lemma 4.8.1. Then there exists S mnk 2 S mnk so that a n (x P ) =Tr(S mnk ).Proof: We x a point x 0 2 M. We choose geodesic polar coordinatescentred at x 0 . In such a coordinate system, all the jets of the metricat x 0 can be computed in terms of the R i1 i 2 i 3 i 4 ... variables. We x aframe s 0 for the ber V 0 over x 0 and extend s 0 by parallel translationalong all the geodesic rays from x 0 to get a frame near x 0 . Then all thederivatives of the connection 1-form at x 0 can be expressed in terms of theR ... and ... variables at x 0 . We can solve the relations of Corollary 4.8.2to express the jets of the symbol of P in terms of the jets of the metric,the jets of the connection 1-form, and the jets of the endomorphism E.These jets can all be expressed in terms of the variables in S at x 0 soany invariant endomorphism which is homogeneous of order n belongs toS mnk . In Lemma 1.7.5, we showed that a n (x P )=Tr(e n (x P )) was thetrace of an invariant endomorphism and this completes the proof we setS mnk = e n (x P ).We use Lemmas 4.8.5 and 4.8.6 to expand a n (x P ). We regard scalar invariantsof the metric as acting on V by scalar multiplication alternatively,such an invariant R ijij could be replaced by R ijij I V .Lemma 4.8.7. Let m = dim M and let k = dim V . Then there existuniversal constants c i (m k) so that if P is as in Lemma 4.8.1,(a) a 2 (x P )=(4) ;m=2 Tr(c 1 (m k)R ijij + c 2 (m k)E),

Of the Heat Equation 319(b)a 4 (x P )=(4) ;m=2 Tr(c 3 (m k)R ijijkk + c 4 (m k)R ijij R klkl+ c 5 (m k)R ijik R ljlk + c 6 (m k)R ijkl R ijkl+ c 7 (m k)E ii + c 8 (m k)E 2+ c 9 (m k)ER ijij + c 10 (m k) ij ij ):This is an important simplication because it shows in particular that termssuch as Tr(E) 2 do not appear in a 4 .The rst observation we shall need is the following:Lemma 4.8.8. The constants c i (m k) of Lemma 4.8.7 can be chosen tobe independent of the dimension m and the ber dimension k.Proof: The leading symbol of P is scalar. The analysis of Chapter 1 inthis case immediately leads to a combinatorial R formula for the coecients interms of certain trignometric integrals e ;jj2 d and the ber dimensiondoes not enter. Alternatively,we could use the additivityofe n (x P 1 P 2 )=e n (x P 1 )e n (x P 2 ) of Lemma 1.7.5 to conclude the formulas involved mustbe independent of the dimension k. We may therefore write c i (m k) =c i (m).There is a natural restriction map r: S mnk ! S m;1nk dened by restrictingto operators of the form P = P 1 1+I V (;@ 2 =@ 2 ) overM = M 1 S 1 . Algebraically, we simply set to zero any variables involvingthe last index. The multiplicative property of Lemma 1.7.5 impliesr(S mnk )=Xp+q=nS m;1pk (P 1 ) S 1q1 (;@ 2 =@ 2 ):Since all the jets of the symbol of ;@ 2 =@ 2 vanish for q > 0, S 1q1 = 0for q>0 and a 0 =(4) ;1=2 by Lemma 4.8.4. Thereforer(S mnk )=(4) ;1=2 S m;1nk :Since wehave included the normalizing constant(4) ;m=2 in our denition,the constants are independent of the dimension m for m 4. If m =1 2 3,then the invariants of Lemma 4.8.6 are not linearly independent so wechoose the constants to agree with c i (m k) in these cases.We remark that if P is a higher order operator with leading symbol givenby a power of the metric tensor, then there a similar theory expressinga n in terms of invariant tensorial expressions. However, in this case, thecoecients depend upon the dimension m in a much more fundamentalway than simply (4) ;m=2 and we refer to (Gilkey, the spectral geometryof the higher order Laplacian) for further details.Since the coecients do not depend on (m k), we drop the somewhatcumbersome notation S mnk and return to the notation e n (x P ) discussedin the rst chapter so Tr(e n (x P )) = a n (x P ). We use the properties ofthe exponential function to compute:

320 4.8. Local Formulas for the InvariantsLemma 4.8.9. Using the notation of Lemma 4.8.7,c 2 =1 c 8 = 1 2 c 9 = c 1 :Proof: Let P be as in Lemma 4.8.1 and let a be a real constant. We constructthe operator P a = P ;a. The metric and connection are unchangedwe must replace E by E + a. Since e ;t(P ;a) = e ;tP e ta we conclude:e n (x P ; a) =Xp+q=ne p (x P )a q =q!by comparing terms in the asymptotic expansion. We shall ignore factorsof (4) ;m=2 henceforth for notational convenience. Then:e 2 (x P ; a) =e 2 (x P )+c 2 a = e 2 (x P )+e 0 (x P )a = e 2 (x P )+a:This implies that c 2 =1as claimed. Next, we havee 4 (x P ; a)=e 4 (x P )+c 8 a 2 +2c 8 aE + c 9 aR ijij= e 4 (x P )+e 2 (x P )a + e 0 (x P )a 2 =2= e 4 (x P )+(c 1 R ijij + c 2 E)a + a 2 =2which implies c 8 =1=2 and c 1 = c 9 as claimed.We now use some recursion relations derived in (Gilkey, Recursion relationsand the asymptotic behavior of the eigenvalues of the Laplacian). Toillustrate these, we rst suppose m =1. We consider two operators:A = @=@x + bA = ;@=@x + bwhere b is a real scalar function. This gives rise to operators:P 1 = A A = ;(@ 2 =@x 2 +(b 0 ; b 2 ))P 2 = AA = ;(@ 2 =@x 2 +(;b 0 ; b 2 ))acting on C 1 (S 1 ). The metric and connection dened by these operatorsis at. E(P 1 )=b 0 ; b 2 and

Of the Heat Equation 321be constructing. Then f A = p g is a complete spectral resolution ofP 2 . We compute:ddt fK(t x x P 1) ; K(t x x P 2 )g= X e ;t f; + A A g= X e ;t f;P 1 + A A g= X e ;t f 00 +(b 0 ; b 2 ) 2 + 0 0 +2b 0 + b 2 2 g= X e ;t ; 12 @=@x (@=@x +2b)f 2 g= 1 2 @=@x(@=@x +2b)K(t x x P 1):We equate terms in the asymptotic expansionsXt(n;3)=2 n ; 1 (e n (x P 1 ) ; e n (x P 2 ))2to complete the proof of the lemma. 1 2Xt (n;1)=2 @=@x(@=@x +2b)e n (x P 1 )We apply this lemma to compute the coecient c 7 . If n = 4, then weconclude:e 4 (x P 1 )=c 7 (b 0 ; b 2 ) 00 + c 8 (b 0 ; b 2 ) 2 = c 7 b 000 + lower order termse 4 (x P 2 )=c 7 (;b 0 ; b 2 ) 00 + c 8 (;b 0 ; b 2 ) 2 = ;c 7 b 000 + lower order termse 2 (x P 1 )=b 0 ; b 2so that:3(e 4 (x P 1 ) ; e 4 (x P 2 )) = 6c 7 b 000 + lower order terms@=@x(@=@x +2b)(b 0 ; b 2 ) = b 000 + lower order termsfrom which it follows that c 7 = 1=6. It is also convenient at this stage toobtain information about e 6 . If we let e 6 = cE 0000 + lower order terms thenwe express:e 6 (x P 1 ) ; e 6 (x P 2 )=2cb (5)@=@x(@=@x +2b)(e 4 ) = c 7 b (5)from which it follows that the constant c is (4) ;1=2 c 7 =10 = (4) ;1=2 =60.We summarize these results as follows:

322 4.8. Local Formulas for the InvariantsLemma 4.8.11. We can expand a n in the form:(a) a 2 (x P )=(4) ;m=2 Tr(c 1 R ijij + E):(b)a 4 (x P )=(4) ;m=2 Tr(c 3 R ijijkk + c 4 R ijij R klkl + c 5 R ijik R ljlk+ c 6 R ijkl R ijkl + E kk =6+E 2 =2+ c 1 R ijij E + c 10 ij ij ):(c)a 6 (x P )=(4) ;m=2 Tr(E kkll =60+ c 11 R ijijkkll +lower order terms):Proof: (a) and (b) follow immediately from the computations previouslyperformed. To prove (c) we argue as in the proof of 4.8.7 to showa 6 (x P )=(4) ;m=2 Tr(cE kk + c 11 R ijijkk + lower order terms) and thenuse the evaluation of c given above.We can use a similar recursion relation if m = 2 to obtain further informationregarding these coecients. We consider the de Rham complex,then:Lemma 4.8.12. If m = 2 and p is the Laplacian on C 1 ( p (T M)),then:(n ; 2)fa n (x 0 ) ; a n (x 1 )+a n (x 2 )g = a n;2 (x 0 ) kk :2Proof: This recursion relationship is due to McKean and Singer. Sincethe invariants are local, we may assume M is orientable and a n (x 0 ) =a n (x 2 ). Let f g be a spectral resolution for the non-zero spectrumof 0 then f g is a spectral resolution for the non-zero spectrum of 2 and f d = p = p g is a spectral resolution for the non-zerospectrum of 1 . Therefore:d ; K(t x x 0 ) ; K(t x x 1 )+K(t x x 2 )dt= X e ;t (;2 + d d + )= X e ;t (;2 0 +2d d )=K(t x x 0 ) kkfrom which the desired identity follows.Before using this identity, we must obtain some additional informationabout 1 .

Of the Heat Equation 323Lemma 4.8.13. Let m be arbitrary and let ij = ;R ijik be the Riccitensor. Then = ; kk + () for 2 C 1 ( 1 ) and where () i = ij j .Thus E( 2 )=;.Proof: We apply Lemma 4.8.3 to concludeE() = 1 2 R abij i e b e a e jfrom which the desired result follows using the Bianchi identities.We can now check at least some of these formulas. If m = 2, thenE( 1 )=R 1212 I and E( 0 )=E( 2 )=0. Therefore:a 2 (x 0 ) ; a 2 (x 1 )+a 2 (x 2 )=(4) ;1 f(1 ; 2+1)c 1 R ijij ; 2R 1212 g= ;R 1212 =2which is, in fact, the integrand of the Gauss-Bonnet theorem. Next, wecompute, supressing the factor of (4) ;1 :a 4 (x 0 ) ; a 4 (x 1 )+a 4 (x 2 )=(c 3 R ijijkk + c 4 R ijij R klkl + c 5 R ijik R ljlk+ c 6 R ijkl R ijkl )(1 ; 2+1)+ 1 6 (;2)(R 1212kk ) ; 2 2 R 1212 R 1212; 4c 1 R 1212 R 1212 ; c 10 Tr( ij ij )= ; 1 3 (R 1212kk ) ; (1 + 4c 1 ; 4c 10 )(R 1212 ) 2a 2 (x 0 ) kk =2c 1 R 1212kkso that Lemma 4.2.11 applied to the case n =4implies:a 4 (x 0 ) ; a 4 (x 1 )+a 4 (x 2 )=a 2 (x 0 ) kkfrom which we derive the identities:c 1 = ; 1 6and 1+4c 1 ; 4c 10 =0from which it follows that c 10 = 1=12. We also consider a 6 and Lemma4.8.11(c)a 6 (x 0 ) ; a 6 (x 1 )+a 6 (x 2 )=c 11 R ijijkkll (1 ; 2+1); 260 R 1212kkll+ lower order termsa 4 (x 0 ) kk =2c 3 R 1212jjkk + lower order terms

324 4.8. Local Formulas for the Invariantsso Lemma 4.2.12 implies that 2(;2=60) = 2c 3 so that c 3 = ;1=30.This leaves only the constants c 4 , c 5 , and c 6 undetermined. We letM = M 1 M 2 be the product manifold with 0 = 1 0 + 2 0. Then theproduct formulas of Lemma 1.7.5 imply:a 4 ( 0 x)=a 4 ( 1 0x 1 )+a 4 ( 2 0x 2 )+a 2 ( 1 0x 1 )a 2 ( 2 0x 2 ):The only term in the expression for a 4 giving rise to cross terms involvingderivatives of both metrics is c 4 R ijij R klkl . Consequently, 2c 4 = c 2 1 =1=36so c 4 =1=72. We summarize these computations as follows:Lemma 4.8.14. We can expand n in the form:(a) a 2 (x P )=(4) ;m=2 Tr(;R kjkj +6E)=6.(b)a 4 (x P )= (4);m=2Tr(;12R ijijkk +5R ijij R klkl + c 5 R ijik R ljlk360+ c 6 R ijkl R ijkl ; 60ER ijij + 180E 2 +60E kk + 30 ij ij ):We have changed the notation slightly to introduce the common denominator360.We must compute the universal constants c 5 and c 6 to complete ourdetermination of the formula a 4 . We generalize the recursion relations ofLemmas 4.8.10 and 4.8.12 to arbitrary dimensions as follows. Let M = T mbe the m-dimensional torus with usual periodic parameters 0 x i 2 fori =1 ...m. We let fe i g be a collection of Cliord matrices so e i e j +e j e i =2 ij . Let h(x) bea real-valued function on T m and let the metric be givenby:ds 2 = e ;h (dx 2 1 + + dx 2 m) dvol = e ;hm=2 dx 1 ...dx m :We let the operator A and A be X dened by:A = e mh=4 @e j e (2;m)h=4@x jand dene:A = e (2+m)h=4 X e jP 1 = A A = ;e (2+m)h=4 X j= ;e h X @2@x 2 j@ 2@x 2 j@@x je ;mh=4e (2;m)h=4+ 1 2 (2 ; m)h @=j@x j+ 1 16 (4(2 ; m)h =jj +(2; m) 2 h =j h =j )P 2 = AA = e mh=4 X je j@@x je h X ke k@@x ke ;mh=4 :

Of the Heat Equation 325Lemma 4.8.15. With the notation given above,hm=2 @2(n ; m)fa n (x P 1 ) ; a n (x P 2 )g = e@x 2 e h(2;m)=2 a n;2 (x P ):kProof: We let P 0 be the scalar operator;e (2+m)h=4 X j@ 2@x 2 e (2;m)h=4 :jIf the representation space on which the e j act has dimension u, thenP 1 = P 0 I u . We let v s be a basis for this representation space. Letf g be a spectral resolution for the operator P 0 then f v s g isa spectral resolution of P 1 . We compute:ddt; Tr K(t x x P1 ) ; Tr(t x x P 2 ) = X se ;t f;(P 1 u s u s )+(A u s A u s )gwhere ( ) denotes the natural inner product (u s u s 0)= ss 0. The e j areself-adjoint matrices. We use the identity (e j u s e k u s )= jk to compute:= X v e ;t e h =kk + 1 2 (2 ; m)h =k =k + 1 4 (2 ; m)h =kk = X + 116 (2 ; m)2 h =k h =k + =k =k + 1 2 (2 ; m)h =k =k + 116 (2 ; m)2 h =k h =k v e ;t e hm=2 1 2= 1 2 ehm=2 X j@ 2Xj@ 2@x 2 e h(2;m)=2 ( )j@x 2 e h(2;m)=2 Tr K(t x x P 1 ):jWe compare coecients of t in the two asymptotic expansions to completethe proof of the lemma.We apply Lemma 4.8.15 to the special case n = m = 6. This impliesthatXj@ 2@x 2 e ;2h a 4 (x P 1 )=0:jSince a x (x P 1 ) is a formal polynomial in the jets of h with coecientswhich are smooth functions of h, and since a 4 is homogeneous of order 4,

326 4.8. Local Formulas for the Invariantsthis identity for all h implies a 4 (x P 1 ) = 0. This implies a 4 (x P 0 ) =1v a 4(x P 1 )=0.Using the formulas of lemmas 4.8.1 and 4.8.2 it is an easy exercise tocalculate R ijkl = 0 if all 4 indices are distinct. The non-zero curvaturesare given by:R iji k = ; 1 4 (2h =jk + h =j h =k ) if j 6= k (don't sum over i)R iji j = ; 1 4 (2h =ii +2h =jj ; ij =0E(P 0 )=e h X kXk6=ik6=j(;h =kk + h =k h =k ):h =k h =k (don't sum over i j)When we contract the curvature tensor to form scalar invariants, we mustinclude the metric tensor since it is not diagonal. This implies: = X ijg ii R iji j = e h X k(;5h =kk +5h =k h =k )=5E(P 1 )which implies the helpful identities:;12 kk +60E kk =0 and 5R ijij R klkl ; 60R ijij E + 180E 2 =5E 2so that we conclude:We expand:5E 2 + c 5 R ijik R ljlk + c 6 R ijkl R ijkl =0:R ijik R ljlk = e 2h ; 152 h =11h =11 +8h =12 h =12 + other terms R ijkl R ijkl = e 2h ; 5h =11 h =11 +8h =12 h =12 + other terms E 2 = e 2h ; h =11 h =11 +0 h =12 h =12 + other terms so we conclude nally:15c 5 +10c 6 +10=0 and c 5 + c 6 =0:We solve these equations to conclude c 5 = ;2 and c 6 = 2 which provesnally:Theorem 4.8.16. Let P be a second order dierential operator withleading symbol given by the metric tensor. Let P = P r ;E be decomposedas in 4.8.1. Let a n (x P ) be the invariants of the heat equation discussedin Chapter 1.

(a) a 0 (x P )=(4) ;m=2 Tr(I):(b) a 2 (x P )=(4) ;m=2 Tr(;R ijij +6E)=6:(c)Of the Heat Equation 327a 4 (x P )= (4);m=2360Tr(;12R ijijkk +5R ijij R klkl ; 2R ijik R ljlk +2R ijkl R ijkl; 60R ijij E + 180E 2 +60E kk + 30 ij ij ):(d)a 6 (x P )=(4) ;m=2 1 ;Tr ;18Rijijkkll +17R ijijk R ululk ; 2R ijikl R ujukl7!; 4R ijikl R ujulk +9R ijkul R ijkul +28R ijij R kukull; 8R ijik R ujukll +24R ijik R ujulkl +12R ijkl R ijkluu+ 19 7!;;35Rijij R klkl R pqpq +42R ijij R klkp R qlqp; 42R ijij R klpq R klpq + 208R ijik R julu R kplp; 192R ijik R uplp R jukl +48R ijik R julp R kulp; 44R ijku R ijlp R kulp ; 80R ijku R ilkp R jlup+ 1360;8ijk ijk +2 ijj ikk +12 ij ijkk ; 12 ij jk ki; 6R ijkl ij kl +4R ijik jl kl ; 5R ijij kl kl+ 1360;6Eiijj +60EE ii +30E i E i +60E 3 +30E ij ij; 10R ijij E kk ; 4R ijik E jk ; 12R ijijk E k ; 30R ijij E 2; 12R ijijkk E +5R ijij R klkl E; 2R ijik R ijkl E +2R ijkl R ijkl E :Proof: We have derived (a){(c) explicitly. We refer to (Gilkey, Thespectral geometry of a Riemannian manifold) for the proof of (d) as itis quite long and complicated. We remark that our sign convention is thatR 1212 = ;1 on the sphere of radius 1 in R 3 .We now begin our computation of a n (x m p ) for n =0 2 4.

328 4.8. Local Formulas for the InvariantsLemma 4.8.17. Let m = 4, and let p be the Laplacian on p-forms.Decomposea 2 (x p )=(4) ;2 c 0 (p) R ijij =6a 4 (x p )=(4) ;2 fc 1 (p)R ijijkk + c 2 (p)R ijij R klklThen the c i are given by the following table:c 3 (p)R ijik R ljlk + c 4 (p)R ijkl R ijkl g=360:c 0 c 1 c 2 c 3 c 4p =0 ;1 ;12 5 ;2 2p =1 2 12 ;40 172 ;22p =2 6 48 90 ;372 132p =3 2 12 ;40 172 ;22p =4 ;1 ;12 5 ;2 2P (;1)p0 0 180 ;720 180Proof: By Poincare duality, a n (x p ) = a n (x 4;p ) so we need onlycheck the rst three rows as the corresponding formulas for 3 and 4 willfollow. In dimension 4 the formula for the Euler form is (4) ;2 (R ijij R klkl ;4R ijik R ljlk +R ijkl R ijkl )=2 so that the last line follows from Theorem 2.4.8.If p =0,then E ==0so the rst line follows from Theorem 4.8.15.If p = 1, then E = ; ij = R ikij is the Ricci tensor by Lemma 4.8.13.Therefore:Tr(;12R ijijkk +60E kk )=;48R ijijkk +60R ijijkk=12R ijijkkTr(5R ijij R klkl ; 60R ijij E)=20R ijij R klkl ; 60R ijij R klkl= ;40R ijij R klklTr(;2R ijik R lklk + 180E 2 )=;8R ijik R ljlk + 180R ijik R ljlk= 172R ijik R ljlkTr(2R ijkl R ijkl + 30 ij ij )=8R ijkl R ijkl ; 30R ijkl R ijkl= ;22R ijkl R ijklwhich completes the proof of the second line. Thus the only unknown isc k (2). This is computed from the alternating sum and completes the proof.More generally, we let m > 4. Let M = M 4 T m;4 be a productmanifold, then this denes a restriction map r m;4 : P mn ! P 4n which is

Of the Heat Equation 329an isomorphism for n 4. Using the multiplication properties given inLemma 1.7.5, it follows:a n (x m p )=Xi+j=np 1 +p 2 =pa i (x 4 p 1)a j (x m;4p 2):On the at torus, all the invariants vanish for j > 0. By Lemma 4.8.4a 0 (x m;4p 2)=(4) ;m=2; m;4 so that:a n (x 4 u)=Xu+v=pp 2(4) (m;4)=2 m ; 4va n (x 4 u) for n 4where a n (x 4 u) is given by Lemma 4.8.16. If we expand this as a polynomialin m for small values of u we conclude:a 2 (x m 1 )= (4);m=2(6 ; m)R ijij6a 4 (x m 1 )= (4);m=2f(60 ; 12m)R ijijkk +(5m ; 60)R ijij R klkl360+ (180 ; 2m)R ijik R ljlk +(2m ; 30)R ijkl R ijkl gand similarly for a 2 (x m 2 ) and a 4 (x m 2 ). In this form, the formulas alsohold true for m = 2 3. We summarize our conclusions in the followingtheorem:Theorem 4.8.18. Let m p denote the Laplacian acting on the space ofsmooth p-forms on an m-dimensional manifold. We let R ijkl denote thecurvature tensor with the sign convention that R 1212 = ;1 on the sphereof radius 1 in R 3 . Then:(a) a 0 (x m p )=(4) ;m=2; mp .n; m;2p;2 +; m;2p ;; 4m;2o(;Rp;1 ijij ).(b) a 2 (x m p )= (4);m=26(c) Let;a 4 (x m p )= (4);m=2c1 (m p)R ijijkk + c 2 (m p)R ijij R klkl360Then for m 4 the coecients are:c 1 (m p) =;12h; m;4pc 2 (m p) = 5h; m;4pc 3 (m p) = ;2h; m;4pc 4 (m p) = 2h; m;4p +; m;4p;4 +; m;4p;4 +; m;4p;4 +; m;4p;4+ c 3 (m p)R ijik R ljlk + c 4 (m p)R ijkl R ijkl:iiii+ 12h; m;4p;1; 40h; m;4p;1+ 172h; m;4p;1; 22h; m;4p;1 +; m;4p;3 +; m;4p;3 +; m;4p;3 +; m;4p;3i ; m;4+ 48p;2i+ 90 ; m;4p;2i; 372 ; m;4p;2i+ 132 ; m;4p;2:

330 4.8. Local Formulas for the Invariants(d)(e)(f )a 0 (x m 0 )=(4) ;m=2a 2 (x m 0 )= (4);m=2(;R ijij )6a 4 (x m 0 )= (4);m=2(;12R ijijkk +5R ijij R klkl ; 2R ijik R ljlk360+2R ijkl R ijkl )a 0 (x m 1 )=(4) ;m=2a 2 (x m 1 )= (4);m=2(6 ; m)R ijij6a 4 (x m 1 )= (4);m=2f(60 ; 12m)R ijijkk +(5m ; 60)R ijij R klkl360+ (180 ; 2m)R ijik R ljlk+(2m ; 30)R ijkl R ijkl )ga 0 (x m 2 )= (4);m=22a 2 (x m 2 )= (4);m=212a 4 (x m 2 )= (4);m=2720m(m ; 1)(;m 2 +13m ; 24)R ijijf(;12m 2 + 108m ; 144)R ijijkkk+(5m 2 ; 115m + 560)R ijij R klkl+(;2m 2 + 358m ; 2144)R ijik R ljlk+(2m 2 ; 58m + 464)R ijkl R ijkl gThese results are, of course, not new. They were rst derived by Patodi.We could apply similar calculations to determine a 0 , a 2 ,anda 4 for any operatorwhich is natural in the sense of Epstein and Stredder. In particular,the Dirac operator can be handled in this way.In principal, we could also use these formulas to compute a 6 , but thelower order terms become extremely complicated. It is not too terriblydicult, however, to use these formulas to compute the terms in a 6 whichinvolve the 6 jets of the metric and which are bilinear in the 4 and 2 jetsof the metric. This would complete the proof of the result concerning a 6 ifm =4discussed in section 4.7.

4.9. Spectral Geometry.Let M be a compact Riemannian manifold without boundary and letspec(M) denote the spectrum of the scalar Laplacian where each eigenvalueis repeated according to its multiplicity. Two manifolds M and Mare said to be isospectral if spec(M) = spec(M). Theleadingterminthe asymptotic expansion of the heat equation is (4) ;m=2 vol(M) t ;m=2so that if M and M are isospectral,dim M =dimM and volume(M) =volume(M)so these two quantitites are spectral R invariants. If P 2P mn0 is an invariantpolynomial, we dene P (M) =MP (G)j dvol j. (This depends on themetric in general.) Theorem 4.8.18 then implies that R ijij (M) is a spectralinvasriant since this appears with a non-zero coecient in the asymptoticexpansion of the heat equation.The scalar Laplacian is not the only natural dierential operator to study.(We use the word \natural" in the technical sense of Epstein and Stredderin this context.) Two Riemannian manifolds M and M are said to bestrongly isospectral if spec(MP) = spec(MP) for all natural operatorsP . Many of the global geometry properties of the manifold are reectedby their spectral geometry. Patodi, for example, proved:Theorem 4.9.1 (Patodi). Let spec(M p ) = spec(M p ) for p =0 1 2. Then:(a)dim M = dim MR ijij (M) =R ijij (M)R ijik R ljlk (M) =R ijik R ljlk (M)volume(M) =volume(M)R ijij R klkl (M) =R ijij R klkl (M)R ijkl R ijkl (M) =R ijkl R ijkl (M):(b) If M has constant scalar curvature c, then so does M.(c) If M is Einstein, then so is M.(d) If M has constant sectional curvature c, then so does M.Proof: The rst three identities of (a) have already been derived.m 4, the remaining 3 integral invariants are independent. We know:Ifa 4 ( m p )=c 2 (m p)R ijij R klkl + c 3 (m p)R ijik R ljlk (M)+ c 4 (m p)R ijkl R ijkl (M):As p = 0 1 2 the coecients form a 3 3 matrix. If we can show thematrix has rank 3, we can solve for the integral invariants in terms of the

332 4.9. Spectral Geometryspectral invariants to prove (a). Let = m ; 4 and c = (4) ;=2 . thenour computations in section 4.8 showc j (m 0) = c c j (4 0)c j (m 1) = c c j (4 0) + c c j (4 1)c j (m 2) = c ( ; 1)=2 c j (4 0) + c c j (4 1) + c c j (4 2)so that the matrix for m > 4 is obtained from the matrix for m = 4 byelementary row operations. Thus it suces to consider the case m = 4.By 4.8.17, the matrix there (modulo a non-zero normalizing constant) is:0B@ 5 ;2 2;40 172 ;2290 ;372 1321CA :The determinant of this matrix is non-zero from which (a) follows. Thecase m =2and m =3can be checked directly using 4.8.18.To prove (b), we note that M has constant scalarcurvature c if and onlyif (2c + R ijij ) 2 (M) = 0 which by (a) is a spectral invariant. (c) and (d)are similar.From (d) follows immediately the corollary:Corollary 4.9.2. Let M and M be strongly isospectral. If M is isometricto the standard sphere of radius r, thenso is M.Proof: If M is a compact manifold with sectional curvature 1=r, thenthe universal cover of M is the sphere of radius r. If vol(M) and vol(S(r))agree, then M and the sphere are isometric.There are a number of results which link the spectral and the globalgeometry of a manifold. We list two of these results below:Theorem 4.9.3. Let M and M be strongly isospectral manifolds. Then:(a) If M is a local symmetric space (i.e., rR =0), then so is M.(b) If the Ricci tensor of M is parallel (i.e., r =0), then the Ricci tensorof M is parallel. In this instance, the eigenvalues of do not depend uponthe particular point of the manifold and they are the same for M and M.Although we have chosen to work in the real category, there are alsoisospectral results available in the holomorphic category:Theorem 4.9.4. Let M and M be holomorphic manifolds and supposespec(M pq ) = spec(M pq ) for all (p q).(a) If M is Kaehler, then so is M.(b) If M is CP n , then so is M.At this stage, the natural question to ask is whether or not the spectralgeometry completely determines M. This question was phrased by Kac in

Spectral Geometry 333the form: Can you hear the shape of a drum? It is clear that if there existsan isometry between two manifolds, then they are strongly isospectral.That the converse need not hold was shown by Milnor, who gave examplesof isospectral tori which were not isometric. In 1978 Vigneras gaveexamples of isospectral manifolds of constant negative curvature which arenot isometric. (One doesn't know yet if they are strongly isospectral.) Ifthe dimension is at least 3, then the manifolds have dierent fundamentalgroups, so are not homotopic. The fundamental groups in question are allinnite and the calculations involve some fairly deep results in quaternionalgebras.In 1983 Ikeda constructed examples of spherical space forms which werestrongly isospectral but not isometric. As de Rham had shown that dieomorphicspherical space forms are isometric, these examples are not dieomorphic.Unlike Vigneras' examples, Ikeda's examples involve nite fundamentalgroups and are rather easily studied. In the remainder of thissection, we will present an example in dimension 9 due to Ikeda illustratingthis phenomenon. These examples occur much more generally, butthisexample is particularly easy to study. We refer to (Gilkey, On sphericalspace forms with meta-cyclic fundamental group which are isospectral butnot equivariant cobordant) for more details.Let G be the group of order 275 generated by two elements A, B subjectto the relations:A 11 = B 25 =1 and BAB ;1 = A 3 :We note that 3 5 1 mod 11. This group is the semi-direct productZ 11 / Z 25 . The center of G is generated by B 5 and the subgroup generatedby A is a normal subgroup. We have short exact sequences:0 ! Z 11 ! G ! Z 25 ! 0 and 0 ! Z 55 ! G ! Z 5 ! 0:We can obtain an explicit realization of G as a subgroup of U(5) as follows.Let = e 2i=11 and = e 2i=25 be primitive roots of unity. Let fe j g bethe standard basis for C 5 and let 2 U(5) be the permutation matrix(e j ) = e j;1 where the index j is regarded as dened mod 5. Dene arepresentation: k (A) = diag( 3 9 5 4 ) and k (B) = k :It is immediate that k (A) 11 = k (B) 25 = 1 and it is an easy computationthat k (B) k (A) k (B) ;1 = k (A) 3 so this extends to a representationof G for k = 1 2 3 4. (If H is the subgroup generated by fA B 5 g, welet (A) = and (B 5 ) = 5k be a unitary representation of H. Therepresentation k = G is the induced representation.)In fact these representations are xed point free:

334 4.9. Spectral GeometryLemma 4.9.5. Let G be the group of order 275 generated by fA Bgwith the relations A 11 = B 25 = 1 and BAB ;1 = A 3 . The center of G isgenerated by B 5 the subgroup generated by A is normal. Let = e 2i=11and = e 2i=25 be primitive roots of unity. Dene representations of G inU(5) for 1 k 4 by: k (A) = diag( 3 9 5 4 ) and k (B) = k where is the permutation matrix dened by (e i )=e i;1 .(a) Enumerate the elements of G in the form A a B b for 0 a < 11 and0 b

of this invariant subspace by:dim e k ()= 1jGjSpectral Geometry 335= 1275Xg2GXabTrfe 0 ()( k (g)gTrfe 0 ()( k (A a B b )g:We apply Lemma 4.9.5 to conclude k (A a B b ) is conjugate to 1 (A a B kb )in U(5) so the two traces are the same anddim e k ()= 1275= 1275XabXabTrfe 0 () 1 (A a B kb )gTrfe 0 () 1 (A a B b )g =dime 1 ()since we are just reparameterizing the group. This completes the proof.Let 2 R 0 (Z 5 ). We regard 2 R 0 (G) by dening (A a B b )=(b). Thisis nothing but the pull-back of using the natural map 0 ! Z 55 ! G !Z 5 ! 0.Lemma 4.9.7. Let 2 R 0 (Z 5 ) and let G be as in Lemma 4.9.5. Let: G ! G be a group automorphism. Then = so this representationof G is independent of the marking chosen.Proof: The Sylow 11-subgroup is normal and hence unique. Thus (A)= A a for some (a 11) = 1 as A generates the Sylow 11-subgroup. Let(B) =A c B d and compute:(A 3 )=A 3a = (B) (A) (B) ;1 = A c B d A a B ;d A ;c = A a3d :Since (a 11) = 1, 3 d 1 (11). This implies d 1 (5). Therefore(A u B v ) = A B dv so that (A u B v ) = (dv) = (v) as 2 R 0 (Z 5 )which completes the proof.These representations are canonical they do not depend on the markingof the fundamental group. This denes a virtual locally at bundle V over each of the M k . Let P be the tangential operator of the signaturecomplex ind( signatureM k ) is an oriented diemorphism of M k . In factmore is true. There is a canonical Z 5 bundle over M k corresponding tothe sequence G ! Z 5 ! 0 which by Lemma 4.9.7 is independent of theparticular isomorphism of 1 (M k ) with G chosen. Lemma 4.6.4(b) showsthis is a Z 5 -cobordism invariant. We apply Lemma 4.6.3 and calculate for 2 R 0 (Z 5 ) that:ind( signatureM k )= 1 X 275 0Tr((b)) defect( k (A a B b ) signature):ab

336 4.9. Spectral GeometryP 0 denotes the sum over 0 a

Spectral Geometry 337Theorem 4.9.8. Let M k = S 9 = k (G) with the notation of Lemma 4.9.5.These four manifolds are all strongly isospectral for k = 1 2 3 4. Thereis a natural Z 5 bundle over each M k . ind( 1 ; 0 signatureM k )=;2 k=5mod Z where k k 1 (5). As these 5 values are all dierent, these 4manifolds are not Z 5 oriented equivariant cobordant. Thus in particularthere is no orientation preserving dieomorphism between any two ofthesemanifolds. M 1 = ;M 4 and M 2 = ;M 3 . There is no dieomorphismbetween M 1 and M 2 so these are dierent topological types.Proof: If we replace by we replace k by 5;k up to unitary equivalence.The map z 7! z reverses the orientation of C 5 and thus M k =;M 5;k as oriented Riemannian manifolds. This change just alters the signof ind( signature ). Thus the given calculation shows M 1 6= M 2 . Thestatement about oriented equivariant cobordism follows from section 4.6.(In particular, these manifolds are not oriented G-cobordant either.) Thisgives an example of strongly isospectral manifolds which are of dierenttopological types.Remark: These examples, of course, generalize we have chosen to workwith a particular example in dimension 9 to simplify the calculations involved.

BIBLIOGRAPHYS. Agmon, \Lectures on Elliptic Boundary Value Problems", Math. Studies No. 2, VanNostrand, 1965.M. F. Atiyah, \K Theory", W. A. Benjamin, Inc., 1967., \Elliptic Operators and Compact Groups", Springer Verlag, 1974., Bott Periodicity and the Index of Elliptic Operators, Quart. J. Math. 19 (1968),113{140., Algebraic Topology and Elliptic Operators, Comm. Pure Appl. Math. 20(1967), 237{249.M. F. Atiyah and R. Bott, ALefschetz Fixed Point Formula for Elliptic Dierential Operators,Bull. Amer. Math. Soc. 72 (1966), 245{250., ALefschetz Fixed Point Formula for Elliptic Complexes I, Ann. of Math. 86(1967), 374{407 II. Applications, 88 (1968), 451{491., The Index Theorem for Manifolds with Boundary, in \Dierential Analysis(Bombay Colloquium)", Oxford, 1964.M. F. Atiyah, R. Bott and V. K. Patodi, On the Heat Equation and Index Theorem, Invent.Math. 19 (1973), 279{330.M. F. Atiyah, R. Bott and A. Shapiro, Cliord Modules, Topology 3 Suppl. 1 (1964),3{38.M. F. Atiyah, H. Donnelly, and I. M. Singer, Geometry and Analysis of Shimizu L-Functions,Proc. Nat. Acad. Sci. USA 79 (1982), p. 5751., Eta Invariants, Signature Defects of Cusps and Values of L-Functions, Ann. ofMath. 118 (1983), 131{171.M. F. Atiyah and F. Hirzebruch, Riemann-Roch Theorems for Dierentiable Manifolds, Bull.Amer. Math. Soc. 69 (1963), 422{433.M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral Asymmetry and Riemannian GeometryI, Math. Proc. Camb. Phil. Soc. 77 (1975), 43{69 II, 78 (1975), 405{432 III,79 (1976), 71{99., Spectral Asymmetry and Riemannian Geometry, Bull. London Math. Soc. 5(1973), 229{234.M. F. Atiyah and G. B. Segal, The Index of Elliptic Operators II, Ann.ofMath.87 (1968),531{545.M. F. Atiyah and I. M. Singer, The Index of Elliptic Operators I, Ann. of Math. 87 (1968),484{530 III, 87 (1968), 546{604 IV, 93 (1971), 119{138 V, 93 (1971), 139{149., The Index of Elliptic Operators on Compact Manifolds, Bull. Amer. Math. Soc.69 (1963), 422{433., Index Theory for Skew-Adjoint Fredholm Operators, Inst.Hautes Etudes Sci.Publ. Math. 37 (1969).P. Baum and J. Cheeger, Innitesimal Isometries and Pontrjagin Numbers, Topology 8(1969), 173{193.P. Berard and M. Berger, Le Spectre d'Une Variete Riemannienne en 1981, (preprint).M. Berger, Sur le Spectre d'Une Variete Riemannienne, C. R. Acad. Sci. Paris Ser. I Math.163 (1963), 13{16., Eigenvalues of the Laplacian, Proc. Symp. Pure Math. 16 (1970), 121{125.M. Berger, P. Gauduchon and E. Mazet, \Le Spectre d'Une Variete Riemannienne",Springer Verlag, 1971.B. Booss, \Topologie und Analysis", Springer-Verlag, 1977.B. Booss and S. Rempel, Cutting and Pasting of Elliptic Operators, C.R. Acad. Sci. ParisSer. I Math. 292 (1981), 711{714.

340 BibliographyB. Booss and B. Schulze, Index Theory and Elliptic Boundary Value Problems, Remarks andOpen Problems, (Polish Acad. Sci. 1978 preprint).R. Bott, Vector Fields and Characteristic Numbers, Michigan Math. J. 14 (1967), 231{244., AResidue Formula for Holomorphic Vector Fields, J. Di. Geo. 1 (1967), 311{330., \Lectures on K(X)", Benjamin, 1969.R. Bott and R. Seeley, Some Remarks on the Paper of Callias, Comm. Math. Physics 62(1978), 235{245.L. Boutet de Monvel, Boundary Problems for Pseudo-Dierential Operators, Acta Math. 126(1971), 11{51.A. Brenner and M. Shubin, Atiyah-Bott-Lefschetz Theorem for Manifolds with Boundary,J. Funct. Anal. (1982), 286{287.C. Callias, Axial Anomalies and Index Theorems on Open Spaces, Comm. Math. Physics 62(1978), 213{234.J. Cheeger, A Lower Bound for the Smallest Eigenvalue of the Laplacian, in \Problems inAnalysis", Princeton Univ. Press, 1970, pp. 195{199., On the Hodge Theory of Riemannian Pseudo-Manifolds, Proc. Symp. in PureMath. 36 (1980), 91{146., Analytic Torsion and the Heat Equation, Ann. of Math. 109 (1979), 259{322., Analytic Torsion and Reidemeister Torsion, Proc. Nat. Acad. Sci. USA 74(1977), 2651{2654., Spectral Geometry of Singular Riemannian Spaces, J. Di. Geo. 18 (1983),575{651., On the Spectral Geometry of Spaces with Cone-like Singularities, Proc. Nat. Acad.Sci. USA 76 No. 5 (1979), 2103-2106.J. Cheeger and M. Taylor, On the Diraction of Waves by Conical Singularities, Comm. PureAppl. Math. 25 (1982), 275{331.J. Cheeger and J. Simons, Dierential Characters and Geometric Invariants, (preprint).S. S. Chern, A Simple Intrinsic Proof of the Gauss-Bonnet Formula for Closed RiemannianManifolds, Ann. of Math. 45 (1944), 741{752., Pseudo-Riemannian Geometry and the Gauss-Bonnet Formula, Separata do Vol35 #1 Dos Anais da Academia Brasileria de Ciencias Rio De Janeiro (1963), 17{26., Geometry of Characteristic Classes, in \Proc. 13th Biennial Seminar", CanadianMath. Congress, 1972, pp. 1{40., On the Curvatura Integra inaRiemannian Manifold, Ann.ofMath.46 (1945),674{684., \Complex Manifolds Without Potential Theory", Springer Verlag, 1968.S. S. Chern and J. Simons, Characteristic Forms and Geometric Invariants, Ann. of Math.99 (1974), 48{69., Some Cohomology Classes in Principal Fiber Bundles and Their Applications toRiemannian Geometry, Proc. Nat. Acad. Sci. USA 68 (1971), 791{794.A. Chou, The Dirac Operator on Spaces with Conical Singularities, Ph. D. Thesis, S. U. N. Y.,Stonybrook, 1982.Y. Colin de Verdierre, Spectre duLaplacian et Longuers des Geodesiques Periodiques II, CompositioMath. 27 (1973), 159{184.G. de Rham, \Varietes Dierentiables", Hermann, 1954, pp. 127{131., Complexes a Automorphismes et Homeomorphie Dierentiable, Annales deL'Institut Fourier 2 (1950), 51{67.J. Dodziuk, Eigenvalues of the Laplacian and the Heat Equation, Amer. Math. Monthly 88(1981), 687{695.A. Domic, An A Priori Inequality for the Signature Operator, Ph. D. Thesis, M. I. T., 1978.H. Donnelly, Eta Invariant of a Fibered Manifold, Topology 15 (1976), 247{252., Eta Invariants for G-Spaces, Indiana Math. J. 27 (1978), 889{918.

Bibliography 341, Spectral Geometry and Invariants from Dierential Topology, Bull. London Math.Soc. 7 (1975), 147{150., Spectrum and the Fixed Point Set of Isometries I, Math. Ann. 224 (1976),161{170., Symmetric Einstein Spaces and Spectral Geometry, Indiana Math. J. 24 (1974),603{606., Eta Invariants and the Boundaries of Hermitian Manifolds, Amer. J. Math. 99(1977), 879{900.H. Donnelly and V. K. Patodi, Spectrum and the Fixed Point Set of Isometries II, Topology16 (1977), 1{11.J. S. Dowker and G. Kennedy, Finite Temperature and Boundary Eects in Static Space-Times, J.Phys A. Math. 11 (1978), 895{920.J. J. Duistermaat, \Fourier Integral Operators", Courant Institute Lecture Notes, 1973.J. J. Duistermaat and V. W. Guillemin, The Spectrum of Positive Elliptic Operators and PeriodicBicharacteristics, Invent. Math. 29 (1975), 39{79., The Spectrum of Positive Elliptic Operators and Periodic Geodesics, Proc. Symp.Pure Math. 27 (1975), 205{209.D. Eck, \Gauge Natural and Generalized Gauge Theories", Memoirs of the Amer. Math.Soc. No. 33, 1981.D. Epstein, Natural Tensors on Riemannian Manifolds, J. Di. Geo. 10 (1975), 631{646.H. D. Fegan, The Laplacian with a Character as a Potential and the Clebsch-Gordon Numbers,(preprint)., The Heat Equation and Modula Forms, J. Di. Geo. 13 (1978), 589{602., The Spectrum of the Laplacian on Forms over a Lie Group, Pacic J. Math. 90(1980), 373{387.H. D. Fegan and P. Gilkey, Invariants of the Heat Equation, Pacic J. Math. (to appear).S. A. Fulling, The Local Geometric Asymptotics of Continuum Eigenfunction Expansions I, SiamJ. Math. Anal. 31 (1982), 891{912.M. Ganey, Hilbert Space Methods in the Theory of Harmonic Integrals, Trans. Amer. Math.Soc. 78 (1955), 426{444.P. B. Gilkey, Curvature and the Eigenvalues of the Laplacian for Elliptic Complexes, Adv. inMath. 10 (1973), 344{381., Curvature and the Eigenvalues of the Dolbeault Complex for Kaehler Manifolds,Adv. in Math. 11 (1973), 311{325., The Boundary Integrand in the Formula for the Signature and Euler Characteristicof a Riemannian Manifold with Boundary, Adv. in Math. 15 (1975), 344{360., The Spectral Geometry of a Riemannian Manifold, J. Di. Geo. 10 (1975),601{618., Curvature and the Eigenvalues of the Dolbeault Complex for Hermitian Manifolds,Adv. in Math. 21 (1976), 61{77., Lefschetz Fixed Point Formulas and the Heat Equation, in \Partial DierentialEquations and Geometry. Proceedings of the Park City Conference", MarcelDekker, 1979, pp. 91{147., Recursion Relations and the Asymptotic Behavior of the Eigenvalues of the Laplacian,Compositio Math. 38 (1979), 281{240., Curvature and the Heat Equation for the de Rham Complex, in \Geometry andAnalysis", Indian Academy of Sciences, 1980, pp. 47{80., The Residue of the Local Eta Function at the Origin, Math. Ann. 240 (1979),183{189., The Spectral Geometry of the Higher Order Laplacian, Duke Math. J. 47 (1980),511{528 correction, 48 (1981), p. 887.

342 Bibliography, Local Invariants of a Pseudo-Riemannian Manifold, Math. Scand. 36 (1975),109{130., The Residue of the Global Eta Function at the Origin, Adv. in Math. 40 (1981),290{307., The Eta Invariant for Even Dimensional Pin-c Manifolds, (preprint)., On Spherical Space Forms with Meta-cyclic Fundamental Group which are Isospectralbut not Equivariant Cobordant, (preprint)., The Eta Invariant and the K-Theory of Odd Dimensional Spherical Space Forms,Invent. Math. (to appear).P. B. Gilkey and L. Smith, The Eta Invariant for a Class of Elliptic Boundary Value Problems,Comm. Pure Appl. Math. 36 (1983), 85{132., The Twisted Index Theorem for Manifolds with Boundary, J. Di. Geo. 18(1983), 393{344.C. Gordon and E. Wilson, Isospectral Deformations of Compact Solvmanifolds, (preprint).A. Gray, Comparison Theorems for the Volumes of Tubes as Generalizations of the Weyl TubeFormula, Topology 21 (1982), 201{228.P. Greinger, An Asymptotic Expansion for the Heat Equation, Proc. Symp. Pure Math. 16(1970), 133{135., An Asymptotic Expansion for the Heat Equation, Archiv Rational MechanicsAnal. 41 (1971), 163{218.M. Gromov and H. Lawson, Positive Scalar Curvature and the Dirac Operator on CompleteRiemannian Manifolds, (preprint)., Classication of simply connected manifolds of positive scalar curvature, Ann. ofMath. 111 (1980), 423{434.P. Gunther and R. Schimming, Curvature andSpectrum of Compact Riemannian Manifolds,J. Di. Geo. 12 (1977), 599{618.S. Helgason, Eigenspaces of the Laplacian, Integral Representations and Irreducibility, J.Funct.Anal. 17 (1974), 328{353.F. Hirzebruch, \Topological Methods in Algebraic Geometry", Springer Verlag, 1966., The Signature Theorem: Reminiscenes and Recreation, in \Prospects in Mathematics",Ann. of Math. Studies 70, Princeton Univ. Press, 1971, pp. 3{31.F. Hirzebruch and D. Zagier, \The Atiyah-Singer Theorem and Elementary NumberTheory", Publish or Perish, Inc., 1974.N. Hitchin, Harmonic Spinors, Adv. in Math. 14 (1974), 1{55.L. Hormander, Fourier Integral Operators I, Acta Math. 127 (1971), 79{183., Pseudo-dierential Operators and Non-Elliptic Boundary Problems, Ann. ofMath. 83 (1966), 129{209., Pseudo-Dierential Operators and Hypoelliptic Equations, Comm. Pure Appl.Math. 10 (1966), 138{183., Pseudo-Dierential Operators, Comm. Pure Appl. Math. 18 (1965), 501{517.A. Ikeda, On Spherical Space Forms which are Isospectral but not Isometric, J. Math. Soc.Japan 35 (1983), 437{444., On the Spectrum of a Riemannian Manifold of Positive Constant Curvature, Osaka J.Math. 17 (1980), 75{93 II, Osaka J. Math. 17 (1980), 691{701., On Lens Spaces which are Isospectral but not Isometric, Ec. Norm. (1980), 303{315., Isospectral Problem for Spherical Space Forms, (preprint).A. Ikeda and Y. Taniguchi, Spectra and Eigenforms of the Laplacian on Sphere and ProjectiveSpace, Osaka J. Math. 15 (1978), 515{546.M. Kac, Can One Hear the Shape of a Drum?, Amer. Math. Monthly 73 (1966), 1{23.T. Kambe, The Structure of K-Lambda-Rings of the Lens Space and Their Application, J. Math.Soc. Japan 18 (1966), 135{146.M. Karoubi, K-theorie Equivariante des Fibres en Spheres,, Topology 12 (1973), 275{281.

Bibliography 343, \K-Theory, AnIntroduction", Springer-Verlag, 1978.K. Katase, Eta Function on Three-Sphere, Proc. Japan. Acad. 57 (1981), 233{237.T. Kawasaki, An Analytic Lefschetz Fixed Point Formula and its Application to V -Manifolds,General Defect Formula and Hirzebruch Signature Theorem, Ph. D. Thesis, Johns HopkinsUniversity, 1976., The Riemann-Roch Formula for Complex V -Manifolds, Topology 17 (1978),75{83.G. Kennedy, Some nite Temperature Quantum Field Theory Calculations in Curved Manifoldswith Boundaries, Ph. D. Thesis, University ofManchester, 1979.J. J. Kohn and L. Nirenberg, An Algebra of Pseudo-dierential Operators, Comm. PureAppl. Math. 18 (1965), 269{305.M. Komuro, On Atiyah-Patodi-Singer Eta Invariant for S-1 Bundles over Riemann Surfaces,J. Fac. Sci. Tokyo 30 (1984), 525{548.T. Kotake, The Fixed Point Theorem of Atiyah-Bott via Parabolic Operators, Comm. PureAppl. Math. 22 (1969), 789{806., AnAnalytic Proof of the Classical Riemann-Roch theorem, Proc. Symp. Pure Math.16 (1970), 137{146.R. Kulkarni, \Index Theorems of Atiyah-Bott-Patodi and Curvature Invariants", MontrealUniversity, 1975.H. Kumano-go, \Pseudo-dierential Operators", M. I. T. Press, 1974.S. C. Lee, ALefschetz Formula for Higher Dimensional Fixed Point Sets, Ph. D. Thesis, BrandeisUniversity, 1975.A. Lichnerowicz, Spineurs Harmoniques, C. R. Acad. Sci. Paris Ser. I Math. 257 (1963),7{9.N. Mahammed, K-theorie des Formes Spheriques Tetradriques, C. R. Acad. Sci. Paris Ser. IMath. 281 (1975), 141{144., APros e la K-theorie des Espaces Lenticulaires, C. R. Acad. Sci. Paris Ser. IMath. 271 (1970), 639{642., K-Theorie des Formes Spheriques, Thesis L'Universite des Sciences et Techniquesde Lille, 1975.H. P. McKean and I. M. Singer, Curvature and the Eigenvalues of the Laplacian, J. Di. Geo.1 (1967), 43{69.E. Miller, Ph. D. Thesis, M. I. T.R. Millman, Manifolds with the Same Spectrum, Amer. Math. Monthly 90 (1983), 553{555.J. Millson, Chern-Simons Invariants of Constant Curvature Manifolds, Ph. D. Thesis, U. C.,Berkeley, 1974., Closed Geodesics and the Eta Invariant, Ann. of Math. 108 (1978), 1{39., Examples of Nonvanishing Chern-Simons Invariant, J. Di. Geo. 10 (1974),589{600.J. Milnor, \Lectures on Characteristic Classes", Princeton Notes, 1967., Eigenvalues of the Laplace Operator on Certain Manifolds, Proc. Nat. Acad. Sci.USA 51 (1964), 775{776.S. Minakshisundaram and A. Pleijel, Some Properties of the Eigenfunctions of the LaplaceOperator on Riemannian Manifolds, Canadian J. Math. 1 (1949), 242{256.A. Nestke, Cohomology of Elliptic Complexes and the Fixed Point Formula of Atiyah and Bott,Math. Nachr. 104 (1981), 289{313.L. Nirenberg, \Lectures on Linear Partial Dierential Equations", Regional ConferenceSeries in Math. No. 17, Amer. Math. Soc., 1972.R. Palais, \Seminar on the Atiyah-Singer Index Theorem", Ann. Math. Study 57,Princeton Univ. Press, 1965.V. K. Patodi, Curvature and the Eigenforms of the Laplace Operator, J.Di. Geo 5 (1971),233{249.

344 Bibliography, An Analytic Proof of the Riemann-Roch-Hirzebruch Theorem for Kaehler Manifolds,J. Di. Geo. 5 (1971), 251{283., Holomorphic Lefschetz Fixed Point Formula, Bull. Amer. Math. Soc. 79(1973), 825{828., Curvature and the Fundamental Solution of the Heat Operator, J. Indian Math.Soc. 34 (1970), 269{285.M. Pinsky, On the Spectrum of Cartan-Hadamard Manifolds, Pacic J. Math. 94 (1981),223-230.D. B. Ray, Reidemeister Torsion and the Laplacian on Lens Spaces, Adv. in Math. 4 (1970),109{126.D. B. Ray and I. M. Singer, RTorsion and the Laplacian on Riemannian Manifolds, Adv. inMath. 7 (1971), 145{210., Analytic Torsion for Complex Manifolds, Ann. of Math. 98 (1973), 154{177.S. Rempel, An Analytical Index formula for Elliptic Pseudo-dierential Boundary Problems,Math. Nachr. 94 (1980), 243{275.S. Rempel and B. Schultze, Complex Powers for Pseudo-dierential Boundary Problems,Math. Nachr. 111 (1983), 41{109.T. Sakai, On Eigenvalues of Laplacian and Curvature ofRiemannian Manifold, Tohoku Math.Journal 23 (1971), 589{603., On the Spectrum of Lens Spaces, Kodai Math. Sem. 27 (1976), 249{257.R. T. Seeley, Complex Powers of an Elliptic Operator, Proc. Symp. Pure Math. 10 (1967),288{307., Analytic Extension of the TraceAssociated with Elliptic Boundary Problems, Amer.J. Math. 91 (1969), 963{983., Topics in Pseudo-Dierential Operators, in \Pseudo-Dierential Operators",C. I. M. E., 1968, pp. 168{305., The Resolvant of an Elliptic Boundary Problem, Amer. J. Math. 91 (1969),889{920., Singular Integrals and Boundary Problems, Amer. J. Math. 88 (1966), 781{809., A Proof of the Atiyah-Bott-Lefschetz Fixed Point Formula, An. Acad. BrasilCience 41 (1969), 493{501., Integro-dierential Operators on Vector Bundles, Trans. Amer. Math. Soc. 117,167{204., Asymptotic Expansions at Cone-like Singularities, (preprint).J. Simons, Characteristic Forms and Transgression II{Characters Associated to a Connection,(preprint).I. M. Singer, Connections Between Geometry, Topology, and Analysis, in \Colloquium Lecturesat 82 Meeting", Amer. Math. Soc., 1976., Elliptic Operators on Manifolds, in \Pseudo-dierential Operators", C. I. M. E,1968, pp. 335{375.D. E. Smith, The Atiyah-Singer Invariant, Torsion Invariants, and Group Actions on Spheres,Trans. Amer. Math. Soc. 177 (1983), 469{488.L. Smith, The Asymptotics of the Heat Equation for a Boundary Value Problem, Invent. Math.63 (1981), 467{494.B. Steer, A Note on Fuchsian Groups, Singularities and the Third Stable Homotopy Group ofSpheres, (preprint).P. Stredder, Natural Dierential Operators on Riemannian Manifolds and Representations of theOrthogonal and Special Orthogonal Group, J. Di. Geo. 10 (1975), 647{660.M. Tanaka, Compact Riemannian Manifolds which are Isospectral to Three-dimensional LensSpaces, in \Minimal Submanifolds and Geodesics", North-Holland, 1979, pp. 273{283 II, Proc. Fac. Sci. Tokai Univ. 14 (1979), 11{34.

Bibliography 345D. Toledo, On the Atiyah-Bott formula for Isolated Fixed Points, J. Di. Geo. 8 (1973),401{436.D. Toledo and Y. L. Tong, A Parametrix for the D-Bar and Riemann-Roch in Cech Theory,Topology 15 (1976), 273{301., The Holomorphic Lefschetz Formula, Bull. Amer. Math. Soc. 81 (1975),1133{1135.Y. L. Tong, De Rham's Integrals and Lefschetz Fixed Point Formula for D-Bar Cohomology,Bull. Amer. Math. Soc. 78 (1972), 420{422.F. Treves, \An Introduction to Pseudo-dierential and Fourier Integral Operators",Plenum Press, 1980.H. Urakawa, Bounded Domains which are Isospectral but not Isometric, Ec. Norm. 15 (1982),441{456.M. F. Vigneras, Varietes Riemanniennes Isospectrales et non Isometriques, Ann. of Math. 112(1980), 21{32.H. Weyl, \The Classical Groups", Princeton Univ. Press, 1946., On the Volume of Tubes, Amer. J. Math. 61 (1939), 461{472.H. Widom, Asymptotic Expansions for Pseudo-dierential Operators on Bounded Domains,(preprint)., ATrace Formula for Wiener-Hopf Operators, J. Operator Th. 8 (1982), 279-298.L. Willis, Invariants of Riemannian Geometry and the Asymptotics of the Heat Equation, Ph.D. Thesis, Univ. of Oregon, 1984.M. Wodzicki, Spectral Asymmetry and Zeta Functions, Invent. Math. 66 (1982), 115{135.K. Wojciechowski, Spectral Asymmetry, Elliptic Boundary Value Problem, and Cutting and Pastingof Elliptic Operators, (preprint)., Spectral Flow and the General Linear Conjugation Problem, (preprint).T. Yoshida, The Eta Invariant of Hyperbolic 3-manifolds, (preprint).D. Zagier, \Equivariant Pontrjagin Classes and Applications to Orbit Spaces", Springer-Verlag, 1972., Higher Dimensional Dedekind Sums, Math. Ann. 202 (1973), 149{172.

INDEXA-roof ( ^A) genus, 99Absolute boundary conditions for thede Rham complex, 243Algebra of operators, 13Allard, William, 45Almost complex structure, 180Arithmetic genus, 190, 200Arzela-Ascoli Theorem, 8Asymptotic expansion of symbol, 51Asymptoticsof elliptic boundary conditions, 72of heat equation, 54Atiyah-Bott Index Theorem, 247Atiyah-Patodi-Singer Theorem, 268, 276Atiyah-Singer Index Theorem, 233Axiomatic characterizationof Chern forms, 202of Euler forms, 121of Pontrjagin forms, 130Bigrading exterior algebra, 107Bott periodicity, 221Bundle of Cauchy data, 70Bundles of positive part of symbol, 76Canonical bundle, 186Cauchy data, 70Cech Z 2 cohomology, 166Change of coordinates, 25Characteristic forms, 91Characteristic ring, 93, 98, 103Cherncharacter, 91, 174, 217class of complex projective space, 111form, 91isomorphism in K-theory, 217Christoel symbols, 104Cliordalgebra, 148matrices, 95multiplication, 148Clutching functions, 218Cohomologyof complex projective space, 109, 200of elliptic complex, 37of Kaehler manifolds, 195Collared manifold, 253Compact operators, 31Complexcharacteristic ring, 93projective space, 109tangent space, 107Connected sum, 154Connection 1-form, 89Continuity of operators, 12Convolution, 2Cotangent space, 25Cup product, 151Curvatureof spin bundle, 1722-form, 90d-bar ( @), 107Defect formulas, 286, 296de Rhamcohomology, 91, 197complex, 39, 149, 246de Rham, signature and spin bundles, 172Dirac matrices, 95Dirichlet boundary conditions, 71, 245Dolbeault complex, 183Dolbeault and de Rham cohomology, 197Donnelly Theorem, 293Double suspension, 220Dual norm, 30Ellipticboundary conditions, 72complex, 37operators, 23Etafunction, 81, 258, 263invariant, 258, 270Eulercharacteristic of elliptic complex, 37form, 100, 121, 174, 308Euler-Poincare characteristic, 246Exterioralgebra, 148multiplication, 39, 148Flat K-theory, 298Form valued polynomial, 129Fourier transform, 3Fredholmindex, 32operators, 32

348 IndexFubini-Study metric, 111, 195, 200Functional calculus, 48Garding's inequality, 23Gauss-Bonnet Theorem, 122, 179, 256Geodesically convex, 166Gradedboundary conditions, 70symbol, 71vector bundle, 37Greens operator, 44Gunther and Schimming, 313Half spin representations, 161Heat equation, 47local formulas, 326HirzebruchL-polynomial, 99, 115, 175Signature Theorem, 154HodgeDecomposition Theorem, 38, 246manifolds, 196 operator, 246Holomorphicconnection, 108manifold, 107Holonomy of representation, 270Homotopy groups of U(n), 221Hyperplane bundle, 109Hypoelliptic, 23Indexand heat equation, 47form in K-theory, 299formula (local), 58in K-theory, 225of representation, 271R=Z valued, 270Index Theoremfor de Rham complex, 122for Dolbeault complex, 189for signature complex, 154for spin complex, 179Instanton bundles, 96Integration along bers, 235Interior multiplication, 40, 148Invariantform valued polynomial, 134polynomials, 119Invariants of heat equation, 326Isospectral manifolds, 331K-theory, 216, 225of locally at bundles, 298of projective spaces, 302of spheres, 221of spherical space forms, 301sphere bundle, 267Kaehlermanifolds, 1952-form, 193Kernelof heat equation, 54of operator, 19L-polynomial, 99, 115, 175Lefschetzxed point formulas, 63, 68, 286number of elliptic complex, 62Lens space, 276, 297Levi-Civita connection, 104Localformula for derivative of eta, 82index formula, 58Locally at bundle, 270Maurer-Cartan form, 275Mellin transform, 78Monopole bundles, 96Neuman series, 29Neumann boundary conditions, 71, 245Nirenberg-Neulander Theorem, 182Operators depending on a complexparameter, 50Orientation form, 150Parametrix, 29, 52Peetre's Inequality, 10Plancherel Formula, 5Poincare duality, 41, 246Pontrjagin form, 98, 174, 308Projective spacearithmetic genus, 200cohomology, 109signature, 152Pseudo-dierential operators, 11Pseudolocal, 20R=Z valued index, 270Real characteristic ring, 99Regularity of eta at 0, 263Relative boundary conditions for thede Rham complex, 243Rellich Lemma, 8Residue of eta at s =0, 84, 258, 266

Index 349Resolvant of operator, 52restriction of invariant, 120, 255Riemann zeta function, 80Riemann-Roch Theorem, 189, 202Ring of invariant polynomials, 119Schwartz class, 2Secondary characteristic classes, 273Serre duality, 199Shape operator, 251Shue formulas, 313Signaturecomplex, 150tangential operator, 261of projective space, 153spin and de Rham bundles, 172Theorem, 154Simple covers, 166Singer's conjecture for the Euler form, 308Snaith's Theorem, 305SobolevLemma, 7spaces, 6Spectralow, 273geometry, 331Spectrumof compact operator, 42of self-adjoint elliptic operator, 43Spherebundle K-theory, 267Chern character, 96Euler form, 101K-theory, 221Spherical space form, 295, 333Spinand other elliptic complexes, 172complex, 176group, 159structures, 165on projective spaces, 170SPIN c ,186Spinors, 159Splitting principal, 190Stable isomorphism of vector bundles, 216Stationary phase, 64Stieel-Whitney classes, 167Suspensionof topological space, 215of symbol, 224Symbolde Rham complex, 40Spin complex, 176Dolbeault complex, 184of operator on a manifold, 25Tangential operator signaturecomplex, 261Tautological bundle of CPn, 109,195Todd form, 97, 114, 236Topological charge, 223Trace heat operator, 47Transgression, 92, 252U(n), homotopy groups of, 221Weyl spanning set, 121Weyl's theorem on invariants of orthogonalgroup, 132Zeta function, 80