Equivariant Cohomological Chern Characters

Equivariant Cohomological Chern CharactersWolfgang Lück ∗Fachbereich MathematikUniversität MünsterEinsteinstr. 6248149 MünsterGermanyAugust 30, 2004AbstractWe construct for an equivariant cohomology theory for proper equivariantCW -complexes an equivariant Chern character, provided that certainconditions about the coefficients are satisfied. These conditions arefulfilled if the coefficients of the equivariant cohomology theory possess aMackey structure. Such a structure is present in many interesting examples.Key words: equivariant cohomology theory, equivariant Chern character.Mathematics Subject Classification 2000: 55N91.0. IntroductionThe purpose of this paper is to construct an equivariant Chern characterfor a proper equivariant cohomology theory H? ∗ with values in R-modules for acommutative associative ring R with unit which satisfies Q ⊆ R. It is a naturaltransformation of equivariant cohomology theoriesch ∗ ? : H ∗ ? → BH ∗ ?for a given equivariant cohomology theory H? ∗ . Here BH∗ ?is the associatedequivariant cohomology theory which is defined by the Bredon cohomology withcoefficients coming from the coefficients of H? ∗ . The notion of an equivariantcohomology theory and examples for it are presented in Section 1 and the associatedBredon cohomology is explained in Section 3. The point is that BH? ∗ is∗ email: lueck@math.uni-muenster.dewww: http://www.math.uni-muenster.de/u/lueck/FAX: 49 251 83383701

much simpler and easier to compute than H? ∗ . If H∗ ?satisfies the disjoint unionaxiom, thench n G(X, A): HG(X, n A) −→ ∼= BHG(X, n A)is bijective for every discrete group G, proper G-CW -pair (X, A) and n ∈ Z.The Chern character ch ∗ ? is only defined if the coefficients of H? ∗ satisfy acertain injectivity condition (see Theorem 4.6). This condition is fulfilled if thecoefficients of H? ∗ come with a Mackey structure (see Theorem 5.5) what is thecase in many interesting examples.The equivariant cohomological Chern character is a generalization to theequivariant setting of the classical non-equivariant Chern character for a (nonequivariant)cohomology theory H ∗ (see Example 4.1)ch n (X, A): H n (X, A) ∼= −→∏p+q=nH p (X, A, H q (∗)).The equivariant cohomological Chern character has already been constructedin the special case, where H? ∗ is equivariant topological K-theory K∗ ?, in [12].Its homological version has already been treated in [8] and plays an importantrole in the computation of the source of the assembly maps appearing in theFarrell-Jones Conjecture for K n (RG) and L 〈−∞〉n (RG) and the Baum-ConnesConjecture for K n (Cr ∗ (G)) (see also [9]).The detailed formulation of the main result of this paper is presented inTheorem 5.5.The equivariant Chern character will play a key role in the proof of thefollowing result which will be presented in [11].Theorem 0.1 (Rational computation of the topological K-theory ofBG). Let G be a discrete group. Suppose that there is a finite G-CW -model forthe classifying space EG for proper G-actions. Then there is a Q-isomorphismch n G : K n (BG) ⊗ Z Q∼ =−→ ∏ i∈ZH 2i+n (BG; Q) × ∏p prime∏(g)∈con p(G)H 2i+n (BC G 〈g〉; Q̂p).Here con p (G) is the set of conjugacy classes (g) of elements g ∈ G of orderp m for some integer m ≥ 1 and C G 〈g〉 is the centralizer of the cyclic subgroupgenerated by g in G.The assumption in Theorem 0.1 that there is a finite G-CW -model for theclassifying space EG for proper G-actions is satisfied for instance, if G is wordhyperbolicin the sense of Gromov, if G is a cocompact subgroup of a Lie groupwith finitely many path components, if G is a finitely generated one-relatorgroup, if G is an arithmetic group, a mapping class group of a compact surfaceor the group of outer automorphisms of a finitely generated free group. Formore information about EG we refer for instance to [1] and [10].A group G is always understood to be discrete and a ring R is always understoodto be associative with unit throughout this paper.2

The paper is organized as follows:1. Equivariant Cohomology Theories2. Modules over a Category3. The Associated Bredon Cohomology Theory4. The Construction of the Equivariant Cohomological Chern Character5. Mackey Functors6. Multiplicative StructuresReferences1. Equivariant Cohomology TheoriesIn this section we describe the axioms of a (proper) equivariant cohomologytheory. They are dual to the ones of a (proper) equivariant homology theory asdescribed in [8, Section 1].Fix a group G and an commutative ring R. A G-CW -pair (X, A) is a pair ofG-CW -complexes. It is proper if all isotropy groups of X are finite. It is relativefinite if X is obtained from A by attaching finitely many equivariant cells, or,equivalently, if G\(X/A) is compact. Basic information about G-CW -pairs canbe found for instance in [7, Section 1 and 2]. A G-cohomology theory HG ∗ withvalues in R-modules is a collection of covariant functors HG n from the categoryof G-CW -pairs to the category of R-modules indexed by n ∈ Z together withnatural transformations δG n (X, A): Hn G (X, A) → Hn+1G(A) := Hn+1G(A, ∅) forn ∈ Z such that the following axioms are satisfied:• G-homotopy invarianceIf f 0 and f 1 are G-homotopic maps (X, A) → (Y, B) of G-CW -pairs, thenH n G (f 0) = H n G (f 1) for n ∈ Z;• Long exact sequence of a pairGiven a pair (X, A) of G-CW -complexes, there is a long exact sequence. . . δn−1 G−−−→ HG(X, n A) Hn G−−−−→ (j)HG(X) n Hn G−−−−→ (i)HG(A) n δn G−→ . . . ,where i: A → X and j : X → (X, A) are the inclusions;• ExcisionLet (X, A) be a G-CW -pair and let f : A → B be a cellular G-map ofG-CW -complexes. Equip (X ∪ f B, B) with the induced structure of a G-CW -pair. Then the canonical map (F, f): (X, A) → (X ∪ f B, B) inducesan isomorphismH n G(F, f): H n G(X, A) ∼= −→ H n G(X ∪ f B, B).3

Sometimes also the following axiom is required.• Disjoint union axiomLet∐{X i | i ∈ I} be a family of G-CW -complexes. Denote by j i : X i →i∈I X i the canonical inclusion. Then the map( )∏∐HG(j n i ): HGn ∼ =X i −→ ∏ HG(X n i )i∈Ii∈I i∈Iis bijective.If HG ∗ is defined or considered only for proper G-CW -pairs (X, A), we callit a proper G-cohomology theory HG ∗ with values in R-modules.The role of the disjoint union axiom is explained by the following result. Itsproof for non-equivariant cohomology theories (see for instance [16, 7.66 and7.67]) carries over directly to G-cohomology theories.Lemma 1.1. Let H ∗ G and K∗ Gbe (proper) G-cohomology theories. Then(a) Suppose that HG ∗ satisfies the disjoint union axiom. Then there existsfor every (proper) G-CW -pair (X, A) with an exhaustion by subcomplexesA = X −1 ⊆ X 0 ⊆ X 1 ⊆ . . . ⊆ ⋃ n≥−1 X n = X a natural short exactsequence0 → lim 1 n→∞H p−1G(X n∪A, A) → H p (X, A) → limn→∞ Hp G (X n∪A, A) → 0;(b) Let T ∗ : HG ∗ → K∗ G be a transformation of (proper) G-cohomology theories,i.e. a collection of natural transformations T n : HG n → Kn G of contravariantfunctors from the category of (proper) G-CW -pairs to the categoryof R-modules indexed by n ∈ Z which is compatible with the boundaryoperator associated to (proper) G-CW -pairs. Suppose that T n (G/H) isbijective for every (proper) homogeneous space G/H and n ∈ Z.Then T n (X, A): HG ∗ (X, A) → K∗ G (X, A) is bijective for all n ∈ Z providedthat (X, A) is relative finite or that both H ∗ and K ∗ satisfy the disjointunion axiom.Remark 1.2 (The disjoint union axiom is not compatible with −⊗ Z Q).Let H ∗ G be a G-cohomology theory with values in Z-modules. Then H∗ G ⊗ Z Q isa G-cohomology theory with values in Q-modules since Q is flat as Z-module.However, even if H ∗ satisfies the disjoint union axiom, H ∗ G ⊗ Z Q does not satisfythe disjoint union axiom since − ⊗ Z Q is not compatible with products overarbitrary index sets.4

Example 1.3 (Rationalizing topological K-theory). Consider for instancethe (non-equivariant) cohomology theory with values in Z-modules satisfyingthe disjoint union axiom given by topological K-theory K ∗ . Let K ∗ (−; Q)be the cohomology theory associated to the rationalization of the K-theoryspectrum. This is a (non-equivariant) cohomology theory with values in Q-modules satisfying the disjoint union axiom. There is a natural transformationT ∗ (X): K ∗ (X) ⊗ Z Q → K ∗ (X; Q)of (non-equivariant) cohomology theory with values in Q-modules. The Q-mapT n ({pt.}) is bijective for all n ∈ Z. Hence T n (X) is bijective for all finiteCW -complexes by Lemma 1.1 (b). Notice that K ∗ (X) ⊗ Z Q does not satisfythe disjoint union axiom in contrast to K ∗ (X; Q). Hence we cannot expectT n (X) to be bijective for all CW -complexes. Consider the case X = BG fora finite group G. Since H p (BG; Q) ∼ = H p ({pt.}; Q) for all p ∈ Z, one obtainsK p (BG; Q) ∼ = K p ({pt.}; Q) for all p ∈ Z. By the Atiyah-Segal CompletionTheorem K p (BG) ⊗ Z Q ∼ = K p ({pt.}) ⊗ Z Q is only true if and only if the finitegroup G is trivial.Let α: H → G be a group homomorphism. Given an H-space X, define theinduction of X with α to be the G-space ind α X which is the quotient of G × Xby the right H-action (g, x) · h := (gα(h), h −1 x) for h ∈ H and (g, x) ∈ G × X.If α: H → G is an inclusion, we also write ind G H instead of ind α .A (proper) equivariant cohomology theory H? ∗ with values in R-modules consistsof a collection of (proper) G-cohomology theory HG ∗ with values in R-modules for each group G together with the following so called induction structure:given a group homomorphism α: H → G and a (proper) H-CW -pair(X, A) such that ker(α) acts freely on X, there are for each n ∈ Z naturalisomorphismsind α : H n G(ind α (X, A))∼ =−→ H n H(X, A) (1.4)satisfying(a) Compatibility with the boundary homomorphismsδ n H ◦ ind α = ind α ◦δ n G ;(b) FunctorialityLet β : G → K be another group homomorphism such that ker(β ◦ α) actsfreely on X. Then we have for n ∈ Zind β◦α = ind α ◦ ind β ◦H n K(f 1 ): H n H(ind β◦α (X, A)) → H n K(X, A),where f 1 : ind β ind α (X, A) ∼= −→ ind β◦α (X, A), (k, g, x) ↦→ (kβ(g), x) is thenatural K-homeomorphism;(c) Compatibility with conjugation5

For n ∈ Z, g ∈ G and a (proper) G-CW -pair (X, A) the homomorphismind c(g) : G→G : H n G (ind c(g) : G→G(X, A)) → H n G (X, A) agrees with Hn G (f 2),where f 2 is the G-homeomorphism f 2 : (X, A) → ind c(g) : G→G (X, A), x ↦→(1, g −1 x) and c(g)(g ′ ) = gg ′ g −1 .This induction structure links the various G-cohomology theories for differentgroups G. It will play a key role in the construction of the equivariant Cherncharacter even if we want to carry it out only for a fixed group G. In all of therelevant examples the induction homomorphism ind α of (1.4) exists for everygroup homomorphism α: H → G, the condition that ker(α) acts freely on X isonly needed to ensure that ind α is bijective. If α is an inclusion, we sometimeswrite ind G H instead of ind α .We say that H? ∗ satisfies the disjoint union axiom if for every group G theG-cohomology theory HG ∗ satisfies the disjoint union axiom.We will later need the following lemma whose elementary proof is analogousto the one in [8, Lemma 1.2].Lemma 1.5. Consider finite subgroups H, K ⊆ G and an element g ∈ Gwith gHg −1 ⊆ K. Let R g −1 : G/H → G/K be the G-map sending g ′ H tog ′ g −1 K and c(g): H → K be the homomorphism sending h to ghg −1 . Letpr: (ind c(g) : H→K {pt.}) → {pt.} be the projection. Then the following diagramcommutesHG n (G/K) H n G (R g−−−−−−−→−1 )HG n (G/H)⏐⏐ind G K↓ ∼ =H n K ({pt.})ind G H↓ ∼ =ind c(g) ◦H n K (pr)−−−−−−−−−−→ H n H ({pt.})Example 1.6 (Borel cohomology). Let K ∗ be a cohomology theory for (nonequivariant)CW -pairs with values in R-modules. Examples are singular cohomologyand topological K-theory. Then we obtain two equivariant cohomologytheories with values in R-modules by the following constructionsH n G(X, A) = K n (G\X, G\A);H n G(X, A) = K n (EG × G (X, A)).The second one is called the equivariant Borel cohomology associated to K.In both cases HG ∗ inherits the structure of a G-cohomology theory from thecohomology structure on K ∗ .The induction homomorphism associated to a group homomorphism α: H →G is defined as follows. Let a: H\X −→ ∼= G\(G × α X) be the homeomorphismsending Hx to G(1, x). Define b: EH × H X → EG × G G × α X by sending (e, x)to (Eα(e), 1, x) for e ∈ EH, x ∈ X and Eα: EH → EG the α-equivariant mapinduced by α. The desired induction map ind α is given by K ∗ (a) and K ∗ (b). Ifthe kernel ker(α) acts freely on X, the map b is a homotopy equivalence andhence in both cases ind α is bijective.If K ∗ satisfies the disjoint union axiom, the same is true for the two equivariantcohomology theories constructed above.6

Example 1.7 (Equivariant K-theory). In [12] G-equivariant topological(complex) K-theory KG ∗ (X, A) is constructed for any proper G-CW -pair (X, A)and shown that KG ∗ defines a proper G-cohomology theory satisfying the disjointunion axiom. Given a group homomorphism α: H → G, it induces an injectivegroup homomorphism α: H/ ker(α) → G. LetInfl H H/ ker(α) : K ∗ H/ ker(α) (ker(α)\X) → K∗ H(X)be the inflation homomorphism of [12, Proposition 3.3] andind α : K ∗ H/ ker(α) (ker(α)\X) ∼= −→ K ∗ G(ind α (ker(α)\X))be the induction isomorphism of [12, Proposition 3.2 (b)]. Define the inductionhomomorphismind α : K ∗ G(ind α X) → K ∗ H(X)by Infl H H/ ker(α) ◦(ind α ) −1 , where we identify ind α X = ind α (ker(α)\X). On thelevel of complex finite-dimensional vector bundles the induction homomorphismind α corresponds to considering for a G-vector bundle E over G × α X the H-vector bundle obtained from E by the pullback construction associated to theα-equivariant map X → G × α X, x ↦→ (1, x).Thus we obtain a proper equivariant cohomology theory K? ∗ with values inZ-modules which satisfies the disjoint union axiom. There is also a real versionKO? ∗ .Example 1.8 (Equivariant cohomology theories and spectra). Denoteby GROUPOIDS the category of small groupoids. Let Ω-SPECTRA be thecategory of Ω-spectra, where a morphism f : E → F is a sequence of mapsf n : E n → F n compatible with the structure maps and we work in the categoryof compactly generated spaces (see for instance [3, Section 1]). A contravariantGROUPOIDS-Ω-spectrum is a contravariant functor E: GROUPOIDS →Ω-SPECTRA.Next we explain how we can associate to it an equivariant cohomology theoryH? ∗ (−; E) satisfying the disjoint union axiom, provided that E respects equivalences,i.e. it sends an equivalence of groupoids to a weak equivalence of spectra.This construction is dual to the construction of an equivariant homology theoryassociated to a covariant GROUPOIDS-spectrum as explained in [13, Section6.2], [14, Theorem 2.10 on page 21].Fix a group G. We have to specify a G-cohomology theory HG ∗ (−; E). LetOr(G) be the orbit category whose set of objects consists of homogeneous G-spaces G/H and whose morphisms are G-maps. For a G-set S we denote byG G (S) its associated transport groupoid. Its objects are the elements of S. Theset of morphisms from s 0 to s 1 consists of those elements g ∈ G which satisfygs 0 = s 1 . Composition in G G (S) comes from the multiplication in G. Thus weobtain for a group G a covariant functorG G : Or(G) → GROUPOIDS, G/H ↦→ G G (G/H), (1.9)7

and a contravariant Or(G)-Ω-spectrum E ◦ G G . Given a G-CW -pair (X, A), weobtain a contravariant pair of Or(G)-CW -complexes (X ? , A ? ) by sending G/Hto (map G (G/H, X), map G (G/H, A)) = (X H , A H ). The contravariant Or(G)-spectrum E ◦ G G defines a cohomology theory on the category of contravariantOr(G)-CW -complexes as explained in [3, Section 4]. It value at (X ? , A ? )is defined to be HG ∗ (X, A; E). Explicitly, (Hn G (X, A; E) is the (−n)-th homotopygroup of the spectrum map Or(G) X + ? ∪ A ?+cone(A ? +), E ◦ G). G We needΩ-spectra in order to ensure that the disjoint union axiom holds.We briefly explain for a group homomorphism α: H → G the definition ofthe induction homomorphism ind α : HG n (ind α X; E) → HH n (X; E) in the specialcase A = ∅. The functor induced by α on the orbit categories is denoted in thesame wayα: Or(H) → Or(G), H/L ↦→ ind α (H/L) = G/α(L).There is an obvious natural transformation of functors Or(H) → GROUPOIDST : G H → G G ◦ α.Its evaluation at H/L is the functor of groupoids G H (H/L) → G G (G/α(L))which sends an object hL to the object α(h)α(L) and a morphism given byh ∈ H to the morphism α(h) ∈ G. Notice that T (H/L) is an equivalence ifker(α) acts freely on H/L. The desired isomorphismind α : H n G(ind α X; E) → H n H(X; E)is induced by the following map of spectramap Or(G)(mapG (−, ind α X + ), E ◦ G G)∼ =−→ map Or(G)(α∗ (map H (−, X + )), E ◦ G G)∼ =−→ map Or(H)(mapH (−, X + ), E ◦ G G ◦ α )map Or(H) (id,E(T ))−−−−−−−−−−−−→ map Or(H)(mapH (−, X + ), E ◦ G H) .Here α ∗ map H (−, X + ) is the pointed Or(G)-space which is obtained from thepointed Or(H)-space map H (−, X + ) by induction, i.e. by taking the balancedproduct over Or(H) with the Or(H)-Or(G) bimodule mor Or(G) (??, α(?)) [3,Definition 1.8]. The second map is given by the adjunction homeomorphism ofinduction α ∗ and restriction α ∗ (see [3, Lemma 1.9]). The first map comes fromthe homeomorphism of Or(G)-spacesα ∗ map H (−, X + ) → map G (−, ind α X + )which is the adjoint of the obvious map of Or(H)-spaces map H (−, X + ) →α ∗ map G (−, ind α X + ) whose evaluation at H/L is given by ind α .8

2. Modules over a CategoryIn this section we give a brief summary about modules over a small categoryC as far as needed for this paper. They will appear in the definition of theequivariant Chern character.Let C be a small category and let R be a commutative ring. A contravariantRC-module is a contravariant functor from C to the category R - MOD of R-modules. Morphisms of contravariant RC-modules are natural transformations.Given a group G, let Ĝ be the category with one object whose set of morphismsis given by G. Then a contravariant RĜ-module is the same as a right RGmodule.Therefore we can identify the abelian category MOD -RĜ with theabelian category of right RG-modules MOD-RG in the sequel. Many of theconstructions, which we will introduce for RC-modules below, reduce in thespecial case C = Ĝ to their classical versions for RG-modules. The reader shouldhave this example in mind. There is also a covariant version. In the sequel RCmodulemeans contravariant RC-module unless stated explicitly differently.The category MOD -RC of RC-modules inherits the structure of an abeliancategory from R - MOD in the obvious way, namely objectwise. For instance asequence 0 → M → N → P → 0 of contravariant RC-modules is called exact ifits evaluation at each object in C is an exact sequence in R - MOD. The notionof an injective and of a projective RC-module is now clear. For a set S denote byRS the free R-module with S as basis. An RC-module is free if it is isomorphicto RC-module of the shape ⊕ i∈I R mor C(?, c i ) for some index set I and objectsc i ∈ C. Notice that by the Yoneda-Lemma there is for every RC-module N andevery object c a bijection of setshom RC (R mor C (?, c), N)∼ =−→ N(c), φ ↦→ φ(id x ).This implies that every free RC-module is projective and a RC-module is projectiveif and only if it is a direct summand in a free RC-module. The categoryof RC-modules has enough projectives and injectives (see [7, Lemma 17.1] and[17, Example 2.3.13]).Given a contravariant RC-module M and a covariant RC-module N, theirtensor product over RC is defined to be the following R-module M ⊗ RC N. It isgiven byM ⊗ RC N =⊕M(c) ⊗ R N(c)/ ∼,c∈Ob(C)where ∼ is the typical tensor relation mf ⊗n = m⊗fn, i.e. for every morphismf : c → d in C, m ∈ M(d) and n ∈ N(c) we introduce the relation M(f)(m) ⊗n − m ⊗ N(f)(n) = 0. The main property of this construction is that it isadjoint to the hom R -functor in the sense that for any R-module L there arenatural isomorphisms of R-moduleshom R (M ⊗ RC N, L)hom R (M ⊗ RC N, L)∼ =−→ hom RC (M, hom R (N, L)); (2.1)∼ =−→ hom RC (N, hom R (M, L)). (2.2)9

Consider a functor F : C → D. Given a RD-module M, define its restrictionwith F to be F ∗ M := M ◦F . Given a contravariant RC-module M, its inductionwith F is the contravariant RD-module F ∗ M given by(F ∗ M)(??) := M(?) ⊗ RC R mor D (??, F (?)), (2.3)and coinduction with F is the contravariant RD-module F ! M given by(F ! M)(??) := hom RC (R mor D (F (?), ??), M(?)). (2.4)Restriction with F can be written as F ∗ N(?) = hom RD (R mor D (??, F (?)), N(??)),the natural isomorphisms sends n ∈ N(F (?)) to the mapR mor D (??, F (?)) → N(??),φ: ?? → F (?) ↦→ N(φ)(n).Restriction with F can also be written as F ∗ N(?) = R mor D (F (?), ??) ⊗ RDN(??), the natural isomorphisms sends φ ⊗ RD n to N(φ)(n). We conclude from(2.2) that (F ∗ , F ∗ ) and (F ∗ , F ! ) form adjoint pairs, i.e. for a RC-module M anda RD-module N there are natural isomorphisms of R-moduleshom RD (F ∗ M, N)hom RD (F ∗ N, M)∼ =−→ hom RC (M, F ∗ N); (2.5)∼ =−→ hom RC (N, F ! M). (2.6)Consider an object c in C. Let aut(c) be the group of automorphism of c.We can think of âut(c) as a subcategory of C in the obvious way. Denote byi(c): âut(c) → Cthe inclusion of categories and abbreviate the group ring R[aut(c)] by R[c] inthe sequel. Thus we obtain functorsThe projective splitting functorsends M to the cokernel of the map⊕M(f):f : c→df not an isomorphismThe injective splitting functori(c) ∗ : MOD -RC → MOD -R[c]; (2.7)i(c) ∗ : MOD -R[c] → MOD -RC; (2.8)i(c) ! : MOD -R[c] → MOD -RC. (2.9)S c : MOD -RC → MOD -R[c] (2.10)⊕f : c→df not an isomorphismM(d) → M(c).T c : MOD -RC → MOD -R[c] (2.11)10

sends M to the kernel of the map∏M(f): M(c) →f : d→cf not an isomorphism∏f : d→cf not an isomorphismM(d).From now on suppose that C is an EI-category, i.e. a small category such thatendomorphisms are isomorphisms. Then we can define the inclusion functorI c : MOD -R[c] → MOD -RC (2.12)by I c (M)(?) = M ⊗ R[c] R mor(?, c) if c ∼ = ? in C and by I c (M)(?) = 0 otherwise.Let B be the RC-R[c]-bimodule, covariant over C and a right module over R[c],given byB(c, ?) = R mor C(c, ?) if c ∼ = ?;0 if c ≁ = ?.Let C be the R[c]-RC-bimodule, contravariant over C and a left module overR[c], given byC(?, c) = R mor C(?, c) if c ∼ = ?;0 if c ≁ = ?.One easily checks that there are natural isomorphismsS c M ∼ = M ⊗ RC B;I c N ∼ = hom R[c] (B, N);T c M ∼ = hom RC (C, M);I c N ∼ = N ⊗ R[c] C.Lemma 2.13. Let C be an EI-category and c, d objects in C.(a) We obtain adjoint pairs (i(c) ∗ , i(c) ∗ ), (i(c) ∗ , i(c) ! ), (S c , I c ) and (I c , T c );∼ = ∼ =(b) There are natural equivalences of functors S c ◦i(c) ∗ −→ id and T c ◦i(c) ! −→id of functors MOD -R[c] → MOD -R[c]. If c ≁ = d, then Sc ◦ i(d) ∗ =T c ◦ i(d) ! = 0;(c) The functors S c and i(c) ∗ send projective modules to projective modules.The functors I c and i(c) ! send injective modules to injective modules.Proof. (a) follows from (2.5), (2.6) and (2.1).(b) This follows in the case T c ◦ i(d) ! from the following chain of canonicalisomorphismsT c ◦ i(d) ! (M) =hom RC (C(?, c), hom R[d] (R mor C (d, ?), M))∼ =−→ hom R[d] (C(?, c) ⊗ RC R mor C (d, ?), M) ∼= −→ hom R[c] (C(c, d), M),and analogously for S d ◦ i(c) ∗ .(c) The functors S c and i(c) ∗ are left adjoint to an exact functor and hencerespect projective. The functors T c and i(c) ! are right adjoint to an exactfunctor and hence respect injective.11

The length l(c) ∈ N ∪ {∞} of an object c is the supremum over all naturalf 1 f 2 f 3numbers l for which there exists a sequence of morphisms c 0 −→ c1 −→ c2 −→f l. . . −→ cl such that no f i is an isomorphism and c l = c. The colength col(c) ∈N ∪ {∞} of an object c is the supremum over all natural numbers l for whichf 1 f 2 f 3 f lthere exists a sequence of morphisms c 0 −→ c1 −→ c2 −→ . . . −→ cl such that nof i is an isomorphism and c 0 = c. If each object c has length l(c) < ∞, we saythat C has finite length. If each object c has colength col(c) < ∞, we say thatC has finite colength.Theorem 2.14. (Structure theorem for projective and injective RCmodules).Let C be an EI-category. Then(a) Suppose that C has finite colength. Let M be a contravariant RC-modulesuch that the R aut(c)-module S c M is projective for all objects c in C.Let σ c : S c M → M(c) be an R aut(c)-section of the canonical projectionM(c) → S c M. Consider the map of RC-modulesµ(M):⊕(c)∈Is(C)i(c) ∗ S c M⊕(c)∈Is(C) i(c)∗σc−−−−−−−−−−−→⊕(c)∈Is(C)i(c) ∗ M(c)⊕(c)∈Is(C) α(c)−−−−−−−−−→ M,where α(c): i(c) ∗ M(c) = i(c) ∗ i(c) ∗ M → M is the adjoint of the identityi(c) ∗ M → i(c) ∗ M under the adjunction (2.5). The map µ(M) is alwayssurjective. It is bijective if and only if M is a projective RC-module;(b) Suppose that C has finite length. Let M be a contravariant RC-modulesuch that the R aut(c)-module T c M is injective for all objects c in C.Let ρ c : M(c) → T c M be an R aut(c)-retraction of the canonical injectionT c M → M(c). Consider the map of RC-modulesν(M): M∏(c)∈Is(C) β(c)−−−−−−−−−→∏(c)∈Is(C)i(c) ! M(c)∏(c)∈Is(C) i(c) !ρ c−−−−−−−−−−−→∏(c)∈Is(C)i(c) ∗ T c Mwhere β(c): M → i(c) ! i(c) ∗ M = i(c) ! M(c) is the adjoint of the identityi(c) ∗ M → i(c) ∗ M under the adjunction (2.6). The map ν(M) is alwaysinjective. It is bijective if and only if M is an injective RC-module.Proof. (a) A contravariant RC-module is the same as covariant RC op -module,where C op is the opposite category of C, just invert the direction of every morphisms.The corresponding covariant version of assertion (a) is proved in [8,Theorem 2.11].(b) is the dual statement of assertion (a). We first show that ν(M) is always12

injective. We show by induction over the length l(x) of an object x ∈ C thatν(M)(x) is injective. Let u be an element in the kernel of ν(M)(x). Considera morphism f : y → x which is not an isomorphism. Then l(y) < l(x) and byinduction hypothesis ν(M)(y) is injective. Since the composite ν(M)(y) ◦ M(f)factorizes through ν(M)(x), we have u ∈ ker(M(f)). This implies u ∈ T x M.Consider the compositeT x M i −→ M(x) ν(M)(x)−−−−−→∏(c)∈Is(C)i(c) ! T c M(x) pr x−−→ i(x) ! T x M(x) −→ j T x M,where i is the inclusion, pr x is the projection onto the factor belonging to theisomorphism class of x and j is the isomorphism hom R[x] (R mor C (x, x), T x M) −→∼=T x M sending φ to φ(id x ). Since this composite is the identity on T x M and ulies in the kernel of ν(M)(x), we conclude u = 0.In particular we see that an injective RC-module M is trivial if and only ifi(d) ! T d M(x) is trivial for all objects d ∈ C.If ν(M) is bijective and each T c M is an injective R[c]-module, then M isan injective RC-module, since i(c) ! sends injective R[c]-modules to injectiveRC-modules by Lemma 2.13 (c) and the product of injective modules is againinjective.Now suppose that M is injective. Let N be the cokernel of ν(M). We havethe exact sequence0 → M ν(M)−−−→ ∏ (c)∈Is(C) i(c) ∗T c M pr−→ N → 0. (2.15)Since M is injective, this is a split exact sequence of injective RC-modules. Fixan object d. The functors i(d) ! and T d send split exact sequences to split exactsequences. Therefore we obtain a split exact sequence if we apply i(d) ! T d to(2.15). Using Lemma 2.13 (b) the resulting exact sequence is isomorphic to theexact sequence0 → i(d) ! T d M id−→ i(d) ! T d M → i(d) ! T d N → 0.Hence i(d) ! T d N vanishes for all objects d. This implies that N is trivial andbecause of (2.15) that ν(M) is bijective.For more details about modules over a category we refer to [7, Section 9A].3. The Associated Bredon Cohomology TheoryGiven a proper equivariant cohomology theory with values in R-modules, wecan associate to it another proper equivariant cohomology theory with values inR-modules satisfying the disjoint union axiom called Bredon cohomology, which13

is much simpler. The equivariant Chern character will identify this simplerproper equivariant cohomology theory with the given one.The orbit category Or(G) has as objects homogeneous spaces G/H and asmorphisms G-maps. Let Sub(G) be the category whose objects are subgroupsH of G. For two subgroups H and K of G denote by conhom G (H, K) theset of group homomorphisms f : H → K, for which there exists an elementg ∈ G with gHg −1 ⊆ K such that f is given by conjugation with g, i.e. f =c(g): H → K, h ↦→ ghg −1 . Notice that f is injective and c(g) = c(g ′ ) holdsfor two elements g, g ′ ∈ G with gHg −1 ⊆ K and g ′ H(g ′ ) −1 ⊆ K if and only ifg −1 g ′ lies in the centralizer C G H = {g ∈ G | gh = hg for all h ∈ H} of H in G.The group of inner automorphisms of K acts on conhom G (H, K) from the leftby composition. Define the set of morphismsmor Sub(G) (H, K) :=Inn(K)\ conhom G (H, K).There is a natural projection pr: Or(G) → Sub(G) which sends a homogeneousspace G/H to H. Given a G-map f : G/H → G/K, we can choosean element g ∈ G with gHg −1 ⊆ K and f(g ′ H) = g ′ g −1 K. Then pr(f)is represented by c(g): H → K. Notice that mor Sub(G) (H, K) can be identifiedwith the quotient mor Or(G) (G/H, G/K)/C G H, where g ∈ C G H actson mor Or(G) (G/H, G/K) by composition with R g −1 : G/H → G/H, g ′ H ↦→g ′ g −1 H.Denote by Or(G, F) ⊆ Or(G) and Sub(G, F) ⊆ Sub(G) the full subcategories,whose objects G/H and H are given by finite subgroups H ⊆ G. BothOr(G, F) and Sub(G, F) are EI-categories of finite length.Given a proper G-cohomology theory HG ∗ with values in R-modules we obtainfor n ∈ Z a contravariant ROr(G, F)-moduleHG n (G/?): Or(G, F) → R - MOD, G/H ↦→ Hn G (G/H). (3.1)Let (X, A) be a pair of proper G-CW -complexes. Then there is a canonicalidentification X H = map(G/H, X) G . Thus we obtain contravariant functorsOr(G, F) → CW -PAIRS, G/H ↦→ (X H , A H );Sub(G, F) → CW -PAIRS, G/H ↦→ C G H\(X H , A H ),where CW -PAIRS is the category of pairs of CW -complexes. If we composethem with the covariant functor CW -PAIRS → Z-CHCOM sending (Z, B)to its cellular Z-chain complex, then we obtain the contravariant ZOr(G, F)-chain complex C∗Or(G,F) (X, A) and the contravariant ZSub(G, F)-chain complexC∗Sub(G,F) (X, A). Both chain complexes are free in the sense that eachchain module is a free ZOr(G, F)-module resp. ZSub(G, F)-module. Namely,if X n is obtained from X n−1 ∪ A n by attaching the equivariant cells G/H i × D nfor i ∈ I n , thenC Or(G,F)n (X, A) ∼ =⊕Z mor Or(G,F) (G/?, G/H i ); (3.2)i∈I n⊕Cn Sub(G,F) (X, A) ∼ = Z mor Sub(G,F) (?, H i ). (3.3)i∈I n14

Given a contravariant ROr(G, F)-module M, the equivariant Bredon cohomology(see [2]) of a pair of proper G-CW -complexes (X, A) with coefficients in Mis defined by( )HOr(G,F) n (X, A; M) := Hn hom ZOr(G,F) (C∗ Or(G,F) (X, A), M) . (3.4)This is indeed a proper G-cohomology theory satisfying the disjoint union axiom.Hence we can assign to a proper G-homology theory HG ∗ another proper G-cohomology theory which we call the associated Bredon cohomology∏BHG(X, n A) := H p Or(G,F) (X, A; Hq G(G/?)). (3.5)p+q=nThere is an obvious ZSub(G; F)-chain mappr ∗ C∗Or(G,F) (X, A) −→ ∼= C∗ Sub(G,F) (X, A)which is bijective because of (3.2), (3.3) and the canonical identificationpr ∗ Z mor Or(G,F) (G/?, G/H i ) = Z mor Sub(G,F) (?, H i ).Given a covariant ZSub(G, F)-module M, we get from the adjunction (pr ∗ , pr ∗ )(see Lemma 2.13 (a)) natural isomorphismsHROr(G,F) n (X, A; res pr M)∼ ( ))=−→ H n hom ZSub(G,F)(C ∗Sub(G,F) (X, A), M . (3.6)This will allow us to work with modules over the category Sub(G; F) which issmaller than the orbit category and has nicer properties from the homologicalalgebra point of view. The main advantage of Sub(G; F) is that the automorphismgroups of every object is finite.Suppose, we are given a proper equivariant cohomology theory H?∗ withvalues in R-modules. We get from (3.1) for each group G and n ∈ Z a covariantRSub(G, F)-moduleHG n (G/?): Sub(G, F) → R - MOD, H ↦→ Hn G (G/H). (3.7)We have to show that for g ∈ C G H the G-map R g −1 : G/H → G/H, g ′ H →g ′ g −1 H induces the identity on HG n (G/H). This follows from Lemma 1.5.We will denote the covariant ROr(G, F)-module obtained by restriction withpr: Or(G, F) → Sub(G, F) from the RSub(G, F)-module HG n (G/?) of (3.7)again by HG n (G/?) as introduced already in (3.1).It remains to show that the collection of G-cohomology theories BHG ∗ (X, A)defined in (3.4) inherits the structure of a proper equivariant cohomology theory,i.e. we have to specify the induction structure. We leave it to the reader to carryout the obvious dualization of the construction for homology in [8, Section 3]and to check the disjoint union axiom.15

4. The Construction of the EquivariantCohomological Chern CharacterWe begin with explaining the cohomological version of the homological Cherncharacter due to Dold [4].Example 4.1 (The non-equivariant Chern character). Consider a (nonequivariant)cohomology theory H ∗ with values in R-modules. Suppose thatQ ⊆ R. For a space X let X + be the pointed space obtained from X by addinga disjoint base point ∗. Since the stable homotopy groups π s p(S 0 ) are finite forp ≥ 1 by results of Serre [15], the condition Q ⊆ R imply that the Hurewiczhomomorphism induces isomorphismshur R : πp(X s + ) ⊗ Z R hur ⊗ Z id−−−−−−→RHp (X) ⊗ Z R −→ ∼= H p (X; R)and that the canonical mapα: H p (X; H q ({pt.})) ∼= −→ hom Q (H p (X; Q), H q ({pt.})) ∼= −→ hom R (H p (X; R), H q ({pt.}))is bijective. Define a mapD p,q : H p+q (X) → hom R (π s p(X + ) ⊗ Z R, H q ({pt.})) (4.2)as follows. Denote in the sequel by σ k the k-fold suspension isomorphism. Givena ∈ H p+q (X) and an element in π s p(X + , ∗) represented by a map f : S p+k → S k ∧X + , we define D p,q (a)([f]) ∈ H q ({pt.}) as the image of a under the compositeH p+q (X) −→ ∼= ˜H p+q (X + ) σk−→ ˜H p+q+k (S k ∧ X + ) ˜H p+q+k (f)−−−−−−−→ ˜H p+q+k (S p+k )(σ p+k ) −1−−−−−−→ ˜H q (S 0 ) ∼= −→ H q ({pt.}).Then the (non-equivariant) Chern character for a CW -complex X is given bythe following compositech n (X): H n (X)∏p+q=n Dp,q−−−−−−−−→∏p+q=n hom R(hur −1R ,id)−−−−−−−−−−−−−−−−→∏p+q=nhom R(πsp (X + , ∗) ⊗ Z R, H q (∗) )∏p+q=nhom R (H p (X; R), H q (∗))∏p+q=n α−1−−−−−−−−→∏p+q=nThere is an obvious version for a pair of CW -complexesch n (X, A): H n (X, A) −→∼= ∏H p (X, A, H q (∗)).p+q=nH p (X, H q (∗)).16

We get a natural transformation ch ∗ of cohomology theories with values in R-modules. One easily checks that it is an isomorphism in the case X = {pt.}.Hence ch n (X, A) is bijective for all relative finite CW -pairs (X, A) and n ∈ Zby Lemma 1.1 (b). If H ∗ satisfies the disjoint union axiom, then ch n (X, A) isbijective for all CW -pairs (X, A) and n ∈ Z by Lemma 1.1 (b).Let R be a commutative ring with Q ⊆ R. Consider an equivariant cohomologytheory H? ∗ with values in R-modules. Let G be a group and let (X, A)be a proper G-CW -pair. We want to construct an R-homomorphism for a finitesubgroup H ⊆ Gch p,qG(X, A)(H): Hp+qG(X, A)(→ hom R Hp (C G H\(X H , A H ); R), H q G (G/H)) . (4.3)We define it only in the case A = ∅, the general case is completely analogous.H p+qGH p+qG (v H)(X)⏐↓H p+qG (ind m HX H )⏐ ⏐↓H p+qG (indm H pr 2 )H p+qG (ind m HEG × X H )ind mH⏐ ⏐↓∼ =H p+qC G H×H (EG × XH )(ind pr : CG H×H→H) −1 ⏐ ⏐↓∼ =H p+qH (EG × C G H X H )⏐↓D p,qH (EG× C G HX H )hom R(πsp ((EG × CG H X H ) + ) ⊗ Z R, H q H ({pt.}))hom R (hur R (EG× CG HX H ),id) −1 ⏐ ⏐↓hom R(Hp (EG × CG H X H ; R), H q H ({pt.}))hom R (H p(pr 1 ;R),id) −1 ⏐ ⏐↓(hom R Hp (C G H\X H ; R), H q H ({pt.}))⏐hom R (id;(ind G H )−1 ) ↓hom R(Hp (C G H\X H ; R), H q G (G/H))Here are some explanations, more details can be found in [8, Section 4].We have a left free C G H-action on EG × X H by g(e, x) = (eg −1 , gx) for g ∈C G H, e ∈ EG and x ∈ X H . The map pr 1 : EG × CG H X H → C G H\X H is the17

canonical projection. Since the projection BL → {pt.} induces isomorphismsH p (BL; R) ∼= −→ H p ({pt.}; R) for all p ∈ Z and finite groups L because of Q ⊆ R,we obtain for every p ∈ Z an isomorphismH p (pr 1 ; R): H p (EG × CG H X H ; R) ∼= −→ H p (C G H\X H ; R).The group homomorphism pr: C G H ×H → H is the obvious projection and thegroup homomorphism m H : C G H × H → G sends (g, h) to gh. The C G H × H-action on EG × X H comes from the obvious C G H-action and the trivial H-action. In particular we equip EG × CG H X H with the trivial H-action. Thekernels of the two group homomorphisms pr and m H act freely on EG × X H .We denote by pr 2 : EG × X H → X H the canonical projection. The G-mapv H : ind mH X H = G × mH X H → X sends (g, x) to gx.Since H is a finite group, a CW -complex Z equipped with the trivial H-action is a proper H-CW -complex. Hence we can think of HH ∗ as an (nonequivariant)homology theory if we apply it to a CW -pair Z with respect to thetrivial H-action. Define the mapD p,qH(Z): Hp+qH(Z) → hom R(πp(Z s + ) ⊗ Z R, H q H ({pt.}))for a CW -complex Z by the map D p,q of (4.2).A calculation similar to the one in [8, Lemma 4.3] shows that the system ofmaps ch p,qG(X, A)(H) (4.3) fit together to an in X natural R-homomorphismch p,q (X, A): Hp+qG G (X, A) (→ hom Sub(G;F) Hp (C G ?\X ? ; R), H q G (G/?)) . (4.4)For any contravariant RSub(G; F)-module M and p ∈ Z there is an in (X, A)natural R-homomorphismα p G (X, A; M): Hp RSub(G;F) (X, A; M) → hom QSub(G;F)(H p (C G ?\X ? ; Q), M)∼ =−→ hom RSub(G;F) (H p (C G ?\X ? ; R), M) (4.5)which is bijective if M is injective as QSub(G; F)-module.Theorem 4.6 (The equivariant Chern character). Let R be a commutativering R with Q ⊆ R. Let H?∗ be a proper equivariant cohomology theory withvalues in R-modules. Suppose that the RSub(G; F)-module H q G(G/?) of (3.7),which sends G/H to H q G(G/H), is injective as QSub(G; F)-module for everygroup G and every q ∈ Z.Then we obtain a transformation of proper equivariant cohomology theorieswith values in R-modulesch ∗ ? : H ∗ ?∼ =−→ BH ∗ ?,if we define for a group G and a proper G-CW -pair (X, A)ch n G(X, A): HG(X, n A) → BHG(X, n A) := ∏H p RSub(G;F) (X, A; Hq G (G/?))p+q=n18

y the composite∏p+q=n chp,q(X,A)HG(X, n GA) −−−−−−−−−−−−→∏p+q=n∏p+q=n αp G (X,A;Hq G (G/?))−1−−−−−−−−−−−−−−−−−−−→hom RSub(G;F)(Hp (C G ?\X ? ; R), H q G (G/?))∏p+q=nH p RSub(G;F) (X, A; Hq G (G/?))of the maps defined in (4.4) and (4.5).The R-map ch n G(X, A) is bijective for all proper relative finite G-CW -pairs(X, A) and n ∈ Z. If H?∗ satisfies the disjoint union axiom, then the R-mapch n G(X, A) is bijective for all proper G-CW -pairs (X, A) and n ∈ Z.Proof. First one checks that ch ∗ G defines a natural transformation of proper G-cohomology theories. One checks for each finite subgroup H ⊆ G and n ∈ Zthat ch n G(G/H) is the identity if we identify for any RSub(G; F)-module M=H p RSub(G;F) (G/H; M) = Hp ( hom RSub(G;F) (C RSub(G;F)∗)(G/H), M{homRSub(G;F)(R morSub(G;F) (?, G/H), M ) = M(G/H) if p = 0;0 if p ≠ 0.Finally apply Lemma 1.1 (b).Remark 4.7 (The Atiyah-Hirzebruch spectral sequence for equivariantcohomology). There exists a Atiyah-Hirzebruch spectral sequence for equivariantcohomology (see [3, Theroem 4.7 (2)]). It converges to H p+qG(X, A) andhas as E 2 -term the Bredon cohomology groups H p RSub(G;F) (X, A; Hq G (G/?)).The conclusion of Theorem 4.6 is that the spectral sequences collapses.Example 4.8 (Equivariant Chern character for K ∗ (G\(X, A))). Let K ∗be a (non-equivariant) cohomology theory with values in R-modules for a commutativering with Q ⊆ R. In Example 1.6 we have assigned to it an equivariantcohomology theory byH n G(X, A) = K n (G\(X, A)).We claim that the assumptions appearing in Theorem 4.6 are satisfied We haveto show that the constant functorK q ({pt.}): Sub(G; F) → Q - MOD,H ↦→ K q ({pt.})is injective. Let i: Sub({1}) → Sub(G; F) the obvious inclusion of categories.Since the object {1} is an initial object in Sub(G; F), the QSub(G; F)-modulesi ! (K q ({pt.})) and K q ({pt.}) are isomorphic. Since i ! sends an injective Q-moduleto an injective RSub(G; F)-module by Lemma2.13 (c) and H q ({pt.}) is injective19

as Q-module, K q ({pt.}) is injective as QSub(G; F)-module. From Theorem 4.6we get a transformation of equivariant cohomology theoriesch n G(X, A): K n (G\(X, A))∼ =−→∏p+q=nH p RSub(G;F) (X, A; Hq ({pt.}))= ∏p+q=nH p (G\(X, A); H q ({pt.})).One easily checks that this is precisely the Chern character of Example 4.1applied to K ∗ and the CW -pair G\(X, A).5. Mackey FunctorsIn Theorem 4.6 the assumption appears that the contravariant RSub(G; F)-module H q G(G/?) is injective for each q ∈ Z. We want to give a criterion whichensures that this assumption is satisfies and which turns out to apply to allcases of interest.Let R be a commutative ring. Let FGINJ be the category of finite groupswith injective group homomorphisms as morphisms. Let M : FGINJ → R - MODbe a bifunctor, i.e. a pair (M ∗ , M ∗ ) consisting of a covariant functor M ∗ anda contravariant functor M ∗ from FGINJ to R - MOD which agree on objects.We will often denote for an injective group homomorphism f : H → G the mapM ∗ (f): M(H) → M(G) by ind f and the map M ∗ (f): M(G) → M(H) by res fand write ind G H = ind f and res H G = res f if f is an inclusion of groups. We callsuch a bifunctor M a Mackey functor with values in R-modules if(a) For an inner automorphism c(g): G → G we have M ∗ (c(g)) = id: M(G) →M(G);(b) For an isomorphism of groups f : G ∼= −→ H the composites res f ◦ ind f andind f ◦ res f are the identity;(c) Double coset formulaWe have for two subgroups H, K ⊆ G∑res K G ◦ ind G H =ind c(g) : H∩g −1 Kg→K ◦ res H∩g−1 KgH,KgH∈K\G/Hwhere c(g) is conjugation with g, i.e. c(g)(h) = ghg −1 .Let G be a group. In the sequel we denote for a subgroup H ⊆ G byN G H the normalizer and by C G H the centralizer of H in G and by W G H thequotient N G H/H · C G H. Notice that W G H is finite if H is finite. Let R be acommutative ring. Let M be a Mackey functor with values in R-modules. Itinduces a contravariant RSub(G, F)-module denoted in the same wayM : Sub(G, F) → R - MOD, (f : H → K) ↦→ (M ∗ (f): M(H) → M(K)) .20

We want to use Theorem 2.14 (b) to show that M is injective and analyse itsstructure. The R[W G H]-module T H M introduced in (2.11) is the same as thekernel of∏M(i K ): M(H) →∏M(K),KHKHwhere for each subgroup K H different from H we denote by i K the inclusion.Suppose that R[W G H]-module T H M is injective for every finite subgroup H ⊆G. For every finite subgroup H ⊆ G choose a retraction ρ H : M(H) → T H Mof the inclusion T H M → M(H). Denote by I = Is(Sub(G, F)) the set ofisomorphism classes of objects in Sub(G; F) which is the same as the set ofconjugacy classes (H) of finite subgroups H of G. Letν = ν(M): M → ∏i(K) ! ◦ T K (M) (5.1)(K)∈Ibe the homomorphism of RSub(G, F)-modules uniquely determined by theproperty that for any (K) ∈ I its composition with the projection onto thefactor indexed by (K) is the adjoint of ρ K : M(K) → T K M for the adjoint pair(i(K) ∗ , i(K) ! ).Theorem 5.2 (Injectivity and Mackey functors). Let G be a group andlet R be a commutative ring such that the order of every finite subgroup of Gis invertible in R. Suppose that the R[W G H]-module T H M is injective for eachfinite subgroup H ⊆ G. Then M is injective as RSub(G, F)-module and themap ν of (5.1) is bijective.Proof. The map ν of (5.1) is the map ν(M) appearing in Theorem 2.14 (b).Because of Theorem 2.14 (b) it suffices to show for each finite subgroup H ⊆ Gthat ν(M)(H) is surjective.Fix for any (K) ∈ I a representative K. Then choose for any W G K · f ∈W G K\ mor(K, H) an element f ∈ conhom(K, H) which represents a morphismf : K → H in Sub(G; F) which belongs to W G K ·f ∈ W G K\ mor(K, H). Noticethat W G K is the automorphism group of the object K in Sub(G; F) and W G K,mor(K, H) and W G K\ mor(K, H) are finite. With these choices we get for everyobject H in Sub(G; F) an identification∏i(K) ! T K M(H) = hom RWG K(R mor(K, H), T K M) =T K M W GK fW G K·f∈W G K\ mor(K,H)where W G K f ⊆ W G K is the isotropy group of f under the W G K-action onmor(K, H). Under this identification ν(H) becomes the mapν(H): M(H) →∏ ∏T K M W GK f(K)∈IW G K·f∈W G K\ mor(K,H)for which the component of ν(H)(m), which belongs (K) ∈ I and W G K · f ∈W G K\ mor(K, H), is ρ K ◦ res f (m) for m ∈ M(H). Notice that the image of21

es f always is contained in M(K) W GK f. Next we define a map⊕ ⊕µ(H):T K M W GK f→ M(H)(K)∈I,(K)≤(H)W G K·f∈W G K\ mor(K,H)by requiring that its restriction to the summand, which belongs to (K) ∈ I andW G K · f ∈ W G K\ mor(K, H), is the composite of the inclusion T K M W GK f→M(K) with ind f : M(K) → M(H). We want to show that the composite→ν(H) ◦ µ(H):∏(K)∈I⊕(K)∈I,(K)≤(H)∏W G K·f∈W G K\ mor(K,H)⊕W G K·f∈W G K\ mor(K,H)T K M W GK f=T K M W GK f⊕(K)∈I,(K)≤(H)⊕W G K·f∈W G K\ mor(K,H)T K M W GK fis bijective. If K is subconjugated to H, we write (K) ≤ (H). Fix (K), (L) ∈I with (K) ≤ (H) and (L) ≤ (H) and W G K · f ∈ mor(K, H) and W G L ·g ∈ mor(L, H). Then the homomorphism T K M W GK f → TL M W GL g given byν(H) ◦ µ(H) and the summands corresponding to (K, f) and (L, g) is inducedby the compositeα (K,f),(L,g) : T K M W GK fi−→ M(K) ind f : K→im(f)−−−−−−−−−→ M(im(f)) indH im(f)−−−−−→ M(H)res im(g)H−−−−−→ M(im(g)) res g : L→im(g)−−−−−−−−→ M(L) ρ L−→ T L M, (5.3)where i is the inclusion. The double coset formula impliesres im(g)H=The compositeT K M W GK f◦ ind H im(f)∑im(g)h im(f)∈im(g)\H/ im(f)ind c(h) : im(f)∩h −1 im(g)h→im(g) ◦ res im(f)∩h−1 im(g)him(f). (5.4)i−→ M(K) ind f : K→im(f)−−−−−−−−−→ M(im(f))res im(f)∩h−1 im(g)him(f)−−−−−−−−−−−−→ M(im(f) ∩ h −1 im(g)h)is trivial by the definition of T K M if im(f) ∩ h −1 im(g)h ≠ im(f) holds. Henceα (K,f),(L,g) ≠ 0 is only possible if im(f) ∩ h −1 im(g)h = im(f) for some h ∈ Hand hence (K) ≤ (L) hold.Suppose that (K) = (L). Then K = L by our choice of representatives.Suppose that α (K,f),(K,g) ≠ 0. We have already seen that this implies im(f) ∩h −1 im(g)h = im(f) for some h ∈ H. Since | im(f)| = |K| = | im(g)| we conclude22

h −1 im(g)h = im(f) and therefore W G K · f = W G K · g in W G K\ mor(K, H).This implies already f = g as group homomorphism K → H by our choiceof representatives. The double coset formula (5.4) implies that α (K,f),(K,f) is|H ∩ N G im(f)| · id TK M W G K fsince for all h ∈ N G im(f) ∩ H the compositeT K M W GK fi−→ M(K) ind f : K→im(f)−−−−−−−−−→ M(im(f))iind c(h) : im(f)→im(f)−−−−−−−−−−−−−→ M(im(f))agrees with T K M W GK f−→ M(K) ind f : K→im(f)−−−−−−−−−→ M(im(f)). Since the order of|H ∩ N G im(f)| is invertible in R by assumption, α (K,f),(K,f) is bijective.We conclude that ν(H)◦µ(H) can be written as a matrix of maps which hasupper triangular form and isomorphisms on the diagonal. Therefore ν(H)◦µ(H)is surjective. This shows that ν(H) is surjective. This finishes the proof ofTheorem 5.2.Theorem 5.5 (The equivariant Chern character and Mackey structures).Let R be a commutative ring with Q ⊆ R. Let H? ∗ be a proper equivariantcohomology theory. Define a contravariant functorH q ?({pt.}): FGINJ → R - MODby sending a homomorphism α: H → K to the compositeH q K ({pt.}) Hq (pr)−−−−→ H q indαK(K/H) −−−→ H q H ({pt.})where pr: H/K = ind α ({pt.}) → {pt.} is the projection and ind α comes fromthe induction structure of H? ∗ . Suppose that it extends to a Mackey functor forevery q ∈ Z. Then(a) For every group G the RSub(G; F)-module H q G(G/?) of (3.7) is injectiveas RSub(G; F)-module, provided that R is semisimple;(b) We obtain a natural transformation of proper equivariant cohomology theorieswith values in R-modulesch ∗ ?(X, A): H ∗ ? → BH ∗ ?.In particular we get for every proper G-CW -pair (X, A) and every n ∈ Za natural R-homomorphismch n G(X, A): H n G(X, A)→ BH n G(X, A) :=∏p+q=nH p RSub(G;F) (X, A; Hq G (G/?)).It is bijective for all proper relative finite G-CW -pairs (X, A) and n ∈ Z.If H? ∗ satisfies the disjoint union axiom, it is bijective for all proper G-CW -pairs (X, A) and n ∈ Z;23

(c) Define for finite subgroup H ⊆ G the R[W G H]-module T H (H q H({pt.})) by⎛⎞ker ⎝ ∏ind K L ◦H q (pr: H/L → {pt.}): H q H ({pt.}) → ∏H q L ({pt.}) ⎠ .LHLHThen the Bredon cohomology BHG n (X, A) of a proper G-CW -pair (X, A)is naturally R-isomorphic to∏ ∏H p W G H (C GH\(X H , A H ); T H (H q H ({pt.}))p+q=n(H),H⊆G finiteprovided that R is semisimple.Proof. (a) This follows from Theorem 5.2 since for every finite subgroup H ⊆ Gthe group W G H is finite and hence the ring R[W G H] is semisimple and everyR[W G H]-module is injective.(b) This follows from assertion (a) applied in the case R = Q together withTheorem 4.6.(c) Since R is semisimple and contains Q and W G H is finite, the ring R[W G H]is semisimple and every R[W G H]-module is injective for every finite subgroupH. Because the map ν of (5.1) is an isomorphism by Theorem 5.2, it remainsto show for a C G H-module NH p RSub(G;F) (X, A; i(H) !N) ∼ = H p W G H (C GH\(X H , A H ); N)This follows from the adjunction (i(H) ∗ , i(H) ! ) of Lemma 2.13 (a).Example 5.6 (Mackey structures for Borel cohomology). Let K ∗ be acohomology theory for (non-equivariant) CW -pairs with values in R-modulesfor a commutative ring R such that Q ⊆ R and R is semisimple. In Example 1.6we have assigned to it an equivariant cohomology theory called equivariant Borelcohomology byH n G(X, A) = K n (EG × G (X, A)).We claim that the assumptions appearing in Theorem 5.5 are satisfied. Namely,the contravariant functorFGINJ → R - MOD,H ↦→ K n (BH)extends to a Mackey functor, the necessary covariant functor comes from theBecker-Gottlieb transfer (see for instance [5] and [6, Corollary 6.4 on page 206]).Hence we get from Theorem 5.5 for every group G and every proper G-CW -pair(X, A) natural R-mapsch n G(X, A): K n ∼ =(EG × G (X, A)) −→∏H p RSub(G;F) (X, A; Kq (B?))∼= ∏p+q=n∏(H),H⊆G finitep+q=nhom RWG H(H p (C G H\X H ; R), T H (K q (BH)),24

if we defineT H (K q (BH)) :=⎛⎞ker ⎝ ∏K q (BK → BH): K q (BH) → ∏K q (BK) ⎠ .KHKHIf (X, A) is relative finite or if K ∗ satisfies the disjoint union axiom, then thesemaps ch n G(X, A) are bijective.Remark 5.7. We mention that this does not prove Theorem 0.1 since we cannotapply it to K ∗ := K ∗ ⊗ Z Q. The problem is that K ∗ ⊗ Z Q defines all axiomsof a cohomology theory but not the disjoint union axiom. But this is neededif we want to deal with classifying spaces BG of groups which are not finiteCW -complexes, for instance of groups containing torsion (see also Remark 1.2and Example 1.3).A proof of Theorem 0.1 will be given in [11].Example 5.8 (Equivariant K-theory and Mackey structures). In Example1.7 we have introduced the equivariant cohomology theory K? ∗ given bytopological K-theory. Recall that it takes values in R-modules for R = Z. Noticethat for a finite group H we get an identification of KH 0 ({pt.}) with thecomplex representation ring R(H) and the associated contravariant functorK q ?: FGINJ → R - MOD, H↦→ Kq H({pt.}) = R(?)sends an injective group homomorphism α: H → G of finite groups to thehomomorphism of abelian groups R(G) → R(H) given by restriction with α.Induction with α induces a covariant functor H ↦→ R(H) and it turns out thatthis defines a Mackey structure on K q ? .For rationalized equivariant topological K-theory K ∗ ? ⊗ Z Q the equivariantChern character of Theorem 5.5 can be identified with the one constructed in[12] for proper relative finite G-CW -pairs (X, A).6. Multiplicative StructuresNext we want to introduce a multiplicative structure on a proper equivariantcohomology theory H? ∗ and show that it induces one on the associated Bredoncohomology BH? ∗ such that the equivariant Chern character is compatible withit.We begin with the non-equivariant case. Let H ∗ be a (non-equivariant) cohomologytheory with values in R-modules. A multiplicative structure assigns to aCW -complex X with CW -subcomplexes A, B ⊆ X natural R-homomorphisms∪: H n (X, A) ⊗ R H n′ (X, B) → H n+n′ (X, A ∪ B). (6.1)25

This product is required to be compatible with the boundary homomorphism ofthe long exact sequence of a pair, to be graded commutative, to be associativeand to have a unit 1 ∈ H 0 ({pt.}).Given a multiplicative structure on H ∗ , we obtain for every p, q ∈ Z a pairing∪: H q ({pt.}) ⊗ R H q′ ({pt.}) → H q+q′ ({pt.}).It yields on singular (or equivalently cellular) cohomology a productH p (X, A; H q ({pt.}))⊗ R H p′ (X, B; H q′ ({pt.})) → H p+p′ (X, A∪B; H q+q′ ({pt.})).The collection of these pairings induce a multiplicative structure on the cohomologytheory given by ∏ p+q=n Hp (X, A; H q ({pt.})). The straightforwardproof of the next lemma is left to the reader.Lemma 6.2. Let R be a commutative ring with Q ⊆ R. Let H ∗ be a (nonequivariant)cohomology theory satisfying the disjoint union axiom which comeswith a multiplicative structure.Then the (non-equivariant) Chern character of Example 4.1ch n (X, A): H n (X, A) ∼= −→∏p+q=nH p (X, A, H q (∗))is compatible with the given multiplicative structure on H ∗ and the induced multiplicativestructure on the target.Next we deal with the equivariant version. We only deal with the propercase, the definitions below make also sense without this condition.Let HG ∗ be a proper G-cohomology theory. A multiplicative structure assignsto a proper G-CW -complex X with G-CW -subcomplexes A, B ⊆ X natural R-homomorphisms∪: H n G(X, A) ⊗ R H n′G (X, B) → H n+n′G(X, A ∪ B). (6.3)This product is required to be compatible with the boundary homomorphismof the long exact sequence of a G-CW -pair, to be graded commutative, to beassociative and to have a unit 1 ∈ HG 0 (X) for every proper G-CW -complex XLet H? ∗ be a proper equivariant cohomology theory. A multiplicative structureon it assigns a multiplicative structure to the associated proper G-cohomologytheory HG ∗ for every group G such that for each group homomorphismα: H → G the maps given by the induction structure of (1.4)ind α : H n G(ind α (X, A))∼ =−→ H n H(X, A)are in the obvious way compatible with the multiplicative structures on HG ∗ andHH ∗ Ṅext we explain how a given multiplicative structure on H ?∗ induces oneon BH? ∗ . We have to specify for every group G a multiplicative structure on26

the G-cohomology theory BHG ∗ . Consider a G-CW -complex X with G-CW -subcomplexes A, B ⊆ X. For two contravariant ROr(G; F)-chain complexes C ∗and D ∗ define the contravariant ROr(G; F)-chain complexes C ∗ ⊗ R D ∗ by sendingG/H to the tensor product of R-chain complexes C ∗ (G/H) ⊗ R D ∗ (G/H).Leta ∗ : C∗ROr(G;F) (X, A) ⊗ R C∗ ROr(G;F) (X, B)∼ =−→ C∗ROr(G;F) ((X, A) × (X, B))be the isomorphism of ROr(G; F)-chain complexes which is given for an objectG/H by the natural isomorphism of cellular R-chain complexesC ∗ (X H , A H ) ⊗ R C ∗ (X H , B H ) ∼= −→ C ∗ ((X H , A H ) × (X H , B H )).The multiplicative structure on HG ∗ yields a map of contravariant ROr(G; F)-modulesc: H q G (G/?) ⊗ R H q G(G/?) → Hq+qG(G/?).Let∆: (X; A ∪ B) → (X, A) × (X, B), x ↦→ (x, x)be the diagonal embedding. Define a R-cochain map by the composite()b ∗ : hom ROr(G;F) C∗ ROr(G;F) (X, A), H q G (G/?)hom ROr(G;F)(⊗ R hom ROr(G;F)(C ROr(G;F)∗ (X, B), H q′G (G/?) )C∗ROr(G;F)hom ROr(G;F)((a ∗) −1 ,c)(−−−−−−−−−−−−−−−→ hom ROr(G;F)⊗R−−→(X, A) ⊗ R C ROr(G;F)∗ (X, B), H q G (G/?) ⊗ R H q′G (G/?) )C∗ROr(G;F)(hom ROr(G;F) (C ROr(G;F)∗ (∆),id)−−−−−−−−−−−−−−−−−−−−−→ hom ROr(G;F) C∗ROr(G;F)There is a canonical R-mapH ∗ (C ∗ ⊗ R D ∗ ) → H ∗ (C ∗ ⊗ R D ∗ )((X, A) × (X, B)), H q+q′G (G/?) )(X, A ∪ B), H q+q′G (G/?) ).for two R-cochain complexes C ∗ and D ∗ . This map together with the mapinduced by b ∗ on cohomology yields an R-homomorphismH p ROr(G;F) (X, A; Hq G (G/?)) ⊗ R H p′ROr(G;F)(X, B; Hq′G (G/?))→ H p+p′ROr(G;F) (X, A ∪ B; Hq G (G/?)) .The collection of these R-homomorphisms yields the desired multiplicative structureon BHG ∗ . We leave it to the reader to check that the axioms of a multiplicativestructure on BHG ∗ are satisfied and that all these are compatible with theinduction structure so that we obtain a multiplicative structure on the equivariantcohomology theory BH? ∗ .We also omit the lengthy but straightforward proof of the following resultwhich is based on Theorem 4.6, Lemma 6.2 and the compatibility of the multiplicativestructure with the induction structure.27

Theorem 6.4 (The equivariant Chern character and multiplicativestructures). Let R be a commutative ring such that Q ⊆ R. Suppose thatH? ∗ is a proper cohomology theory with values in R-modules which comes witha multiplicative structure. Suppose that the RSub(G; F)-module H q G(G/?) of(3.7), which sends G/H to H q G(G/H), is injective for each q ∈ Z.Then the natural transformation of equivariant cohomology theories appearingin Theorem 4.6ch ∗ ? : H? ∗ → BH?∗is compatible with the given multiplicative structure on H? ∗ and the induced multiplicativestructure on BH? ∗ .Remark 6.5 (External products and restriction structures). One canalso define an external product for a proper equivariant cohomology theory H ∗ ?with values in R-modules. It assigns to every two groups G and H, a proper G-CW -pair (X, A), a proper H-CW -pair(Y, B) and p, q ∈ Z an R-homomorphism×: H p G (X, A) ⊗ R H q H(Y, B) → Hp+qG×H((X, A) × (Y, B)).One requires graded commutativity, associativity, the existence of a unit 1 ∈H{1} 0 ({pt.}) and compatibility with the induction structure and the boundaryhomomorphism associated to a pair. One can show that BH? ∗ inherits an externalproduct and prove the analogon of Theorem 6.4 for external products.One can also introduce the notion of a restriction structure on H? ∗ . It yieldsfor every injective group homomorphism α: H → G, every proper G-CW -pair(X, A) and p ∈ Z an R-homomorphismres α : H p G (X, A)) → Hp H (res α(X, A)).Again certain axioms are required such as compatibility with the boundaryhomomorphism associated to pair, compatibility with induction for group isomorphismsα: H −→ ∼= G, compatibility with conjugation, the double coset formulaand compatibility for projections onto quotients under free actions. Onecan show that BH? ∗ inherits a restriction structure and prove the analogon ofTheorem 6.4 for restriction structures.An external product together with a restriction structure yields a multiplicativestructure as follows. Consider G-CW -pairs (X, A) and (X, B). Letd: G → G × G and D : (X, A ∪ B) → (X, A) × (X, B) be the diagonal maps.Defineto be the composite∪: H m G (X, A) ⊗ R H n G(X, A) → H m+nG(X, A ∪ B) (6.6)H m G (X, A) ⊗ R H n G(X, A)res dG×−→ H m+n ((X, A) × (X, B))G×GH m+nG (D)−−→ Hm+n((X, A) × (X, B)) −−−−−−→ H m+n (X, A ∪ B).G28

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