Simplicial Structures in Topology

More documents

Recommendations

Info

IV.1 Cohomology with Coefficients in G 155 for which the following sequence of cohomology groups ...→ H n (K;Z) Hn (i) −→ H n (L;Z) ˜ λ n −→ H n+1 (K,L;Z) q∗ (n+1) −→ H n+1 (K;Z) → ... is exact. The proof of this theorem follows the steps taken in proving the corresponding theorem in homology Theorem (II.3.1); we only wish to point out that the main result needed for this proof is the following lemma: (IV.1.4) Lemma. If A �� f �� B is a short exact sequence of free groups and G is an Abelian group, then also the sequence Hom(C,G) �� g g �� Hom(B,G) �� C �f �� Hom(A,G) is exact. The reader may take the proof of Lemma (II.5.1) as a basis for proving this one; actually, the only tensor product property used in Lemma (II.5.1) is that −⊗G is a functor which takes sums of morphisms into sums of morphisms; this very same property holds also for the (contravariant) functor Hom(−,G). The cohomology determined by an oriented chain simplex K with coefficients in an Abelian group G is defined in the same manner as the one with coefficients in Z, except that we apply to the chain complex (C(K),∂) the contravariant functor Hom(−,G) instead of Hom(−,Z). We wish to determine the cohomology with coefficients in G of a simplicial complex, based on its homology with coefficients in Z and the Abelian group G. We seek to use the same previous ideas to our benefit, bearing in mind that the functor A ↦→ A ⊗ G is covariant, whereas A ↦→ Hom(A,G) is contravariant. We begin by reviewing what has been done so far in general terms. A cochain complex (or cocomplex) is a graded Abelian group {Cn } together with an endomorphism of degree +1, called coboundary homomorphism ∂ ∗ = {∂ n : Cn → Cn+1 ,},forwhich∂n+1∂n = 0. A cochain homomorphism between two cochain complexes (C,∂ ∗ ) and (C ′ ,∂ ′∗ ) is a homomorphism between the graded Abelian groups C and C ′ that commutes with the coboundary homomorphisms. Also the other concepts defined for chain complexes have their dual correspondents: the elements of Cn are called n-cochains; the elements of are n-cocycles, and those of Z n (C) := ker(∂ n : C n → C n+1 ) B n (C) := im(∂ n−1 : C n−1 → C n ) are n-coboundaries; finally, the quotient H n (C) := Z n (C)/B n (C) is the cohomology group of the cocomplex (C,∂ ∗ ) in dimension n.
156 IV Cohomology If, starting from a cocomplex (C ∗ ,∂ ∗ ),wedefineCn := C −n and ∂n := ∂ −n , it becomes clear that (C∗,∂) is a chain complex and all concepts have their respective equivalents. Specifically, the cohomology of (C ∗ ,∂ ∗ ) is exactly the homology of the complex (C∗,∂ ). This gives us an indication of how to write all results in cohomological terms (and we shall do so from now on, without further comments). For every complex (C,∂) and for every Abelian group G, wemaydefineacocomplex (C ∗ ,∂ ∗ ) where C n := Hom(Cn,G) is the homomorphism group from Cn into G, and∂ n : C n → C n+1 is the map adjoint to ∂n+1 : Cn+1 → Cn. This cocomplex cohomology is called cohomology of the complex (C,∂) with coefficients in G. We therefore assume that the complex (C,∂) is free and, so, both ZnC and BnC are also free; for every n, the sequence of free Abelian groups Zn(C) �� Cn ∂n �� Bn−1(C) is short exact; hence, by Lemma (IV.1.4), the sequence Hom(Bn−1(C),G) �� Hom(Cn,G) �� Hom(Zn(C),G) is exact. This enables us to construct the short exact complex sequence where (Hom( � B(C),G),0) �� (Hom(C,G),∂) �B(C)={ � B(C) n } := {Bn−1(C)} . �� (Hom(Z(C),G),0) The Cohomology Long Exact Sequence Theorem, in its general form (in terms of chain complexes), allows us to write the long exact sequence ··· �� Hom(Zn−1(C),G) jn �� Hom(Zn(C),G) �in−1 �� Hom(Bn−1(C),G) �in �� Hom(Bn(C),G) hn �� n H (C;G) �� ··· (where �in is the adjoint of the inclusion in : Bn(C) → Zn(C)) that breaks down into short exact sequences imhn �� n H (C;G) �� im �jn . Since we have the isomorphisms imhn ∼ = Hom(Bn−1(C),G)/kerhn and kerhn = im �in−1, wehaveimhn = coker(�in−1); besides, im �jn = ker �in and so these last short exact sequences become the short exact sequences coker(�in−1) �� H n (C;G) �� ker �in .
Page 2:
Canadian Mathematical Society Soci
Page 5 and 6:
Dr. Davide L. Ferrario Università
Page 8:
Foreword to the English Edition Exc
Page 11 and 12:
x Preface same qualitative properti
Page 13 and 14:
xii Preface homology groups, also w
Page 16 and 17:
Contents I Fundamental Concepts ...
Page 18 and 19:
Chapter I Fundamental Concepts I.1
Page 20 and 21:
I.1 Topology 3 We need to show that
Page 22 and 23:
I.1 Topology 5 Let X be a topologic
Page 24 and 25:
I.1 Topology 7 (x, y) (x, −y) Fig
Page 26 and 27:
I.1 Topology 9 that f is a homeomor
Page 28 and 29:
I.1 Topology 11 We recall that an i
Page 30 and 31:
I.1 Topology 13 Cx = � Xj j∈J i
Page 32 and 33:
I.1 Topology 15 A Fig. I.5 Example
Page 34 and 35:
I.1 Topology 17 with the topology i
Page 36 and 37:
I.1 Topology 19 are closed in Y. Th
Page 38 and 39:
I.1 Topology 21 is the diagonal of
Page 40 and 41:
I.1 Topology 23 (I.1.38) Lemma. Let
Page 42 and 43:
I.1 Topology 25 Here is an example.
Page 44 and 45:
I.2 Categories 27 5. Prove that a f
Page 46 and 47:
I.2 Categories 29 3. For every pair
Page 48 and 49:
I.2 Categories 31 If conditions 2.
Page 50 and 51:
I.2 Categories 33 Conversely, given
Page 52 and 53:
I.2 Categories 35 q: B ⊔C → B
Page 54 and 55:
I.2 Categories 37 (I.2.5) Lemma. Gi
Page 56 and 57:
I.3 Group Actions 39 I.3 Group Acti
Page 58:
I.3 Group Actions 41 S2 = {(x,y,z)
Page 61 and 62:
44 II Simplicial Complexes (0, 1) (
Page 63 and 64:
46 II Simplicial Complexes II.2 Abs
Page 65 and 66:
48 II Simplicial Complexes II.2.1 T
Page 67 and 68:
50 II Simplicial Complexes Let us r
Page 69 and 70:
52 II Simplicial Complexes r = tp+(
Page 71 and 72:
54 II Simplicial Complexes In a sim
Page 73 and 74:
56 II Simplicial Complexes (not nec
Page 75 and 76:
58 II Simplicial Complexes Let K3,3
Page 77 and 78:
60 II Simplicial Complexes ordering
Page 79 and 80:
62 II Simplicial Complexes are two
Page 81 and 82:
64 II Simplicial Complexes 1-simple
Page 83 and 84:
66 II Simplicial Complexes An infin
Page 85 and 86:
68 II Simplicial Complexes 2. λn i
Page 87 and 88:
70 II Simplicial Complexes Please n
Page 89 and 90:
72 II Simplicial Complexes (II.3.7)
Page 91 and 92:
74 II Simplicial Complexes (II.3.10
Page 93 and 94:
76 II Simplicial Complexes If we do
Page 95 and 96:
78 II Simplicial Complexes In the c
Page 97 and 98:
80 II Simplicial Complexes Φi = {
Page 99 and 100:
82 II Simplicial Complexes obtained
Page 101 and 102:
84 II Simplicial Complexes (that is
Page 103 and 104:
86 II Simplicial Complexes Therefor
Page 105 and 106:
88 II Simplicial Complexes Exercise
Page 107 and 108:
90 II Simplicial Complexes Hn(C;G)=
Page 109 and 110:
92 II Simplicial Complexes Since im
Page 111 and 112:
94 II Simplicial Complexes and cons
Page 113 and 114:
96 II Simplicial Complexes In parti
Page 116 and 117:
Chapter III Homology of Polyhedra I
Page 118 and 119:
III.1 The Category of Polyhedra 101
Page 120 and 121:
III.1 The Category of Polyhedra 103
Page 122 and 123: III.1 The Category of Polyhedra 105
Page 124 and 125: III.1 The Category of Polyhedra 107
Page 126 and 127: III.2 Homology of Polyhedra 109 Now
Page 128 and 129: III.2 Homology of Polyhedra 111 whe
Page 130 and 131: III.2 Homology of Polyhedra 113 A s
Page 132 and 133: III.2 Homology of Polyhedra 115 Fro
Page 134 and 135: III.3 Some Applications 117 where H
Page 136 and 137: III.3 Some Applications 119 we now
Page 138 and 139: III.3 Some Applications 121 has no
Page 140 and 141: III.3 Some Applications 123 Proof.
Page 142 and 143: III.4 Relative Homology 125 6. Let
Page 144 and 145: III.4 Relative Homology 127 such th
Page 146 and 147: III.5 Real Projective Spaces 129 We
Page 148 and 149: III.5 Real Projective Spaces 131 0
Page 150 and 151: III.5 Real Projective Spaces 133 No
Page 152 and 153: III.5 Real Projective Spaces 135 Le
Page 154 and 155: III.5 Real Projective Spaces 137 he
Page 156 and 157: III.5 Real Projective Spaces 139 We
Page 158 and 159: III.6 Homology of the Product of Tw
Page 166: III.6 Homology of the Product of Tw
Page 169 and 170: 152 IV Cohomology (IV.1.1) Theorem.
Page 171: 154 IV Cohomology (IV.1.2) Remark.
Page 175 and 176: 158 IV Cohomology When we apply thi
Page 177 and 178: 160 IV Cohomology groups Q and R ar
Page 179 and 180: 162 IV Cohomology (IV.2.2) Corollar
Page 181 and 182: 164 IV Cohomology Since g and π r
Page 183 and 184: 166 IV Cohomology It is easily prov
Page 185 and 186: 168 IV Cohomology ∩: B p (K;Z) ×
Page 188 and 189: Chapter V Triangulable Manifolds V.
Page 190 and 191: V.1 Topological Manifolds 173 is a
Page 192 and 193: V.1 Topological Manifolds 175 |Ki|
Page 194 and 195: V.2 Closed Surfaces 177 we define C
Page 196 and 197: V.2 Closed Surfaces 179 Fig. V.6 Co
Page 198 and 199: V.2 Closed Surfaces 181 a a b a a d
Page 200 and 201: V.2 Closed Surfaces 183 where i = 1
Page 202 and 203: V.2 Closed Surfaces 185 Before proc
Page 204 and 205: V.2 Closed Surfaces 187 Simplicial
Page 206 and 207: V.3 Poincaré Duality 189 to the co
Page 208 and 209: V.3 Poincaré Duality 191 (V.3.3) T
Page 210: V.3 Poincaré Duality 193 therefore
Page 213 and 214: 196 VI Homotopy Groups and define t
Page 215 and 216: 198 VI Homotopy Groups is a homotop
Page 217 and 218: 200 VI Homotopy Groups Proof. First
Page 219 and 220: 202 VI Homotopy Groups (VI.1.10) Ex
Page 221 and 222: 204 VI Homotopy Groups (VI.1.13) Th
Page 223 and 224:
206 VI Homotopy Groups to the simpl
Page 225 and 226:
208 VI Homotopy Groups Proof. For e
Page 227 and 228:
210 VI Homotopy Groups Exercises 1.
Page 229 and 230:
212 VI Homotopy Groups Proof. Since
Page 231 and 232:
214 VI Homotopy Groups is precisely
Page 233 and 234:
216 VI Homotopy Groups The group π
Page 235 and 236:
218 VI Homotopy Groups 4. R ′ α
Page 237 and 238:
220 VI Homotopy Groups and � y0 H
Page 239 and 240:
222 VI Homotopy Groups Let f := F(
Page 241 and 242:
224 VI Homotopy Groups be two homot
Page 243 and 244:
226 VI Homotopy Groups (VI.3.16) Re
Page 245 and 246:
228 VI Homotopy Groups where iA is
Page 247 and 248:
230 VI Homotopy Groups f : I n g: I
Page 249 and 250:
232 VI Homotopy Groups 8. Prove tha
Page 251 and 252:
234 VI Homotopy Groups This derives
Page 253 and 254:
236 VI Homotopy Groups Proof. The f
Page 256 and 257:
References 1. J.W. Alexander - A pr
Page 258 and 259:
Index H-space, 219 n-simple, 225 ab
Page 260:
Index 243 Euclidean, 43 join, 47 su
show all

Simplicial Structures in Topology

Create successful ePaper yourself

Delete template?

Save as template?