Simplicial Structures in Topology

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II.3 Homological Algebra 65 II.3 Introduction to Homological Algebra In the previous section, we have seen that we can associate a graded Abelian group C(K) ={Cn(K)} with any simplicial complex K and a homomorphism ∂n : Cn(K) → Cn−1(K), such that ∂n−1∂n = 0, to each integer n; these homomorphisms define a graded Abelian group H∗(K;Z) ={Hn(K;Z)}. All this can be viewed in the framework of a more general and more useful context. A chain complex (C,∂) is a graded Abelian group C = {Cn} together with an endomorphism ∂ = {∂n} of degree −1, called boundary homomorphism3 ∂ = {∂n : Cn → Cn−1}, such that ∂ 2 = 0; this means that, for every n ∈ Z, ∂n∂n+1 = 0. Hence Bn = im∂n+1 ⊂ Zn = ker∂n and so we can define the graded Abelian group H∗(C)={Hn(C)=Zn/Bn | n ∈ Z}; this is the homology of C. A chain homomorphism between two chain complexes (C,∂) and (C ′ ,∂ ′ ) is a graded group homomorphism f = { fn : Cn → C ′ n} of degree 0 commuting with the boundary homomorphism, that is to say, for every n ∈ Z, fn−1∂n = ∂ ′ n fn. Chain complexes and chain homomorphisms form a category C, thecategory of chain complexes. It is costumary to visualize chain complexes as diagrams ··· �� Cn+1 ∂n+1 �� Cn and their morphisms as commutative diagrams ··· ··· �� Cn+1 �� ′ C n+1 fn+1 ∂n+1 �� Cn �� ∂ ′ n+1 �� ′ C n fn ∂n �� Cn−1 ∂n �� Cn−1 �� ∂ ′ n �� ′ C n−1 fn−1 �� ··· �� ··· �� ··· The previous definitions are clearly inspired by what we did to define the homology groups of a simplicial complex; indeed, we emphasize the fact that, for every simplicial complex X, the graded Abelian group {Cn(K)|n ∈ Z} together with its boundary homomorphism ∂ K = {∂ K n |n ∈ Z} is a chain complex (C(K),∂ K ). The chain complex C(K) is said to be positive because its terms of negative index are 0. In particular, for every simplicial function f : K → M, the homomorphism is a chain homomorphism. C( f ): C(K) → C(M) 3 In some textbooks, it is called differential operator.
66 II Simplicial Complexes An infinite sequence of Abelian groups ··· �� Gn+1 fn+1 �� Gn fn �� Gn−1 is said to be exact if and only if, for every n ∈ Z,imfn+1 = ker fn. The exact sequences with only three consecutive nontrivial groups ... �� 0 �� Gn+1 fn+1 �� Gn fn �� Gn−1 �� 0 �� ··· �� ... are particularly important; in that case, fn+1 is injective and fn is surjective. These sequences are called short exact sequences. The previous short exact sequence is also written up in the form Gn+1 �� fn+1 �� Gn fn �� Gn−1. The concept of short exact sequence of groups can be easily exported to the category C of chain complexes: a sequence of chain complexes (C,∂) �� f �� ′ ′ (C ,∂ ) g �� ′′ ′′ (C ,∂ ) is exact if every horizontal line of its representative diagram . �� ∂n+2 . fn+1 Cn+1 �� C ′ n+1 �� Cn �� ∂n+1 ∂n �� fn �� C ′ n �� ∂ ′ n+2 ∂ ′ n+1 ∂ ′ n fn−1 Cn−1 �� C ′ n−1 �� . . ∂n−1 �� . . ∂ ′ n−1 gn+1 �� gn �� gn−1 �� . �� C ′′ n+1 �� C ′′ n �� ∂ ′′ n+2 ∂ ′′ n+1 ∂ ′′ n C ′′ n−1 �� . . ∂ ′′ n−1
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Canadian Mathematical Society Soci
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Dr. Davide L. Ferrario Università
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Foreword to the English Edition Exc
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x Preface same qualitative properti
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xii Preface homology groups, also w
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Contents I Fundamental Concepts ...
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Chapter I Fundamental Concepts I.1
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I.1 Topology 3 We need to show that
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I.1 Topology 5 Let X be a topologic
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I.1 Topology 7 (x, y) (x, −y) Fig
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I.1 Topology 9 that f is a homeomor
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I.1 Topology 11 We recall that an i
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I.1 Topology 13 Cx = � Xj j∈J i
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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
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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
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Page 75 and 76: 58 II Simplicial Complexes Let K3,3
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Page 79 and 80: 62 II Simplicial Complexes are two
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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
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Page 116 and 117: Chapter III Homology of Polyhedra I
Page 118 and 119: III.1 The Category of Polyhedra 101
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
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III.2 Homology of Polyhedra 115 Fro
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III.3 Some Applications 117 where H
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III.3 Some Applications 119 we now
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III.3 Some Applications 121 has no
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III.3 Some Applications 123 Proof.
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III.4 Relative Homology 125 6. Let
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III.4 Relative Homology 127 such th
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III.5 Real Projective Spaces 129 We
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III.5 Real Projective Spaces 131 0
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III.5 Real Projective Spaces 133 No
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III.5 Real Projective Spaces 135 Le
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III.5 Real Projective Spaces 137 he
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III.5 Real Projective Spaces 139 We
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III.6 Homology of the Product of Tw
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152 IV Cohomology (IV.1.1) Theorem.
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154 IV Cohomology (IV.1.2) Remark.
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156 IV Cohomology If, starting from
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158 IV Cohomology When we apply thi
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160 IV Cohomology groups Q and R ar
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162 IV Cohomology (IV.2.2) Corollar
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164 IV Cohomology Since g and π r
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166 IV Cohomology It is easily prov
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168 IV Cohomology ∩: B p (K;Z) ×
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Chapter V Triangulable Manifolds V.
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V.1 Topological Manifolds 173 is a
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V.1 Topological Manifolds 175 |Ki|
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V.2 Closed Surfaces 177 we define C
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V.2 Closed Surfaces 179 Fig. V.6 Co
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V.2 Closed Surfaces 181 a a b a a d
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V.2 Closed Surfaces 183 where i = 1
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V.2 Closed Surfaces 185 Before proc
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V.2 Closed Surfaces 187 Simplicial
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V.3 Poincaré Duality 189 to the co
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V.3 Poincaré Duality 191 (V.3.3) T
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V.3 Poincaré Duality 193 therefore
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196 VI Homotopy Groups and define t
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198 VI Homotopy Groups is a homotop
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200 VI Homotopy Groups Proof. First
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202 VI Homotopy Groups (VI.1.10) Ex
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204 VI Homotopy Groups (VI.1.13) Th
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206 VI Homotopy Groups to the simpl
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208 VI Homotopy Groups Proof. For e
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210 VI Homotopy Groups Exercises 1.
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212 VI Homotopy Groups Proof. Since
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214 VI Homotopy Groups is precisely
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216 VI Homotopy Groups The group π
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218 VI Homotopy Groups 4. R ′ α
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220 VI Homotopy Groups and � y0 H
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222 VI Homotopy Groups Let f := F(
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224 VI Homotopy Groups be two homot
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226 VI Homotopy Groups (VI.3.16) Re
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228 VI Homotopy Groups where iA is
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230 VI Homotopy Groups f : I n g: I
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232 VI Homotopy Groups 8. Prove tha
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234 VI Homotopy Groups This derives
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236 VI Homotopy Groups Proof. The f
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References 1. J.W. Alexander - A pr
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Index H-space, 219 n-simple, 225 ab
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Index 243 Euclidean, 43 join, 47 su
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Simplicial Structures in Topology

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