Simplicial Structures in Topology

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II.5 Homology with Coefficients 91 and (r ⊗ 1G)( f ⊗ 1G)=1A⊗G (g ⊗ 1G)(s ⊗ 1G)=1C⊗G ( f ⊗ 1G)(r ⊗ 1G)+(s ⊗ 1G)(g ⊗ 1G)=1B⊗G. The first of these relations tells us that ( f ⊗1G) is injective; the second, that (g⊗1G) is surjective, and the third, that im( f ⊗ 1G) =ker(g ⊗ 1G) because, if we take x ∈ B ⊗ G such that (g ⊗ 1G)(x)=0, then x =(f ⊗ 1G)(r ⊗ 1G)(x)+(s ⊗ 1G)(g ⊗ 1G)(x) =(f ⊗ 1G)((r ⊗ 1G)(x)) ∈ im( f ⊗ 1G). Returning to our free chain complex (C,∂), we notice that for each integer n,the sequence Zn(C) ⊗ G �� Cn ⊗ G ∂n ⊗ 1G�� Bn−1(C) ⊗ G is short exact. In addition, we observe that the graded Abelian groups Z(C) = {Zn(C)|n ∈ Z} and B(C)={Bn(C)|n ∈ Z} may be viewed as chain complexes with trivial boundary operator 0; we then construct the chain complexes 1. (Z(C) ⊗ G,0 ⊗ 1G); 2. (C ⊗ G,∂ ⊗ 1G); 3. ( � B(C) ⊗ G,0 ⊗ 1G), where � B(C) n = Bn−1(C) and observe that in view of the preceding short exact sequence of Abelian groups, we have a short exact sequence of chain complexes (Z(C) ⊗ G,0 ⊗ 1G) �� (C ⊗ G,∂ ⊗ 1G) �� ( B(C) � ⊗ G,0 ⊗ 1G). By the Long Exact Sequence Theorem (II.3.1), we obtain the long exact sequence of homology groups ··· Hn( � B(C) ⊗ G) �� Hn(Z(C) ⊗ G) �� Hn(C ⊗ G) �� Hn−1(Z(C) ⊗ G) �� ··· in other words, by considering the format of the boundary operators, we have the following exact sequence of Abelian groups: Hn(C;G) ··· �� Bn(C) ⊗ G in ⊗ 1G �� Zn(C) ⊗ G hn �� Bn−1(C) ⊗ G in−1 ⊗ 1G �� Zn−1(C) ⊗ G �� jn �� ··· Note that in is the inclusion of Bn(C) in Zn(C) and jn is the induced homomorphism by the inclusion of Zn(C) in Cn; the reader is also asked to notice that the connecting homomorphism λn+1 in Theorem (II.3.1) coincides with in ⊗ 1G. �
92 II Simplicial Complexes Since im jn = kerhn, we conclude that, for every n ≥ 0, the sequence im jn �� Hn(C;G) hn �� imhn is short exact. (II.5.2) Lemma. If the group G is free, the short exact sequence splits, 4 and so im jn �� Hn(C;G) Hn(C;G) ∼ = im jn ⊕ imhn hn �� imhn (however, one should note that this isomorphism is not canonic). Proof. Let us take the homomorphism of Abelian groups hn : Hn(C;G) → Bn−1(C) ⊗ G. As a subgroup of the free Abelian group Cn−1, Bn−1(C) is free and by hypothesis G is also free; then Bn−1(C) ⊗ G is free and it follows that imhn is free. We now choose, for every generator x ∈ imhn,anelementy ∈ Hn(C;G) such that hn(y)=x; by linearity, we obtain a homomorphism s: imhn −→ Hn(C;G) such that hns = 1imhn .Exercise1in Sect. II.3 completes the proof. The homomorphism s depends on the choice of the elements y for the generators x; therefore, s is not canonically determined. � We now give another interpretation of the groups im jn and imhn. Note that the quotient group im jn ∼ = Zn(C) ⊗ G/ker jn = Zn(C) ⊗ G/im(in ⊗ 1G); Zn(C) ⊗ G/im(in ⊗ 1G) := coker(in ⊗ 1G) is called cokernel of in ⊗ 1G. Since imhn = ker(in−1 ⊗ 1G), the exact sequence is written as im jn �� Hn(C;G) coker(in ⊗ 1G) �� Hn(C;G) hn �� imhn �� ker(in−1 ⊗ 1G) 4 The definition of splitting short exact sequence can be found in Exercise 1, Sect. II.3.
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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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I.1 Topology 15 A Fig. I.5 Example
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I.1 Topology 17 with the topology i
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I.1 Topology 19 are closed in Y. Th
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I.1 Topology 21 is the diagonal of
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I.1 Topology 23 (I.1.38) Lemma. Let
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I.1 Topology 25 Here is an example.
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I.2 Categories 27 5. Prove that a f
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I.2 Categories 29 3. For every pair
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I.2 Categories 31 If conditions 2.
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I.2 Categories 33 Conversely, given
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I.2 Categories 35 q: B ⊔C → B
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I.2 Categories 37 (I.2.5) Lemma. Gi
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I.3 Group Actions 39 I.3 Group Acti
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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
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Page 75 and 76: 58 II Simplicial Complexes Let K3,3
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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
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Page 105 and 106: 88 II Simplicial Complexes Exercise
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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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Page 134 and 135: III.3 Some Applications 117 where H
Page 136 and 137: III.3 Some Applications 119 we now
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Page 140 and 141: III.3 Some Applications 123 Proof.
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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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