Simplicial Structures in Topology

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III.6 Homology of the Product of Two Polyhedra 145 The diagram is commutative. In fact, F(X)0 η0 �� G(X)0 ε �� Z ε ′ �� Z ¯f ε(F( f )0(x i 0 )) = ε′ G( f )0(y i 0 )=ε′ η0(F( f )0(x i 0 )) and therefore ¯f ε = ε ′ η0. We now prove that η0 is natural. For any given g ∈ C(X,Y), that is to say, the diagram η0F(g)0(F( f )0(x i 0 )) = η0F(gf)0(x i 0 ) G(gf)0(y i 0)=G(g)0G( f )0(y i 0)=G(g)0η0(F( f )0(x i 0)), F(X)0 η0 �� G(X)0 G(g)0 F(g)0 �� F(Y )0 ¯f η0 �� G(Y )0 is commutative. We now construct η1. Let us consider the diagram F(M i 1)1 G(M i 1 )1 d �� i F(M1)0 d ′ for a model M i 1 ∈ M1 and let us take η0 �� i G(M1 )0 η0d(x i 1) ∈ G(M i 1)1 ε �� Z for every universal element x i 1 ∈ F(Mi 1 )1. Here is where the hypothesis on the acyclicity of G on the models of the set M is needed. Indeed, since εd = 0, we have ε ′ η0d(x i 1 ) ∈ kerε′ = imd ′ ¯f �� Z
146 III Homology of Polyhedra and, consequently, there is yi 1 ∈ G(Mi 1 )1 such that d ′ (yi 1 )=η0d(xi 1 ); we then define η1(F( f )1(x i 1 )) := G( f )1(y i 1 ) for every f : M i 1 → X. We ask the reader to prove that η0d = d ′ η1 and that η1 is natural. The functor ηn is obtained by induction. We now suppose θ : F → G to be a natural transformation extending ¯f . We wish to construct the homotopy E : F → G that connects η to θ. We begin with E0 : F(X)0 → G(X)1 for an arbitrary X. Let us take a model M i 0 ∈ M0 and a universal element x i 0 ∈ F(Mi 0 )0. From ε ′ [η(x i 0 ) − θ(xi0 )] = 0 and since G is acyclic for the model M i 0 , it follows that there exists zi 0 ∈ G(Mi 0 )1 such that d ′ (z i 0 )=η(xi 0 ) − θ(xi 0 ); we now define E0 : F(X)0 → G(X)1 on the generators of F(X)0 by the formula E0(F( f )0(x i 0 )) := G( f )1(z i 0 ) and then extend it on F(X)0 by linearity. We now suppose that the morphisms E1,...,En−1 have been defined; our aim is to define En. As usual, we take a model M i n and a universal element xi n ∈ F(Mi n )n. We note that d ′ (ηn−1d(x i n ) − θn−1d(x i n ) − En−1d(x i n )) = 0 because ηn−1d(x i n) − θn−1d(x i n) − En−1d(x i n)) = En−2d(d(x i n)). Therefore, since G is acyclic on the model, (∃t i n ∈ G(Mi n )n+1)d ′ t i n = ηn−1d(x i n ) − θn−1d(x i n ) − En−1d(x i n ). We define En on the generators F( f )n(x i n) of F(X)n by En(F( f )n(x i n )) := G( f )n+1(t i n ) and linearly extend it. We leave to the reader the final details of the proof. � The next result, known as the Eilenberg–Zilber Theorem, is a direct consequence of the Acyclic Models Theorem. (III.6.3) Theorem. Let C : Csim −→ Clp be the functor that takes each simplicial complex K to the augmented chain complex C(K)={Cn(K),∂n}.
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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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I.3 Group Actions 41 S2 = {(x,y,z)
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44 II Simplicial Complexes (0, 1) (
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46 II Simplicial Complexes II.2 Abs
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48 II Simplicial Complexes II.2.1 T
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50 II Simplicial Complexes Let us r
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52 II Simplicial Complexes r = tp+(
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54 II Simplicial Complexes In a sim
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56 II Simplicial Complexes (not nec
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58 II Simplicial Complexes Let K3,3
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60 II Simplicial Complexes ordering
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62 II Simplicial Complexes are two
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64 II Simplicial Complexes 1-simple
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66 II Simplicial Complexes An infin
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68 II Simplicial Complexes 2. λn i
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70 II Simplicial Complexes Please n
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72 II Simplicial Complexes (II.3.7)
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74 II Simplicial Complexes (II.3.10
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76 II Simplicial Complexes If we do
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78 II Simplicial Complexes In the c
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80 II Simplicial Complexes Φi = {
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82 II Simplicial Complexes obtained
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84 II Simplicial Complexes (that is
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86 II Simplicial Complexes Therefor
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88 II Simplicial Complexes Exercise
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90 II Simplicial Complexes Hn(C;G)=
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92 II Simplicial Complexes Since im
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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
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
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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
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Page 169 and 170: 152 IV Cohomology (IV.1.1) Theorem.
Page 171 and 172: 154 IV Cohomology (IV.1.2) Remark.
Page 173 and 174: 156 IV Cohomology If, starting from
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
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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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