Simplicial Structures in Topology

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III.2 Homology of Polyhedra 115 From this, we conclude that |g ′ (π (r,s) (σ))|⊂|s( f (b(σ)))| and, consequently, g ′ (π (r,s) (σ)) ⊂ s( f (b(σ))). � (III.2.7) Corollary. Amap f: |K|→|L| defines a unique homomorphism of graded groups H∗( f ,Z): H∗(|K|,Z) → H∗(|L|,Z) . Proof. For any simplicial approximation g: K (r) → L of f ,letusset H∗( f ,Z)=H∗(g,Z)(H∗(π r )) −1 (recall that (H∗(π r )) −1 = H∗(ℵ r ) ). Suppose g ′ : K (s) → L to be another simplicial approximation of f , with s < r. The contiguity of the simplicial functions g and g ′ π (r,s) (which follows from the previous theorem) ensures that H∗(g ′ π (r,s) )=H∗(g) (cf. Corollary (III.2.1)). But π r = π s π (r,s) and therefore H∗(g ′ ,Z)(H∗(π s ,Z)) −1 = H∗(g,Z)(H∗(π r ,Z)) −1 . Corollary (III.2.7) completes the definition of functor H∗(−,Z) on morphisms; we repeat it here: for every n ≥ 0 and any map f : |K|→|L|, Hn( f ,Z) := Hn(g,Z)(Hn(π r ,Z)) −1 : Hn(|K|,Z) → Hn(|L|,Z) where g: K (r) → L is any simplicial approximation of f . We now prove that the homomorphism H∗( f ,Z),definedbyamap f : |K|→|L|, is a homotopy invariant, meaning that the next result is true. (III.2.8) Theorem. Let f ,g: |K|→|L| be homotopic maps. Then H∗( f ,Z)=H∗(g,Z) . Proof. Let H : |K|×I →|L| be the homotopy that links f to g; suppose that Hi0 = f and Hi1 = g,wherei0 and i1 are the maps iα : |K|→|K|×I , p ↦→ (p,α) , α = 0,1 . We represent the intervall I =[0,1] as the geometric realization of the complex I =(Y,ϒ ) , Y = {0,1} , ϒ = {{0},{1},{0,1}} . �
116 III Homology of Polyhedra With the appropriate identification |K × I |≡|K|×I (cf. Theorem (III.1.1)), we may assume that i0 and i1 are the geometric realizations of the simplicial functions �ıα : K → K × I that take each 0-simplex {x} of K to the 0-simplex {(x,α)} of K × I . To verify this theorem, it is sufficient to prove that�ı0 and�ı1 are contiguous. We order the set X = {x0,x1 ...,xr} of all vertices of K by setting x0 < x1
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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
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 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 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
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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
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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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