Simplicial Structures in Topology

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III.1 The Category of Polyhedra 105 (III.1.5) Theorem. For every real number ε > 0, there is a positive integer r such that diam |K (r) | < ε. Proof. Suppose that dimK = n and take a 1-simplex {σ0,σ1} of K (1) arbitrarily; we may assume that and σ0 = {xi 0 ,...,xiq } σ1 = {xi 0 ,...,xiq ,xj 0 ,...,x jp } where p + q + 1 ≤ n. Since|K| ∼ = |K (1) | (actually, it is useful to make the identification |K|≡|K (1) |), the length of the 1-simplex of |K (1) | represented abstractly by {σ0,σ1} is computed through the formula because = Therefore, d � q ∑ k=0 � � 1 q + 1 − 1 q + 1 xi , k q ∑ k=0 1 p + q + 2 xi + k �2 1 (q + 1)+ p + q + 2 � p ∑ ℓ=0 1 p + q + 2 x � jℓ = 1 p + q + 2 � 1 (p + 1)(p + q + 2) = = p + q + 2 q + 1 = p + q + 1 � (p + 1)(p + q + 2) p + q + 2 (p + q + 1) 2 n √ < 2 (q + 1) n + 1 p + q + 1 n (p + 1)(p + q + 2) < , p + q + 2 n + 1 (p + q + 1) 2 < 2 . (q + 1) �2 �1/2 (p + 1) = diam |K (1) | < n √ 2 . � n + 1 Note that the dimension of a simplicial complex is not affected by successive barycentric subdivisions. Besides, by definition, the dimension of a polyhedron |K| is the dimension of the simplicial complex K,thatistosay,dim|K| = dimK. Let L be a simplicial subcomplex of K; the pair of polyhedra (|K|,|L|) has the Homotopy Extension Property: for every topological space Z and every pair of maps f : |K|×{0}−→Z G: |L|×I −→ Z with the same restriction to |L|×{0}, there is a map (not necessarily unique) F : |K|×I → Z whose restrictions to |K|×{0} and |L|×I coincide with f and G (see the following commutative diagram).
106 III Homology of Polyhedra |L|×{0} i × 1 {0} �� |K|×{0} �� |L|×I i × 1I �� G |K|×I �� F � f �� Z Before proving this property of pairs of polyhedra, we prove a lemma that characterizes the homotopy extension property. (III.1.6) Lemma. Let A be a closed subspace of X. Then, (X,A) has the homotopy extension property if and only if there exists a retraction r : X × I −→ �X = X ×{0}∪A × I (that is to say, a map r whose restriction to �X is the identity). Proof. We note that, since A is closed in X, the following diagram is a pushout: A ×{0} 1A × i0 �� i × 1 {0} �� X ×{0} 1A × i0 A × I i × 1 {0} �� X In the diagram, i0 is the inclusion of {0} in I and i is the inclusion of A in X. Let us suppose that the pair (X,A) has the homotopy extension property. Then, there is a map r : X × I −→ �X such that r(i × 1I)=i × 10 and r(1X × i0)=1A × i0 . Let ι be the inclusion of �X in X × I. By the universal property of pushouts, rι = 1 �X and so r is a retraction. Conversely, let us suppose that r : X ×I → �X is a retraction and let f : X ×{0}→ Z and G: A × I → Z be maps such that f (i × i0) =G(1A × i0); by the definition of pushout, there is a map H : �X → Z that completes the following commutative diagram:
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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+(
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 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
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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.
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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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