Simplicial Structures in Topology

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VI.1 Fundamental Group 205 We take a disk D 2 with center c and divide its boundary ∂D 2 into n + 1partsby means of the (distinct) points b0,...,bn. LetD =(Y,Ψ) be the simplicial complex with vertices Y = {c,b0,...,bn} and simplexes Ψ = {c,b0,...,bn;b0b1,b1b2,...,bnb0,cb0,cb1,...,cbn, cb0b1,cb1b2,...,cbnb0} (we omit the curly brackets for the simplexes). The one-dimensional simplicial complex ∂D formed by the simplexes is a subcomplex of D; note that {b0,...,bn;b0b1,b1b2,...,bnb0} |D| ∼ = D 2 and |∂ D| ∼ = ∂D 2 ∼ = S 1 . The construction is nothing but a cone (with vertex c) on a simplicial subdivision of S 1 . We now consider the simplicial function f : ∂D → K , (∀i = 0,1,...,n) f (bi)=ai and an adjunction space X defined by the pushout S 1 ∼ = |∂ D| �� D 2 ∼ = |D| With these conditions we have the following: (VI.1.15) Theorem. The fundamental group π(X,a0) of the space X, obtained by adjoining a 2-cell to |K| by means of the path α, is obtained from the group π(|K|,a0) together with the relation defined by the homotopy class of the same path α. Proof. We consider the barycentric subdivision of the triangulation of D 2 just described, in other words, before gluing D 2 to K, we refine the triangulation of D 2 by adding 2(n + 1) new vertices | f | | f | �� |K| �� X c0,c1,...,cn; d0,d1,...,dn
206 VI Homotopy Groups to the simplicial complex D, where each ci is the barycenter of the simplex {c,bi} and each di is the barycenter of the triangle {c,bi,bi+1} (here the index i = 0...n is understood as modulo n). See in Fig. VI.2 such a subdivision, for n = 6. Let then b 3 d 3 d 4 b 2 b 4 c 3 d 2 c2 c 1 c 4 c5 b 1 d 1 c 0 d 0 b 0 d5 Fig. VI.2 Barycentric subdi- b5 vision of D2 M =(Z,Θ) be the simplicial complex of the barycentric subdivision. The space obtained by gluing |M| ∼ = D 2 to |K| is still X; note that, naturally, X may be viewed as the geometric realization of an abstract simplicial complex W. We now consider the 2-simplex σ = {b0,c0,d0} (indicated in Fig. VI.2) and the pushout |W � σ|∩|σ| iW �σ �� iσ �� |σ| iW �σ |W � σ| By Theorem (VI.1.13), we conclude that π(X,a0) is a pushout of the diagram π(|W � σ|∩|σ|,a0) π(iW�σ ) �� π(iσ) �� π(|σ|,a0) and thus π(X,a0) is obtained from the free product together with the relations �� X π(|σ|,a0) ∗ π(|W � σ|,a0) π(iW�σ )(c)(π(iσ )(c)) −1 , iσ π(|W � σ|,a0)
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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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94 II Simplicial Complexes and cons
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96 II Simplicial Complexes In parti
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Chapter III Homology of Polyhedra I
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III.1 The Category of Polyhedra 101
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III.2 Homology of Polyhedra 109 Now
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III.2 Homology of Polyhedra 111 whe
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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.
Page 171 and 172: 154 IV Cohomology (IV.1.2) Remark.
Page 173 and 174: 156 IV Cohomology If, starting from
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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
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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
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Page 200 and 201: V.2 Closed Surfaces 183 where i = 1
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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
Page 213 and 214: 196 VI Homotopy Groups and define t
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Page 217 and 218: 200 VI Homotopy Groups Proof. First
Page 219 and 220: 202 VI Homotopy Groups (VI.1.10) Ex
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Page 225 and 226: 208 VI Homotopy Groups Proof. For e
Page 227 and 228: 210 VI Homotopy Groups Exercises 1.
Page 229 and 230: 212 VI Homotopy Groups Proof. Since
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Page 233 and 234: 216 VI Homotopy Groups The group π
Page 235 and 236: 218 VI Homotopy Groups 4. R ′ α
Page 237 and 238: 220 VI Homotopy Groups and � y0 H
Page 239 and 240: 222 VI Homotopy Groups Let f := F(
Page 241 and 242: 224 VI Homotopy Groups be two homot
Page 243 and 244: 226 VI Homotopy Groups (VI.3.16) Re
Page 245 and 246: 228 VI Homotopy Groups where iA is
Page 247 and 248: 230 VI Homotopy Groups f : I n g: I
Page 249 and 250: 232 VI Homotopy Groups 8. Prove tha
Page 251 and 252: 234 VI Homotopy Groups This derives
Page 253 and 254: 236 VI Homotopy Groups Proof. The f
Page 256 and 257: References 1. J.W. Alexander - A pr
Page 258 and 259: Index H-space, 219 n-simple, 225 ab
Page 260: Index 243 Euclidean, 43 join, 47 su
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Simplicial Structures in Topology

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