Simplicial Structures in Topology

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Chapter III Homology of Polyhedra III.1 The Category of Polyhedra In Sect. II.2, we have defined polyhedra as geometric realizations of (finite) simplicial complexes. We have also seen that polyhedra are compact topological spaces. We shall indicate with P the category of polyhedra and continuous functions between them; clearly, P is a subcategory of Top. We start by proving that the category P is closed under finite products, thatisto say, the Cartesian product of a finite number of polyhedra is (homeomorphic to) a polyhedron; it is enough to prove this result for two polyhedra. (III.1.1) Theorem. Let two simplicial complexes K =(X,Φ) and L =(Y,Ψ ) , be given. There exists a simplicial complex K × L such that |K × L| ∼ = |K|×|L| . Proof. The simplicial complex K × L is defined as follows: 1. The vertices of K × L are the elements of X ×Y. 2. To define the simplexes of K × L, we first order the sets X and Y ;wethengive X ×Y the lexicographical order. 3. A n-simplex of K × L isgivenbyaset{(xi0 ,yj 0 ),...,(xin ,yjn )} of elements of X ×Y satisfying the following conditions: (a)(xi0 ,yj0 )
100 III Homology of Polyhedra (the vertices xi k and y j k are not necessarily different). As a consequence, the projections pr1 : X ×Y → X , pr2 : X ×Y → Y are morphisms between simplicial complexes determining therefore a map φ = |pr1|×|pr2|: |K × L|→|K|×|L| . We wish to prove that φ is a homeomorphism. Let p ∈|K| and q ∈|L| be given by and We define and p = q = m ∑ αixi , αi > 0 , i=0 n ∑ j=0 β jy j , β j > 0 , as = bt = s ∑ i=0 t ∑ j=0 m ∑ αi = 1 , x0 < ···< xm i=0 n ∑ j=0 β j = 1 , y0 < ···< yn . αi , s = 0,1,...,m β j , t = 0,1,...,n ; we then take the set {0,a0,...,am−1,b0,...,bn = 1}, rename and reorder its m + n + 2 elements so to obtain the ordered set {c−1,c0,...,cm+n} such that 0 = c−1 < c0 < c1
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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
Page 65 and 66: 48 II Simplicial Complexes II.2.1 T
Page 67 and 68: 50 II Simplicial Complexes Let us r
Page 69 and 70: 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 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
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
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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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