Simplicial Structures in Topology

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Chapter II The Category of Simplicial Complexes II.1 Euclidean Simplicial Complexes Let us recall that a subset C ⊂ R n is convex if x,y ∈C,t ∈ [0,1] =⇒ tx+(1−t)y ∈C. The convex hull of a subset X ⊂ R n is the smallest convex subset of R n ,which contains X. We say that d + 1 points x0,x1,...,xd belonging to the Euclidean space R n are linearly independent (from the affine point of view) if the vectors x1 − x0, x2 − x0, ..., xd − x0 are linearly independent. A vector x − x0 of the vector space generated by these vectors can be written as a sum x − x0 = ∑ d i=1 ri(xi − x0) with real coefficients ri; notice that if we write x as x = ∑ d i=0 αixi, then∑ d i=0 αi = 1. If {x0,...,xd} ∈X are affinely independent, the convex hull of X is said to be an (Euclidean) simplex of dimension d contained in R n ; its points x can be written in a unique fashion as linear combinations x = d ∑ i=1 λixi, with real coefficients λi. The coefficients λi are called barycentric coordinates of x; they are nonnegative real numbers and satisfy the equality ∑ d i=0 λi = 1. The points xi are the vertices of the simplex. The standard n-simplex is the simplex obtained by taking the convex hull of the n+1 points of the standard basis of R n+1 (see Figs. II.1 and II.2 for dimensions n = 1andn = 2, respectively). The faces of a simplex s ⊂ R n are the convex hulls of the subsets of its vertices; the faces which do not coincide with s are the proper faces. We can define the interior of a simplex s as the set of all points of s with positive barycentric coordinates λi > 0. We indicate the interior of s with ˚s. If the dimension of s is at least 1, ˚s coincides with the topological interior. At any rate, it is not hard to prove that we obtain the interior of a simplex by removing all of its proper faces. An Euclidean simplicial complex is a finite family of simplexes of an Euclidean space R n , which satisfies the following properties: D.L. Ferrario and R.A. Piccinini, Simplicial Structures in Topology, 43 CMS Books in Mathematics, DOI 10.1007/978-1-4419-7236-1 II, © Springer Science+Business Media, LLC 2011
44 II Simplicial Complexes (0, 1) (1, 0, 0) (0, 0, 1) (1, 0) (0, 1, 0) Fig. II.1 Fig. II.2 S1 If s ∈ K, then every face of s is in K. S2 If s1 and s2 are simplexes of K with nondisjoint interiors ˚s1 ∩ ˚s2 �= /0, then s1 = s2. The dimension of K is the maximal dimension of its simplexes. 1 Figure II.3 repre- Fig. II.3 sents a two-dimensional simplicial complex of R 2 ;Fig. II.4 is a set of simplexes, which is not a simplicial complex. 1 It is possible to define Euclidean complexes with infinitely many simplexes, provided we add the local finiteness property that is to say, we ask that each point of a simplex has a neighborhood, which intersects only finitely many simplexes of K. We do this so that the topology of the (infinite) Euclidean complex K coincides with the topology of the geometric realization | �K| (we are referring to the topology defined by Remark (II.2.13)) of the abstract simplicial complex | �K| associated in a natural fashion to K (we shall give the definition of abstract simplicial complex in a short while).
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Canadian Mathematical Society Soci
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Dr. Davide L. Ferrario Università
Page 8: Foreword to the English Edition Exc
Page 11 and 12: x Preface same qualitative properti
Page 13 and 14: xii Preface homology groups, also w
Page 16 and 17: Contents I Fundamental Concepts ...
Page 18 and 19: Chapter I Fundamental Concepts I.1
Page 20 and 21: I.1 Topology 3 We need to show that
Page 22 and 23: I.1 Topology 5 Let X be a topologic
Page 24 and 25: I.1 Topology 7 (x, y) (x, −y) Fig
Page 26 and 27: I.1 Topology 9 that f is a homeomor
Page 28 and 29: I.1 Topology 11 We recall that an i
Page 30 and 31: I.1 Topology 13 Cx = � Xj j∈J i
Page 32 and 33: I.1 Topology 15 A Fig. I.5 Example
Page 34 and 35: I.1 Topology 17 with the topology i
Page 36 and 37: I.1 Topology 19 are closed in Y. Th
Page 38 and 39: I.1 Topology 21 is the diagonal of
Page 40 and 41: I.1 Topology 23 (I.1.38) Lemma. Let
Page 42 and 43: I.1 Topology 25 Here is an example.
Page 44 and 45: I.2 Categories 27 5. Prove that a f
Page 46 and 47: I.2 Categories 29 3. For every pair
Page 48 and 49: I.2 Categories 31 If conditions 2.
Page 50 and 51: I.2 Categories 33 Conversely, given
Page 52 and 53: I.2 Categories 35 q: B ⊔C → B
Page 54 and 55: I.2 Categories 37 (I.2.5) Lemma. Gi
Page 56 and 57: I.3 Group Actions 39 I.3 Group Acti
Page 58: I.3 Group Actions 41 S2 = {(x,y,z)
Page 63 and 64: 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)=
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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.
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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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