Simplicial Structures in Topology

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II.1 Euclidean Simplicial Complexes 45 Fig. II.4 (II.1.1) Example (Euclidean polyhedra). 1. Every simplex of R n together with all its faces is a simplicial complex. 2. The set of all proper faces of a d-dimensional simplex in R n is a (d − 1)dimensional simplicial complex. 3. The set of all closed intervals [1/n,1/(n + 1)], with n ∈ N, is a simplicial complex (with infinitely many simplexes) of R. 4. Let Pm be the regular polygonal line contained in C ∼ = R 2 , whose vertices are the mth-roots of the unity {z ∈ C | z m = 1}. The corresponding simplicial complex is homeomorphic to the circle S 1 and is depicted in Fig. II.5. Fig. II.5 5. The Platonic solids can be subdivided by triangles; they give rise to simplicial complexes of R 3 . An example is given by the icosahedron of Fig. II.6. Fig. II.6
46 II Simplicial Complexes II.2 Abstract Simplicial Complexes In this section, we shall give the definition of the category Csim of simplicial complexes and simplicial maps; furthermore, we shall define two important functors with domain Csim, namely, the geometric realization functor and the homology functor. An (abstract) simplicial complex is a pair K =(X,Φ) given by a finite set X and a set of nonempty subsets of X such that: K1 (∀x ∈ X) , {x}∈Φ, K2 (∀σ ∈ Φ)(∀σ ′ ⊂ σ , σ ′ �= /0) , σ ′ ∈ Φ. The elements of X are the vertices of K. The elements of Φ are the simplexes of K. If σ is a simplex of K, every non-empty σ ′ ⊂ σ is a face of σ. According to condition K2, we can say that all faces of a simplex are simplexes. A simplex σ with n + 1elements(n ≥ 0) is an n-simplex (we also say that σ is a simplex of dimension n); we adopt the notation dimσ = n. It follows that the 0-simplexes are vertices. The dimension of K is the maximal dimension of its simplexes; if the dimensions of all simplexes of K have a maximum n, we say that K has dimension n or that K is n-dimensional. (II.2.1) Remark. We explicitly observe that in this book all simplicial complexes have a finite number of vertices. Before we present some examples and constructions with simplicial complexes, we give a definition: a simplicial complex L =(Y,Ψ) is a subcomplex of K =(X,Φ) if Y ⊂ X and Ψ ⊂ Φ. (II.2.2) Remark. Let K0 =(X0,Φ0) and K1 =(X1,Φ1) be subcomplexes of a simplicial complex K; we observe that the union K0 ∪ K1 =(X0 ∪ X1,Φ0 ∪ Φ1) and the intersection K0 ∩K1 =(X0 ∩X1,Φ0 ∩Φ1) (with X0 ∩X1 �= /0) are subcomplexes of K. In particular, the union of two disjoint simplicial complexes K0 and K1 (that is to say, such that X0 ∩ X1 = /0) is a simplicial complex. Let us now give some examples. 1. Let X be a finite set and let ℘(X)=2 X be the set of all subsets of X; clearly, the pair K =(X,℘(X) � /0) is a simplicial complex. 2. The set of all simplexes of an Euclidean simplicial complex is an abstract simplicial complex if we forget the fact that its vertices are points of R n . The set X is the set of all vertices, while Φ is the set of simplexes. Thus, the examples of Euclidean polyhedra on p. 45 are examples of abstract simplicial complexes. 3. Let Γ be a graph (that is to say, a set of vertices X and a symmetric subset Φ of X × X, called set of edges). It is not hard to prove that (X,Φ) is a simplicial complex if we assume that σ ∈ Φ ⊂ 2 X whenever σ is a set with just one element or is the set of the two vertices at the ends of an edge.
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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
Page 11 and 12: x Preface same qualitative properti
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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
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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
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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
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Page 99 and 100: 82 II Simplicial Complexes obtained
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Page 105 and 106: 88 II Simplicial Complexes Exercise
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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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