Simplicial Structures in Topology

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II.2 Abstract Simplicial Complexes 53 (iii): Let ℓ be a ray with origin p, andletq be the point of ℓ determined by the condition d(p,q)=sup{d(p,q ′ ) | q ′ ∈ ℓ ∩ D(p)}. Then, |s(q)| ∈S(p); otherwise, we could extend ℓ in D(p) beyond q and thus we would have q ∈ S(p). On the other hand, because s(q) is a face of a simplex containing p, the vertices of s(p) and s(q) define a simplex of which s(p) is a face (in fact, s(p) ∪ s(q) is a simplex of the simplicial complex D(p)). The points of ℓ beyond q cannot be in S(p) and the open segment (p,q) is contained in D(p) � S(p). Hence, ℓ intersects S(p) in one point only. � Notice that D(p) is not necessarily convex; at any rate, as we have seen in part (ii) of the previous theorem, D(p) is endowed with a certain kind of convexity in the sense that, for every q ∈ D(p), the segment [p,q] is entirely contained in D(p). We say that D(p) is p-convex (star convex). Theorem (II.2.10) allows us to define amap πp : D(p) � {p}→S(p) , q ↦→ ℓp,q ∩ S(p) where ℓp,q is the ray with origin p and containing q; the function πp is the radial projection with center p from D(p) onto S(p). Let i: S(p) → D(p) � {p} be the inclusion map; then πpi = 1 S(p),andiπp is homotopic to the identity map of D(p)� {p} onto itself with homotopy given by the map H : (D(p) � {p}) × I → D(p) � {p} , (q,t) ↦→ (1 − t)q + tπp(q). Hence, S(p) is a deformation retract of D(p) � {p} (see Exercise 2, Sect. I.2). This shows another similarity between the spaces D(p), S(p) and, respectively, the n-dimensional Euclidean disk and its boundary. The next result (cf. [24]) will be used only when studying triangulable manifolds (Sect. V.1); the reader could thus leave it for later on. (II.2.11) Theorem. Let f : |K|→|L| be a homeomorphism. Then, for every p ∈|K|, S(p) and S( f (p)) are of the same homotopy type. Proof. Assume that s( f (p)) = {y0,...,yn} and let U = |s( f (p))| � | s( f (p))| be the interior of |s( f (p))|, that is to say, the set of all q ∈|L| such that q(yi) > 0, for every i = 0,...,n. Notice that U is an open set of D( f (p)); moreover, f −1 (U) is an open set of |K| containing p. The bounded, compact set D(p) can be shrunk at will: in fact, for any real number 0 < λ ≤ 1wedefinethecompression λD(p) as the set of all points r =(1− λ )p + λ q, foreveryq∈D(p); observe that λD(p) is a closed subset of |K|, and is homeomorphic to D(p). Let λ ∈ (0,1] be such that p ∈ λD(p) ⊂ f −1 (U); then f (p) ∈ f (λD(p)) ⊂ U ⊂ D( f (p)). •
54 II Simplicial Complexes In a similar fashion, we can find two other real numbers μ,ν ∈ (0,1] such that f (p) ∈ f (νD(p)) ⊂ μD( f (p)) ⊂ f (λD(p)) ⊂ D( f (p)). Because f (νD(p)) ⊂ μD( f (p)), we can define the radial projection with center f (p) ψ : f (νS(p)) → μS( f (p)) , f (q) ↦→ π f (p)( f (q)) for every q ∈ νS(p). We also define the map φ : μS( f (p)) → f (νS(p)) , q ↦→ f (ν(πp( f −1 (q))) where πp is the radial function with center p in λD(p) (notice that f −1 (q) �= p, for every q ∈ μS( f (p)) and moreover, f −1 (μD( f (p)) ⊂ λD(p)). Since the spaces f (νS(p)) and μS( f (p)) are contained in D( f (p)) and this last space is f (p)-convex, we can define the homotopy H1 : μS( f (p)) × I −→ D( f (p)) � 1 (1 − 2t)ψφ(q)+2tf(p) , 0 ≤ t ≤ H1(q,t)= 2 (2 − 2t) f (p)+(2t− 1)φ(q) , 1 2 ≤ t ≤ 1 for every q ∈ μS( f (p)). Strictly speaking, H1 is a homotopy between φ composed with the inclusion map f (νS(p) ⊂ D( f (p)) and ψφ composed with μS( f (p)) ⊂ D( f (p)). We now take the maps f −1 φ : μS( f (p)) → νD(p) and f −1 : μS( f (p)) → λD(p). Because D(p) is p-convex, we can construct the homotopy � (1 − 2t) f −1 1 φ(q)+2tp, 0 ≤ t ≤ H2(q,t)= 2 (2 − 2t)p +(2t− 1) f −1 (q) , 1 2 ≤ t ≤ 1. which, when composed with the homeomorphism f , gives rise to a homotopy finally, we consider the homotopy fH2 : μS( f (p)) × I → D( f (p)); F : μS( f (p) × I → D( f (p)) defined by the formula � H1(q,2t) , 0 ≤ t ≤ F(q,t)= 1 2 fH2(q,2t − 1) , 1 2 ≤ t ≤ 1.
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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
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)
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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
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Page 75 and 76: 58 II Simplicial Complexes Let K3,3
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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
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Page 97 and 98: 80 II Simplicial Complexes Φi = {
Page 99 and 100: 82 II Simplicial Complexes obtained
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Page 105 and 106: 88 II Simplicial Complexes Exercise
Page 107 and 108: 90 II Simplicial Complexes Hn(C;G)=
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Page 116 and 117: Chapter III Homology of Polyhedra I
Page 118 and 119: III.1 The Category of Polyhedra 101
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III.1 The Category of Polyhedra 103
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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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