Simplicial Structures in Topology

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III.4 Relative Homology 125 6. Let M + 3×3 be the set of all real square matrices with no negative elements. Apply the preceeding exercise to prove that every matrix A ∈ M + 3×3 has at least one nonnegative eigenvalue. III.4 Relative Homology for Polyhedra In Sect. II.4, we have studied the relative homology of a pair of simplicial complexes (K,L); specifically, we have proved that given two simplicial complexes K =(X,Φ) and L =(Y,Θ) with Y ⊂ X and Θ ⊂ Φ, then (∀n ≥ 1) Hn(K,L;Z) ∼ = Hn(K ∪CL;Z) where CL is the abstract cone vL Theorem (II.4.7). In Sect. III.2, wehavestudied the homology functor in the category of polyhedra, giving the definition of H∗(|K|,Z) := H(K,Z) for any polyhedron |K| (the definition of functor H∗(−,Z) on morphisms is more intricate and depends on the Simplicial Approximation Theorem). Nevertheless, we may say that (∀n ≥ 1) Hn(|K|,|L|;Z) ∼ = Hn(|K ∪CL|;Z) for every pair of polyhedra (|K|,|L|). In this section, we prove the following result: (III.4.1) Theorem. For every pair of polyhedra (|K|,|L|) (with L a subcomplex of K) and every integer n ≥ 1, Hn(|K|,|L|;Z) ∼ = Hn(|K|/|L|;Z) . This theorem is a direct consequence of a more general result in the category of topological spaces: (III.4.2) Theorem. Let (X,A) be a pair of topological spaces with A closed in X; suppose that (X,A) has the Homotopy Extension Property. Let CA be the cone (A × I)/(A ×{0}). Then the adjunction space X ∪CA and the quotient space X/A are of the same homotopy type. Proof. The reader may intuitively come to this result by considering the cone CA as a space contractible to a point; here is a rigorous proof of the statement. In this theorem, we have two pushouts: one for constructing X ∪CA and the other for X/A; these are their corresponding diagrams: i1 A �� CA i �� i i1 X �� X ∪CA
126 III Homology of Polyhedra with i1 : A → CA given by x ↦→ [x,1] and i: A → X the inclusion; c A �� ∗ i i �� X c �� X/A with c: A →∗the constant map and c = q: X → X/A the quotient map. Let p: CA → X/A be the constant map to the point [a0] that is identified with A in X/A. Sinceqi = pi1, there exists a unique map ℓ: X ∪CA → X/A such that the following diagram is commutative: A i1 �� CA i �� X i1 ¯ ī �� q �� X ∪CA �� ℓ �� p �� X/A Now we must find a function � ℓ: X/A → X ∪CA such that � ℓℓ ∼ 1X∪CA and ℓ � ℓ ∼ 1 X/A. With this in mind, we consider the homotopy H : A × I → X ∪CA , (x,t) ↦→ [x,1 − t] and apply the Homotopy Extension Property of (X,A) to get a continuous function G: A × I −→ X ∪CA whose restriction to A×{0} coincides with i1 andsuchthatG(i×1I)=H. Themap g: A −→ X ∪CA , g = G(−,1) is homotopic to the inclusion i: X → X ∪ CA andissuchthat,foreveryx ∈ A, g(x)=[x,0], the vertex of the cone CA. SinceX/A is a pushout space, there exists a unique map �ℓ: X/A −→ X ∪CA
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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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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
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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
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 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
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Page 154 and 155: III.5 Real Projective Spaces 137 he
Page 156 and 157: III.5 Real Projective Spaces 139 We
Page 158 and 159: III.6 Homology of the Product of Tw
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Page 169 and 170: 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
Page 175 and 176: 158 IV Cohomology When we apply thi
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
Page 185 and 186: 168 IV Cohomology ∩: B p (K;Z) ×
Page 188 and 189: Chapter V Triangulable Manifolds V.
Page 190 and 191: 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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