Simplicial Structures in Topology

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II.4 Simplicial Homology 77 such that k : K → K ′ and ℓ: L → L ′ is the restriction of k to L. The reader can easily verify that the relative homology determines a covariant functor H(−,−;Z): CCsim −→ Ab Z . The next result, which is an immediate application of the Long Exact Sequence Theorem (II.3.1), is called Long Exact Homology Sequence Theorem; it relates the homology groups of L, K, and(K,L) to each other. (II.4.1) Theorem. Let (K,L) be a pair of simplicial complexes. For every n > 0, there is a homomorphism λn : Hn(K,L;Z) → Hn−1(L;Z) (connecting homomorphism) that causes the following sequence of homology groups ...→ Hn(L;Z) Hn(i) −→ Hn(K;Z) q∗(n) −→ Hn(K,L;Z) λn −→ Hn−1(L;Z) → ... , to be exact; here, Hn(i) is the homomorphism induced by the inclusion i: L → K and q∗(n) is the homomorphism induced by the quotient homomorphism qn : Cn(K) → Cn(K)/Cn(L). Proof. For every n > 0, let qn : Cn(K) → Cn(K)/Cn(L) be the quotient homomorphism. With the given definitions, it is easily proved that and therefore, (∀n ≥ 0) ∂ K,L n qn = qn−1∂ K n q = {qn}: C(K) → C(K,L) is a homomorphism of chain complexes. We note furthermore that for each n ≥ 0, the sequence of Abelian groups Cn(L) �� Cn(i) �� Cn(K) qn �� Cn(K)/Cn(L) is a short exact sequence and therefore, we have a short exact sequence of chain complexes C(L) � C(i) � �� C(K) q �� C(K,L); the result follows from Theorem (II.3.1). � The exact sequence of homology groups described in the statement of Theorem (II.4.1) is the exact homology sequence of the pair (K,L).
78 II Simplicial Complexes In the context of the categories Csim and CCsim, the naturality of the connecting homomorphism λn : Hn(K,L;Z) → Hn−1(L;Z) can be explained as follows. We start with a result whose proof is easily obtained from the given definitions and is left to the reader. (II.4.2) Theorem. Let (k,ℓ): (K,L) → (K ′ ,L ′ ) be a given simplicial function. Then, for every n ≥ 1, the following diagram commutes. Let be the functor defined by Hn(K,L;Z) Hn(k,ℓ) �� Hn(K ′ ,L ′ ;Z) λn �� λn Hn−1(L;Z) Hn−1(ℓ) �� Hn−1(L ′ ;Z) pr2 : CCsim → Csim (∀(K,L) ∈ CCsim) pr2(K,L)=L and (∀(k,ℓ) ∈ CCsim((K,L),(K ′ ,L ′ ))) pr2(k,ℓ)=ℓ. For each n ≥ 0, take the covariant functors Hn(−,−): CCsim → Gr and Hn−1(−) ◦ pr2 : CCsim → Gr . Theorem (II.3.3) states that λn : Hn(−,−;Z) → Hn−1(−;Z) ◦ pr2 is a natural transformation (see the definition of natural transformation of functors in Sect. I.2). Computing the homology of a complex K can be made easier by the exact homology sequence, provided that we can compute the homology of L and the relative homology of (K,L). Another very useful technique for computing the homology of a simplicial complex is using the Mayer–Vietoris sequence. Consider two simplicial complexes K1 =(X1,Φ1) and K2 =(X2,Φ2) such that K1 ∩ K2 and K1 ∪ K2 are simplicial complexes; in addition, K1 ∩K2 must be a subcomplex of both K1 and K2. The inclusions Φ1 ∩ Φ2 ↩→ Φα , Φα ↩→ Φ1 ∪ Φ2 , α = 1,2
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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.
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
Page 58: I.3 Group Actions 41 S2 = {(x,y,z)
Page 61 and 62: 44 II Simplicial Complexes (0, 1) (
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)
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Page 105 and 106: 88 II Simplicial Complexes Exercise
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Page 109 and 110: 92 II Simplicial Complexes Since im
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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 142 and 143: 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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