Simplicial Structures in Topology

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II.5 Homology with Coefficients 89 Cn(K) being the free Abelian groups of formal linear combinations with coefficients in Z of the n-simplexes of K. We now wish to generalize our homology with coefficients in the Abelian group Z to a homology with coefficients drawn from any Abelian group G. We begin by reviewing the construction of the tensor product of two Abelian groups A, B: by definition, A ⊗ B is the Abelian group generated by the set of elements {a ⊗ b|a ∈ A , b ∈ B} where (∀a,a ′ ∈ A , b,b ′ ∈ B) 1. (a + a ′ ) ⊗ b = a ⊗ b + a ′ ⊗ b , 2. a ⊗ (b + b ′ )=a ⊗ b + a ⊗ b ′ . We notice that the function A ⊗ Z → A , a ⊗ n ↦→ na is a group isomorphism, that is to say, A⊗Z ∼ = A (similarly, Z⊗A ∼ = A). The reader may easily prove that (A ⊕ B) ⊗C ∼ = (A ⊗C) ⊕ (B ⊗C) for any three Abelian groups A, B,andC. Finally, given two group homomorphisms φ : A → A ′ and ψ : B → B ′ , the function φ ⊗ ψ : A ⊗ B → A ′ ⊗ B ′ defined by φ ⊗ ψ(a ⊗ b)=φ(a) ⊗ ψ(b) is a homomorphism of Abelian groups. In this way, by fixing an Abelian group G we are able to construct a covariant functor −⊗G: Ab → Ab that transforms a group A into A ⊗ G and a morphism φ : A → B into the morphism φ ⊗ 1G. We extend this functor to chain complexes. We transform a given chain complex (C,∂) ∈ C in (C ⊗ G,∂ ⊗ 1G), by setting (C ⊗ G)n := Cn ⊗ G for every n ∈ Z, and by defining the homomorphisms Since (∂ ⊗ 1G)n := ∂n ⊗ 1G : Cn ⊗ G → Cn−1 ⊗ G . (∂ ⊗ 1G)n−1(∂ ⊗ 1G)n =(∂n−1 ⊗ 1G)(∂n ⊗ 1G)=∂n−1∂n ⊗ 1G = 0 , we conclude that (C ⊗ G,∂ ⊗ 1G) is a chain complex whose homology groups are the homology groups of (C,∂ ) with coefficients in G. Thenth-homology group of (C,∂) with coefficients in G is defined by the quotient group
90 II Simplicial Complexes Hn(C;G)=ker(∂n ⊗ 1G)/im(∂n+1 ⊗ 1G); the graded Abelian group H∗(C;G) is the graded homology group of C with coefficients in G. In particular, if (C,∂ )=(C(K),∂), the chain complex of the oriented complex K, thenH∗(C(K);G) – simply denoted by H∗(K;G) – is the homology of K with coefficients in G. We recall that the chain complex (C(K),∂) is positive, free, and has an augmentation homomorphism ε : C0(K) → Z. To continue with our work, we only need one of these properties, namely, that (C,∂ ) be free. For every free chain complex (C,∂) and for each n ∈ Z, we have a short exact sequence of free Abelian groups Zn(C) �� Cn ∂n �� Bn−1(C) . The main point is that, by taking the tensor product of each component of this exact sequence with G, we obtain again a short exact sequence. (II.5.1) Lemma. If A �� f �� B is a short exact sequence of free Abelian groups and G is an Abelian group, then also the sequence is exact. g �� C A ⊗ G �� f ⊗ 1G g ⊗ 1G �� B ⊗ G �� C ⊗ G Proof. We begin by noting that the group C is free; therefore, we may define a map s: C → B simply by choosing for each element of a basis of C an element of its antiimage under g, and by extending this operation linearly; through this procedure, we obtain a homomorphism of Abelian groups s: C −→ B such that gs = 1C, the identity homomorphism of C onto itself. It follows that g(1B − sg)=g −(gs)g = 0, in other words, the image of 1B − sg is contained in the kerg = im f ; for this reason, we may define the map r := f −1 (1B − sg): B → A that also satisfies rf = f −1 (1B − sg) f = 1A. We thus obtain the relations rf = 1A , gs = 1C , and fr+ sg = 1B. We know that the tensor product by G is a functor and that it transforms sums of homomorphisms into sums of transformed homomorphisms; consequently, tensorization gives us the relations
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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
Page 56 and 57: I.3 Group Actions 39 I.3 Group Acti
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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
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Page 75 and 76: 58 II Simplicial Complexes Let K3,3
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Page 99 and 100: 82 II Simplicial Complexes obtained
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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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Page 146 and 147: III.5 Real Projective Spaces 129 We
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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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