Simplicial Structures in Topology

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I.2 Categories 33 Conversely, given ˆf ∈ M∗(X,ΩY ), its adjoint f : I × X → Y is continuous because I is compact Hausdorff (see Lemma (I.1.39))and f ∈ M∗(ΣX,Y); this allows us to construct a function Φ ′ : M∗(X,ΩY) → M∗(ΣX,Y) such that ΦΦ ′ and Φ ′ Φ be equal to the respective identity functions. In other words, Φ : M∗(ΣX,Y ) → M∗(X,ΩY ) is a bijection (injective and surjective). For this reason, we say that Σ is left adjoint to Ω. Given two functors F,G: C → C ′ ,a natural transformation η : F → G is a correspondence that takes each object A ∈ C to a morphism and such that, for every f ∈ C(A,B), η(A): FA → GA G( f )η(A)=η(B)F( f ), in other words, such that the following diagram is commutative: F( f ) FA η(A) �� FB η(B) GA �� GB G( f ) Two functors F,G: C → C ′ are equivalent (and we write F . = G)iftherearetwo natural transformations η : F → G and τ : G → F such that τη = 1F and ητ = 1G where 1F and 1G equal the natural transformations given by the identity. I.2.2 Pushouts Given two morphisms f : A → B and g: A → C of a category C,apushout of ( f ,g) is a pair of morphisms f ∈ C(C,D) and g ∈ C(B,D) such that gf = g f and satisfying the following
34 I Fundamental Concepts universal property: given h ∈ C(B,X) and k ∈ C(C,X) such that hf = kg, there exists a unique morphism ℓ ∈ C(D,X) such that ℓ f = k and ℓg = h. We depict this situation with the commutative diagram A �� C g f ¯f �� B ¯g �� h D k The symbol at the right lower angle of the commutative square indicates that we have a pushout diagram. As a consequence of the universal property of pushouts, a pushout of ( f ,g) is unique up to isomorphism. In fact, suppose that ( f ′ ,g ′ ) with f ′ : C → D ′ and g ′ : B → D ′ is a pushout; then by the universal property, there is a unique ℓ ′ : D ′ → D such that ℓ ′ g ′ = g and ℓ ′ f ′ = f ; hence, ℓ ′ ℓ = 1D. Similarly, we conclude that ℓℓ ′ = 1C. The morphism ℓ is an isomorphism and its inverse is ℓ ′ . We say that a category C is closed by pushouts or closed regarding pushouts if every pair of morphisms f : A → B and g: A → C of C has a pushout. Not every category is closed by pushouts; we now prove that the category Top of topological spaces and the category Gr of groups are closed by pushouts. Let us start with Top.Wedefinethedisjoint union of two topological spaces B and C by taking a set of two points, say, {i, j} and constructing the spaces B ×{i} and C ×{j}, homeomorphic to B and C, respectively. We then define the union B ⊔C = B ×{i}∪C ×{j} with the inclusions ℓ �� X ιB : B → B ⊔C , b ↦→ (b,i) , ιC : C → B ⊔C , c ↦→ (c, j) and give B ⊔C the topology defined by the open sets ιB(U) ∪ ιC(V )=(U ×{i}) ∪ (V ×{j}) with U open in B and V open in C; in this way, we obtain the topological space called disjoint union of B and C. By constructing a quotient of the disjoint union, it is possible to prove that Top is closed regarding pushouts. (I.2.2) Theorem. The category Top is closed by pushouts. Proof. Let f : A → B and g: A → C be any two maps in Top. In the disjoint union B ⊔C, identify f (a) with g(a) for every a ∈ A; take the quotient set B ⊔ f ,g C obtained by the identifications f (a) =g(a) and then the canonic surjection
Page 2: Canadian Mathematical Society Soci
Page 5 and 6: Dr. Davide L. Ferrario Università
Page 8: Foreword to the English Edition Exc
Page 11 and 12: x Preface same qualitative properti
Page 13 and 14: xii Preface homology groups, also w
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
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 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)
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 71 and 72: 54 II Simplicial Complexes In a sim
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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
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
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84 II Simplicial Complexes (that is
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86 II Simplicial Complexes Therefor
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88 II Simplicial Complexes Exercise
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90 II Simplicial Complexes Hn(C;G)=
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92 II Simplicial Complexes Since im
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94 II Simplicial Complexes and cons
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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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