Simplicial Structures in Topology

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III.6 Homology of the Product of Two Polyhedra 149 and so, for every n ≥ 1, we obtain the short exact sequence cokerλn+1 �� Hn(K × L;Z) �� kerλn. As for changing the group of coefficients in homology (Sect. II.5), we prove that cokerλn+1 ∼ = ∑ (Hi(K;Z) ⊗ Hj(L;Z)) i+ j=n kerλn ∼ = ∑ i+ j=n−1 Tor(Hi(K;Z),Hj(L;Z)) and so we have the next result, known as the Künneth Theorem: (III.6.4) Theorem. For every pair of polyhedra |K| and |L| and every n ≥ 1, the following short sequence of Abelian groups is exact. Exercises ∑ (Hi(K;Z) ⊗ Hj(L;Z)) �� Hn(K × L;Z) i+ j=n �� ∑ i+ j=n−1 Tor(Hi(K;Z),Hj(L;Z)) 1. Prove that, for every C,C ′ ∈ Clp, the tensor product C ⊗C ′ belongs to Clp. 2. Let f ∈ C(C,D) and g ∈ C(C ′ ,D ′ ) be two morphisms of given chain complexes; prove that f ⊗ g: C ⊗C ′ → D ⊗ D ′ defined on the generators by ( f ⊗ g)(x ⊗ y)= f (x) ⊗ g(y) is a morphism of chain complexes. 3. Let C,C ′ ∈ C be two chain complexes. Prove that defined on generators by the formula is an isomorphism of chain complexes. μ : C ⊗C ′ → C ′ ⊗C μ(x ⊗ y)=(−1) |x||y| y ⊗ x
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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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74 II Simplicial Complexes (II.3.10
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76 II Simplicial Complexes If we do
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78 II Simplicial Complexes In the c
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80 II Simplicial Complexes Φi = {
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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
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
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
Page 152 and 153: III.5 Real Projective Spaces 135 Le
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
Page 168 and 169: Chapter IV Cohomology IV.1 Cohomolo
Page 170 and 171: IV.1 Cohomology with Coefficients i
Page 178 and 179: IV.2 The Cohomology Ring 161 is suc
Page 180 and 181: IV.2 The Cohomology Ring 163 (for Z
Page 182 and 183: IV.2 The Cohomology Ring 165 and th
Page 184 and 185: IV.3 The Cap Product 167 Proof. Let
Page 186: IV.3 The Cap Product 169 Therefore
Page 189 and 190: 172 V Triangulable Manifolds (V.1.2
Page 191 and 192: 174 V Triangulable Manifolds other
Page 193 and 194: 176 V Triangulable Manifolds are li
Page 195 and 196: 178 V Triangulable Manifolds and ta
Page 197 and 198: 180 V Triangulable Manifolds We vie
Page 199 and 200: 182 V Triangulable Manifolds may be
Page 201 and 202: 184 V Triangulable Manifolds Fig. V
Page 203 and 204: 186 V Triangulable Manifolds we tak
Page 205 and 206: 188 V Triangulable Manifolds a tree
Page 207 and 208: 190 V Triangulable Manifolds The de
Page 209 and 210: 192 V Triangulable Manifolds and th
Page 212 and 213: Chapter VI Homotopy Groups VI.1 Fun
Page 214 and 215: VI.1 Fundamental Group 197 The func
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VI.1 Fundamental Group 199 Proof. L
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VI.1 Fundamental Group 201 We now d
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VI.1 Fundamental Group 203 are redu
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VI.1 Fundamental Group 205 We take
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VI.1 Fundamental Group 207 c being
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VI.1 Fundamental Group 209 We now c
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VI.2 Fundamental Group and Homology
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VI.3 Homotopy Groups 213 We now set
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VI.3 Homotopy Groups 215 for every
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VI.3 Homotopy Groups 217 such that
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VI.3 Homotopy Groups 219 is a bijec
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VI.3 Homotopy Groups 221 VI.3.1 Act
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VI.3 Homotopy Groups 223 We now con
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VI.3 Homotopy Groups 225 The follow
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VI.3 Homotopy Groups 227 Since the
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VI.3 Homotopy Groups 229 (VI.3.22)
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VI.3 Homotopy Groups 231 Z ��
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VI.4 Obstruction Theory 233 c n+1 f
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VI.4 Obstruction Theory 235 3. (VI.
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VI.4 Obstruction Theory 237 Proof.
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240 References 24. C.R.F. Maunder -
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242 Index free presentation, 93 fun
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Simplicial Structures in Topology

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