Simplicial Structures in Topology

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II.5 Homology with Coefficients 97 0 A �� B �� C �� f g h ′ A ′ B ′ C Prove that there exists a homomorphism d that turns the sequence ker f �� kerg �� kerh d �� coker f �� cokerg �� 0 �� cokerh into an exact sequence; also, by using this Lemma, give an alternative proof to Theorem (II.3.1). 4. (General form of the Five Lemma)Let A α �� A ′ f f ′ �� B β �� ′ B g g ′ �� C γ �� ′ C h �� h ′ D δ �� ′ D k �� be a commutative diagram of Abelian groups with exact rows. Prove that: • If α is surjective and β , δ are injective, then γ is injective; • If ε is injective and β, δ are surjective, then γ is surjective. It follows directly from these results that, if α, β, δ, andεare isomorphisms, then also γ is an isomorphism. k ′ E ε �� ′ E
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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) (
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
Page 69 and 70: 52 II Simplicial Complexes r = tp+(
Page 71 and 72: 54 II Simplicial Complexes In a sim
Page 73 and 74: 56 II Simplicial Complexes (not nec
Page 75 and 76: 58 II Simplicial Complexes Let K3,3
Page 77 and 78: 60 II Simplicial Complexes ordering
Page 79 and 80: 62 II Simplicial Complexes are two
Page 81 and 82: 64 II Simplicial Complexes 1-simple
Page 83 and 84: 66 II Simplicial Complexes An infin
Page 85 and 86: 68 II Simplicial Complexes 2. λn i
Page 87 and 88: 70 II Simplicial Complexes Please n
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
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 117 and 118: 100 III Homology of Polyhedra (the
Page 119 and 120: 102 III Homology of Polyhedra where
Page 121 and 122: 104 III Homology of Polyhedra such
Page 123 and 124: 106 III Homology of Polyhedra |L|×
Page 125 and 126: 108 III Homology of Polyhedra We no
Page 127 and 128: 110 III Homology of Polyhedra Let K
Page 129 and 130: 112 III Homology of Polyhedra the g
Page 131 and 132: 114 III Homology of Polyhedra that
Page 133 and 134: 116 III Homology of Polyhedra With
Page 135 and 136: 118 III Homology of Polyhedra linea
Page 137 and 138: 120 III Homology of Polyhedra (III.
Page 139 and 140: 122 III Homology of Polyhedra (III.
Page 141 and 142: 124 III Homology of Polyhedra 1-sim
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Page 145 and 146: 128 III Homology of Polyhedra III.5
Page 147 and 148: 130 III Homology of Polyhedra such
Page 149 and 150: 132 III Homology of Polyhedra all,
Page 151 and 152: 134 III Homology of Polyhedra For t
Page 153 and 154: 136 III Homology of Polyhedra since
Page 159 and 160: 142 III Homology of Polyhedra Note
Page 161 and 162: 144 III Homology of Polyhedra We ma
Page 163 and 164: 146 III Homology of Polyhedra and,
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148 III Homology of Polyhedra We no
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Chapter IV Cohomology IV.1 Cohomolo
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IV.1 Cohomology with Coefficients i
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IV.2 The Cohomology Ring 161 is suc
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IV.2 The Cohomology Ring 163 (for Z
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IV.2 The Cohomology Ring 165 and th
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IV.3 The Cap Product 167 Proof. Let
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IV.3 The Cap Product 169 Therefore
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172 V Triangulable Manifolds (V.1.2
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174 V Triangulable Manifolds other
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176 V Triangulable Manifolds are li
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178 V Triangulable Manifolds and ta
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180 V Triangulable Manifolds We vie
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182 V Triangulable Manifolds may be
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184 V Triangulable Manifolds Fig. V
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186 V Triangulable Manifolds we tak
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188 V Triangulable Manifolds a tree
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190 V Triangulable Manifolds The de
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192 V Triangulable Manifolds and th
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Chapter VI Homotopy Groups VI.1 Fun
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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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