Simplicial Structures in Topology

More documents

Recommendations

Info

IV.3 The Cap Product 169 Therefore d ∩Cp+q( f )(c)=d({ f (x0,..., f (xp)}){ f (xp),..., f (xp+q)} . On the other hand, C p ( f )=dCp( f ) and so, Cq( f )(C p ( f )(d) ∩ c)=d({ f (x0,..., f (xp)}){ f (xp),..., f (xp+q)} . If there are i, j for which f (xi)= f (x j),thenCp+q( f )(c)=0 and at least one of the two assertions C p ( f )(d)=0 , Cq( f )({xp,...,xp+q})=0 is true; therefore, the theorem holds also in this case. � Theorem (IV.3.5) and what we have previously done enable us to state the following result: (IV.3.6) Theorem. Let |L|,|K|∈P and a continuous function f : |K|→|L| be given. The following diagram: H p (|L|;Z) × Hp+q(|K|;Z) H p ( f ;Z) × 1 �� 1 × Hp+q( f ;Z) is commutative. �� H p (|L|;Z) × Hp+q(|L|;Z) ∩ H p (|K|;Z) × Hp+q(|K|;Z) ∩ �� Hq(|K|;Z) Hq( f ;Z) �� Hq(|L|;Z) Proof. We leave it to the reader. � Exercises 1. Let f : |K|→|L| be a given map, z ∈ Hn(|K|,Z), andu ∈ H q (|L|,Z). Provethat 2. Prove that Hn−q( f )(u ∩ Hn( f )(z)) = H q ( f )(u) ∩ z . (∀z ∈ Hn(|K|,Z)) 1 ∩ z = z .
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 50 and 51:
I.2 Categories 33 Conversely, given
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
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
Page 113 and 114:
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 120 and 121:
Page 122 and 123:
Page 124 and 125:
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 166: III.6 Homology of the Product of Tw
Page 169 and 170: 152 IV Cohomology (IV.1.1) Theorem.
Page 171 and 172: 154 IV Cohomology (IV.1.2) Remark.
Page 173 and 174: 156 IV Cohomology If, starting from
Page 175 and 176: 158 IV Cohomology When we apply thi
Page 177 and 178: 160 IV Cohomology groups Q and R ar
Page 179 and 180: 162 IV Cohomology (IV.2.2) Corollar
Page 181 and 182: 164 IV Cohomology Since g and π r
Page 183 and 184: 166 IV Cohomology It is easily prov
Page 185: 168 IV Cohomology ∩: B p (K;Z) ×
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
Page 216 and 217: VI.1 Fundamental Group 199 Proof. L
Page 218 and 219: VI.1 Fundamental Group 201 We now d
Page 220 and 221: VI.1 Fundamental Group 203 are redu
Page 222 and 223: VI.1 Fundamental Group 205 We take
Page 224 and 225: VI.1 Fundamental Group 207 c being
Page 226 and 227: VI.1 Fundamental Group 209 We now c
Page 228 and 229: VI.2 Fundamental Group and Homology
Page 230 and 231: VI.3 Homotopy Groups 213 We now set
Page 232 and 233: VI.3 Homotopy Groups 215 for every
Page 234 and 235: VI.3 Homotopy Groups 217 such that
Page 236 and 237:
VI.3 Homotopy Groups 219 is a bijec
Page 238 and 239:
VI.3 Homotopy Groups 221 VI.3.1 Act
Page 240 and 241:
VI.3 Homotopy Groups 223 We now con
Page 242 and 243:
VI.3 Homotopy Groups 225 The follow
Page 244 and 245:
VI.3 Homotopy Groups 227 Since the
Page 246 and 247:
VI.3 Homotopy Groups 229 (VI.3.22)
Page 248 and 249:
VI.3 Homotopy Groups 231 Z ��
Page 250 and 251:
VI.4 Obstruction Theory 233 c n+1 f
Page 252 and 253:
VI.4 Obstruction Theory 235 3. (VI.
Page 254:
VI.4 Obstruction Theory 237 Proof.
Page 257 and 258:
240 References 24. C.R.F. Maunder -
Page 259 and 260:
242 Index free presentation, 93 fun
show all

Simplicial Structures in Topology

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?