Topological methods in Euclidean spaces

Naber, G.L. Topological Methods in Euclidean Spaces(CUP,1980)(ISBN 0521227461)(600dpi)(240p) ......Page 3
Copyright ......Page 4
Contents vii ......Page 6
Preface ix ......Page 8
1 Introduction 1 ......Page 10
2 Preliminaries 6 ......Page 15
3 Open sets, closed sets, and continuity 10 ......Page 19
4 Compact spaces 21 ......Page 30
5 Connectivity properties 25 ......Page 34
6 Real-valued continuous functions 34 ......Page 43
7 Retracts 38 ......Page 47
8 Topological dimension 41 ......Page 50
Supplementary exercises 44 ......Page 53
2 Hyperplanes in Rn 46 ......Page 55
3 Simplexes and complexes 49 ......Page 58
4 Sample triangulations 55 ......Page 64
5 Simplicial maps 60 ......Page 69
6 Barycentric subdivision 63 ......Page 72
7 The Simplicial Approximation Theorem 70 ......Page 79
8 Sperner’s Lemma 73 ......Page 82
9 The Brouwer Fixed Point Theorem 75 ......Page 84
10 Topological dimension of compact subsets of Rn 77 ......Page 86
Supplementary exercises 79 ......Page 88
1 Introduction 81 ......Page 90
2 The homotopy relation, nullhomotopic maps, and contractible spaces 84 ......Page 93
3 Maps of spheres 86 ......Page 95
4 The fundamental group 90 ......Page 99
5 Fundamental groups of the spheres 99 ......Page 108
Supplementary exercises 106 ......Page 115
1 Introduction 108 ......Page 117
2 Oriented complexes and chains 111 ......Page 120
3 Boundary operators 116 ......Page 125
4 Cycles, boundaries, and homology groups 118 ......Page 127
5 Elementary examples 122 ......Page 131
6 Cone complexes, augmented complexes, and the homology groups Hp(K(sn+1)) 125 ......Page 134
7 Incidence numbers and the homology groups Hp(Kn(sn+1)) 128 ......Page 137
8 Elementary homological algebra 130 ......Page 139
9 The homology complex of a geometric complex 133 ......Page 142
10 Acyclic carrier functions 136 ......Page 145
11 Invariance of homology groups under barycentric subdivision 138 ......Page 147
12 Homomorphisms induced by continuous maps 141 ......Page 150
13 Homology groups of topological polyhedra 144 ......Page 153
14 The Hopf Trace Theorem 147 ......Page 156
15 The Lefschetz Fixed Point Theorem 150 ......Page 159
Supplementary exercises 151 ......Page 160
1 Introduction 154 ......Page 163
2 Smooth maps 161 ......Page 170
3 The Stone-Weierstrass Theorem 163 ......Page 172
4 Derivatives as linear transformations 166 ......Page 175
5 Differentiable manifolds 173 ......Page 182
6 Tangent spaces and derivatives 175 ......Page 184
7 Regular and critical values of smooth maps 179 ......Page 188
8 Measure zero and Sard’s Theorem 183 ......Page 192
9 Morse functions 190 ......Page 199
10 Manifolds with boundary 201 ......Page 210
11 One-dimensional manifolds 205 ......Page 214
12 Topological characterization of Sk 208 ......Page 217
13 Smooth tangent vector fields 211 ......Page 220
Supplementary exercises 217 ......Page 226
Guide to further study 220 ......Page 229
Bibliography 222 ......Page 231
List of symbols and notation 224 ......Page 233
Index 228 ......Page 237
cover......Page 1