拓扑流形引论PDF电子书下载

1 Introduction 1

What Are Manifolds？ 1

Why Study Manifolds？ 4

2 Topological Spaces 17

Topologies 17

Bases 27

Manifolds 30

Problems 36

3 New Spaces from Old 39

Subspaces 39

Product Spaces 48

Quotient Spaces 52

Group Actions 58

Problems 62

4 Connectedness and Compactness 65

Connectedness 65

Compactness 73

Locally Compact Hausdorff Spaces 81

Problems 88

5 Simplicial Complexes 91

Euclidean Simplicial Complexes 92

Abstract Simplicial Complexes 96

Triangulation Theorems 102

Orientations 105

Combinatorial Invariants 109

Problems 114

6 Curves and Surfaces 117

Classification of Curves 118

Surfaces 119

Connected Sums 126

Polygonal Presentations of Surfaces 129

Classification of Surface Presentations 137

Combinatorial Invariants 142

Problems 146

7 Homotopy and the Fundamental Group 147

Homotopy 148

The Fundamental Group 150

Homomorphisms Induced by Continuous Maps 158

Homotopy Equivalence 161

Higher Homotopy Groups 169

Categories and Functors 170

Problems 176

8 Circles and Spheres 179

The Fundamental Group of the Circle 180

Proofs of the Lifting Lemmas 183

Fundamental Groups of Spheres 187

Fundamental Groups of Product Spaces 188

Fundamental Groups of Manifolds 189

Problems 191

9 Some Group Theory 193

Free Products 193

Free Groups 199

Presentations of Groups 201

Free Abelian Groups 203

Problems 208

10 The Seifert-Van Kampen Theorem 209

Statement of the Theorem 210

Applications 212

Proof of the Theorem 221

Distinguishing Manifolds 227

Problems 230

11 Covering Spaces 233

Definitions and Basic Properties 234

Covering Maps and the Fundamental Group 239

The Covering Group 247

Problems 253

12 Classification of Coverings 257

Covering Homomorphisms 258

The Universal Covering Space 261

Proper Group Actions 266

The Classification Theorem 283

Problems 289

13 Homology 291

Singular Homology Groups 292

Homotopy Invariance 300

Homology and the Fundamental Group 304

The Mayer-Vietoris Theorem 308

Applications 318

The Homology of a Simplicial Complex 323

Cohomology 329

Problems 334

Appendix：Review of Prerequisites 337

Set Theory 337

Metric Spaces 347

Group Theory 352

References 359

Index 362