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