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Chapter 0.Some Underlying Geometric Notions 1

Homotopy and Homotopy Type 1

Cell Complexes 5

Operations on Spaces 8

Two Criteria for Homotopy Equivalence 10

The Homotopy Extension Property 14

Chapter 1.The Fundamental Group 21

1.1.Basic Constructions 25

Paths and Homotopy 25

The Fundamental Group of the Circle 29

Induced Homomorphisms 34

1.2.Van Kampen's Theorem 40

Free Products of Groups 41

The van Kampen Theorem 43

Applications to Cell Complexes 50

1.3.Covering Spaces 56

Lifting Properties 60

The Classification of Covering Spaces 63

Deck Transformations and Group Actions 70

Additional Topics 83

1.A.Graphs and Free Groups 83

1.B.K（G,1）Spaces and Graphs of Groups 87

Chapter 2.Homology 97

2.1.Simplicial and singular Homology 102

△-Complexes 102

Simplicial Homology 104

Singular Homology 108

Homotopy Invariance 110

Exact Sequences and Excision 113

The Equivalence of Simplicial and Singular Homology 128

2.2.Computations and Applications 134

Degree 134

Cellular Homology 137

Mayer-Vietoris Sequences 149

Homology with Coefficients 153

2.3.The Formal Viewpoint 160

Axioms for Homology 160

Categories and Functors 162

Additional Topics 166

2.A.Homology and Fundamental Group 166

2.B.Classical Applications 169

2.C.Simplicial Approximation 177

Chapter 3.Cohomology 185

3.1.Cohomology Groups 190

The Universal Coefficient Theorem 190

Cohomology of Spaces 197

3.2.Cup Product 206

The Cohomology Ring 211

A Künneth Formula 218

Spaces with Polynomial Cohomology 224

3.3.Poincaré Duality 230

Orientations and Homology 233

The Duality Theorem 239

Connection with Cup Product 249

Other Forms of Duality 252

Additional Topics 261

3.A.Universal Coefficients for Homology 261

3.B.The General Künneth　Formula 268

3.C.H-Spaces and Hopf Algebras 281

3.D.The Cohomology of SO（n） 292

3.E.Bockstein Homomorphisms 303

3.F.Limits and Ext 311

3.G.Transfer Homomorphisms 321

3.H. Local Coefficients 327

Chapter 4.Homotopy Theory 337

4.1.Homotopy Groups 339

Definitions and Basic Constructions 340

Whitehead,s Theorem 346

Cellular Approximation 348

CW Approximation 352

4.2.Elementary Methods of Calculation 360

Excision for Homotopy Groups 360

The Hurewicz Theorem 366

Fiber Bundles 375

Stable Homotopy Groups 384

4.3.Connections with Cohomology 393

The Homotopy Construction of Cohomology 393

Fibrations 405

Postnikov Towers 410

Obstruction Theory 415

Additional Topics 421

4.A.Basepoints and Homotopy 421

4.B.The Hopf Invariant 427

4.C.Minimal Cell Structures 429

4.D.Cohomology of Fiber Bundles 431

4.E.The Brown Representability Theorem 448

4.F.Spectra and Homology Theories 452

4.G.Gluing Constructions 456

4.H.Eckmann-Hilton Duality 460

4.I.Stable Splittings of Spaces 466

4.J.The Loopspace of a Suspension 470

4.K.The Dold-Thom Theorem 475

4.L.Steenrod Squares and Powers 487

Appendix 519

Topology of Cell Complexes 519

The Compact-Open Topology 529

Bibliography 533

Index 539