Since the beginning of time, or at least the era of A'rchimedes, smooth　manifolds (curves, surfaces, mechanical configurations, the universe) have　been a central focus in mathematics. They have always been at the core of　interest in topology. After the seminal work of Milnor, Smale, and many　others, in the last half of this century, the topological aspects of smooth　manifolds, as distinct from the differential geometric aspects, became a subject in its own right. While the major portion of this book is devoted to algebraic topology, I attempt to give the reader some glimpses into the beautiful and important realm of smooth manifolds along the way, and to instill the tenet that the algebraic tools are primarily intended for the understanding of the geometric world.

Preface

Acknowledgments

CHAPTER Ⅰ General Topology

1. Metric Spaces

2. Topological Spaces

3. Subspaces

4. Connectivity and Components

5. Separation Axioms

6. Nets (Moore-Smith Convergence)

7. Compactness

8. Products

9. Metric Spaces Again

10. Existence of Real Valued Functions

11. Locally Compact Spaces

12. Paracompact Spaces

13. Quotient Spaces

14. Homotopy

15. Topological Groups

16. Convex Bodies

17. The Baire Category Theorem

CHAPTER Ⅱ Differentiable Manifolds

1. The Implicit Function Theorem

2. Differentiable Manifolds

3. Local Coordinates

4. Induced Structures and Examples

5. Tangent Vectors and Differentials

6. Sard's Theorem and Regular Values

7. Local Properties of Immersions and Submersions

8. Vector Fields and Flows

9. Tangent Bundles

10. Embedding in Euclidean Space

11. Tubular Neighborhoods and Approximations

12. Classical Lie Groups

13. Fiber Bundles

14. Induced Bundles and Whitney Sums

15. Transversality

16. Thom-Pontryagin Theory

CHAPTER Ⅲ  Fundamental Group

1. Homotopy Groups

2. The Fundamental Group

3. Covering Spaces

4. The Lifting Theorem

5. The Action of nl on the Fiber

6. Deck Transformations

7. Properly Discontinuous Actions

8. Classification of Covering Spaces

9. The Seifert-Van Kampen Theorem

10. Remarks on SO(3)

CHAPTER Ⅳ  Homology Theory

1. Homology Groups

2. The Zeroth Homology Group

3. The First Homology Group

4. Functorial Properties

5. Homological Algebra

6. Axioms for Homology

7. Computation of Degrees

8. CW-Complexes

9. Conventions for CW-Complexes

10. Cellular Homology

11. Cellular Maps

12. Products of CW-Complexes

13. Euler's Formula

14. Homology of Real Projective Space

15. Singular Homology

16. The Cross Product

17. Subdivision

18. The Mayer-Vietoris Sequence

19. The Generalized Jordan Curve Theorem

20. The Borsuk-Ulam Theorem

21. Simplicial Complexes

22. Simplicial Maps

23. The Lefschetz-Hopf Fixed Point Theorem

CHAPTER Ⅴ Cohomology

1. Multilinear Algebra

2. Differential Forms

3. Integration of Forms

4. Stokes' Theorem

5. Relationship to Singular Homology

6. More Homologicat Algebra

7. Universal Coefficient Theorems

8. Excision and Homotopy

9. de Rham's Theorem

10. The de Rham Theory of CPn

11. Hopf's Theorem on Maps to Spheres

12. Differential Forms on Compact Lie Groups

CHAPTER Ⅵ Products and Duality

1. The Cross Product and the Kfinneth Theorem

2. A Sign Convention

3. The Cohomology Cross Product

4. The Cup Product

5. The Cap Product

6. Classical Outlook on Duality

7. The Orientation Bundle

8. Duality Theorems

9. Duality on Compact Manifolds with Boundary

10. Applications of Duality

11. Intersection Theory

12. The Euler Class, Lefschetz Numbers, and Vector Fields

13. The Gysin Sequence

14. Lefschetz Coincidence Theory

15. Steenrod Operations

16. Construction of the Steenrod Squares

17. Stiefel-Whitney Classes

18. Plumbing

CHAPTER Ⅶ IHomotopy Theory

1. Cofibrations

2. The Compact-Open Topology

3. H-Spaces, H-Groups, and H-Cogroups

4. Homotopy Groups

5. The Homotopy Sequence of a Pair

6. Fiber Spaces

7. Free Homotopy

8. Classical Groups and Associated Manifolds

9. The Homotopy Addition Theorem

10. The Hurewicz Theorem

11. The Whitehead Theorem

12. Eilenberg-Mac Lane Spaces

13. Obstruction Theory

14. Obstruction Cochains and Vector Bundles

Appendices

App. A. The Additivity Axiom

App. B. Background in Set Theory

App. C. Critical Values

App. D. Direct Limits

App. E. Euclidean Neighborhood Retracts

Bibliography

Index of Symbols

Index