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