This time of writing is the hundredth anniversary of the publication （1892） of Poincare's first note on topology， which arguably marks the beginning of the subject of algebraic， or "combinatorial，" topology. There was earlier scattered work by Euler， Listing （who coined the word "topology"）， M/Sbius and his band， Riemann， Klein， and Betti. Indeed， even as early as 1679， Leibniz indicated the desirability of creating a geometry of the topological type. The establishment of topology （or "analysis situs" as it was often called at the time） as a coherent theory， however， belongs to Poincar6.Curiously， the beginning of general topology， also called "point settopology，" dates fourteen years later when Fr6chet published the first abstracttreatment of the subject in 1906. 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.
作者簡(jiǎn)介
暫缺《拓?fù)渑c幾何（英文版）》作者簡(jiǎn)介
圖書目錄
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 …… CHAPTER V Cohomology CHAPTER VI Products and Duality CHAPTER VII Homotopy theory Appendices Bibliography Index of Symbols Index
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