拓扑与几何
2008-1
北京世图
布里登
557
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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 abstract treatment 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：(美国)布里登
PrefaceAcknowledgmentsCHAPTER 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 TheoremCHAPTER 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 CohomologyCHAPTER VI Products and DualityCHAPTER VII Homotopy theoryAppendicesBibliographyIndex of SymbolsIndex
本书是一部比较原始但又不失趣味性的拓扑与几何课本，完全是从现代观点研究问题，可以说是25年以来，继Spanier之后真正的一本全新的拓扑书。很适合作为一年级研究生的代数拓扑教科书。内容安排紧凑、合理，从一般拓扑开始，讲述了微分流形，上同调，乘积和对偶，基础群，同调理论和同伦理论。包括了面理论，群理论，和纤维丛理论这些大多数拓扑学家想让学拓扑的学生了解的知识点。并且有很多内容很具有启发性，这些内容并不是所有传统的课本中都包含的。通过这本书的阅读也可以提高数学学习能力。尽管这本书具有很强的综合性，但并没有过分去去囊括多余的综合材料，而是这些材料真正地提高了表述的效率和清晰度。
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刚买来，还没看，应该不错
这是我见过的本科以下最好的数学英语教材。
内容广泛但是写的感觉有点乱
以为是新书，封面有一折痕，背面有圆珠笔画了一杠。2008年一月印刷。像换货了，结果只剩一本
在图书馆随便翻过。然后买了。还没看。罪过。