拓?fù)渑c幾何

內(nèi)容概要

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.

作者簡介

作者：(美國)布里登

書籍目錄

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

編輯推薦

本書是一部比較原始但又不失趣味性的拓?fù)渑c幾何課本，完全是從現(xiàn)代觀點(diǎn)研究問題，可以說是25年以來，繼Spanier之后真正的一本全新的拓?fù)鋾?。很適合作為一年級研究生的代數(shù)拓?fù)浣炭茣?。?nèi)容安排緊湊、合理，從一般拓?fù)溟_始，講述了微分流形，上同調(diào)，乘積和對偶，基礎(chǔ)群，同調(diào)理論和同倫理論。包括了面理論，群理論，和纖維叢理論這些大多數(shù)拓?fù)鋵W(xué)家想讓學(xué)拓?fù)涞膶W(xué)生了解的知識點(diǎn)。并且有很多內(nèi)容很具有啟發(fā)性，這些內(nèi)容并不是所有傳統(tǒng)的課本中都包含的。通過這本書的閱讀也可以提高數(shù)學(xué)學(xué)習(xí)能力。盡管這本書具有很強(qiáng)的綜合性，但并沒有過分去去囊括多余的綜合材料，而是這些材料真正地提高了表述的效率和清晰度。

圖書封面

圖書標(biāo)簽Tags

無

評論、評分、閱讀與下載

---

    拓?fù)渑c幾何 PDF格式下載

---