代数拓扑
1997-9
世界图书出版公司
W.Fulton
430
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To the Teacher. This book is designed to introduce a student to some of the important ideas of algebraic topology by emphasizing the relations of these ideas with other areas of mathematics. Rather than choosing one point of view of modem topology (homotopy theory,simplicial complexes, singular theory, axiomatic homology, differential topology, etc.), we concentrate our attention on concrete problems in low dimensions, introducing only as much algebraic machinery as necessary for the problems we meet. This makes it possible to see a.wider variety of important features of the subject than is usual in a beginning text. The book is designed for students of mathematics or science who are not aiming to become practicing algebraic topologists--without, we hope, discouraging budding topologists. We also feel that this approach is in better harmony with the historical development of the subject.
PrefacePART I CALCULUS IN THE PLANE CHAPTER 1 Path Integrals 1a. Differential Forms and Path Integrals 1b. When Are Path Integrals Independent of Path 1c. A Criterion for Exactness CHAPTER 2 Angles and Deformations 2a. Angle Functions and Winding Numbers 2b. Reparametrizing and Deforming Paths 2e. Vector Fields and Fluid FlowPART II WINDING NUMBERS CHAPTER 3 The Winding Number 3a. Definition of the Winding Number 3b. Homotopy and Reparametrization 3c. Varying the Point 3d. Degrees and Local Degrees CHAPTER 4 Applications of Winding Numbers 4a. The Fundamental Theorem of Algebra 4b. Fixed Points and Retractions 4c. Antipodes 4d. SandwichesPART III COHOMOLOGY AND HOMOLOGY, I CHAPTER 5 De Rham Cohomology and the Jordan Curve Theorem 5a. Definitions of the De Rham Groups 5b. The Coboundary Map 5c. The Jordan Curve Theorem 5d. Applications and Variations CHAPTER 6 Homology 6a. Chains, Cycles, and HoU 6b. Boundaries, H1U, and Winding Numbers 6c. Chains on Grids 6d. Maps and Homology 6e. The First Homology Group for General SpacesPART IV VECTOR FIELDS CHAPTER 7 Indices of Vector Fields 7a. Vector Fields in the Plane 7b. Changing Coordinates 7c. Vector Fields on a Sphere CHAPTER 8 Vector Fields on Surfaces 8a. Vector Fields on a Torus and Other Surfaces 8b. The Euler CharacteristicPART V COHOMOLOGY AND HOMOLOGY, II CHAPTER 9 Holes and Integrals 9a. Multiply Connected Regions 9b. Integration over Continuous Paths and Chains 9c. Periods of Integrals 9d. Complex Integration CHAPTER 10 Mayer-Vietoris 10a. The Boundary Map 10b. Mayer-Vietoris for Homology 10c. Variations and Applications 10d. Mayer-Vietoris for CohomologyPART VI COVERING SPACES AND FUNDAMENTAL GROUPS, I CHAPTER 11 Coveting Spaces CHAPTER 12 The Fundamental GroupPART VII COVERING SPACES AND FUNDAMENTAL GROUPS, II CHAPTER 13 The Fundamental Group and Covering Spaces CHAPTER 14 The Van Kampen TheoremPART VIII COHOMOLOGY AND HOMOLOGY, III CHAPTER 15 CohomologyCHAPTER 16 VariationsPART IX TOPOLOGY OF SURFACES CHAPTER 17 The Topology of Surfaces CHAPTER 18 Cohomology on SurfacesPART X RIEMANN SURFACES CHAPTER 19 Riemann Surfaces CHAPTER 20 Riemann Surfaces and Algebraic Curves CHAPTER 21 The Riemann-Roch TheoremPART XI HIGHER DIMENSIONS CHAPTER 22 Toward Higher Dimensions CHAPTER 23 Higher Homology CHAPTER 24 DualityAPPENDICES APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX EIndex of symbolsIndex
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这是一本很好的代数拓扑方面的书籍个人以为他同其他的spanier&munkres或者是其他人写的代数拓扑是不一样的有点偏几何的感觉相信结合Armstrong的书来读会达到一个不错的效果是很好的一个选择希望大家买来看看还有hatcher&maunder他们写的书也是值得一看的至于那个dold的书我们老师说过这本书讲同调的部分堪称经典因为继承了汉堡学派的作风此书错误较少适合初学者……
这本Fulton的书跟同学讨论过前几章。展开的比较慢，内容比较具体。比较适合本科生阅读。但是里面的拓扑知识对于学习代数几何也是一个比较好的准备了，注意Fulton的很多工作都是在代数几何的。