代数拓扑

出版时间：1997-9  出版社：世界图书出版公司  作者：W.Fulton  页数：430  
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内容概要

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.

书籍目录

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

用户评论 (总计2条)

•   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。