《从微积分到上同调》PDF下载

  • 购买积分:11 如何计算积分?
  • 作  者:(丹)马森(Madsen,I.H.),(丹)托内哈弗(Tornehave,J.)著
  • 出 版 社:北京:清华大学出版社
  • 出版年份:2003
  • ISBN:7302075638
  • 页数:286 页
图书介绍:De Rham 上同调是微分形式的上同调。本书在不要求读者具备代数拓扑或上同调的知识的前提下,以较低的起点系统地讨论了De Rham上同调问题,并从曲率的观点讨论了示性类理论。书中的前10章研究Euclidean空间中开集的上同调,论述光滑流形及其上同调,最后以流形上的积分告一段落。后11章内容涵盖了Morse理论、向量场的指数、Poincare对偶、向量丛、连通与曲率、陈省身示性类和Euler示性类、Thom同构,最后以广义的Gauss-Bonnet定理结束。全书包含150个习题,并给出在规范理论和4维几何中必要的引申背景说明,本书也可作为代数拓扑的初级教程。本书对于任何想了解上同调、曲率及它们应用的读者都是非常有益的。

Chapter 1 Introduction 1

Chapter 2 The Alternating Algebra 7

Chapter 3 de Rham Cohomology 15

Chapter 4 Chain Complexes and their Cohomology 25

Chapter 5 The Mayer-Vietoris Sequence 33

Chapter 6 Homotopy 39

Chapter 7 Applications of de Rham Cohomology 47

Chapter 8 Smooth Manifolds 57

Chapter 9 Differential Forms on Smoth Manifolds 65

Chapter 10 Integration on Manifolds 83

Chapter 11 Degree,Linking Numbers and Index of Vector Fields 97

Chapter 12 The Poincaré-Hopf Theorem 113

Chapter 13 PoincaréDuality 127

Chapter 14 The Complex Projective Space CPn 139

Chapter 15 Fiber Bundles and Vector Bundles 147

Chapter 16 Operations on Vector Bundles and their Sections 157

Chapter 17 Connections and Curvature 167

Chapter 18 Characteristic Classes of Complex Vector Bundles 181

Chapter 19 The Euler Class 193

Chapter 20 Cohomology of Projective and Grassmannian Bundles 199

Chapter 21 Thom Isomorphism and the General Gauss-Bonnet Formula 211

Appendix A Smooth Partition of Unity 221

Appendix B Invariant Polynomials 227

Appendix C Proof of Lemmas 12.12 and 12.13 233

Appendix D Exercises 243

References 281

Index 283