代数拓扑的微分形式

出版时间：1999-11  出版社：世界图书出版公司  作者：R.BottBottLoringW.Tu  页数：331  
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内容概要

The guiding principle in this book is to use differential forms as an aid in exploring some of the less digestible aspects of algebraic topology. Accordingly, we move primarily in the realm of smooth manifolds and use the de Rham theory as a prototype of all of cohomology. For applications to homotopy theory we also discuss by way of analogy cohomology with arbitrary coefficients. Although we have in mind an audience with prior exposure to algebraic or differential topology, for the most part a good knowledge of linear algebra, advanced calculus, and point-set topology should suffice. Some acquaintance with manifolds, simplicial complexes, singular homology and cohomology, and homotopy groups is helpful, but not really necessary. Within the text itself we have stated with care the more advanced results that are needed, so that a mathematically mature reader who accepts these background materials on faith should be able to read the entire book with the minimal prerequisites.

书籍目录

IntroductionCHAPTERⅠ  De Rham Theory  1 The de Rham Complex on W    The de Rham complex    Compact supports  2 The Mayer-Vietoris Sequence    The functor    The Mayer-Vietoris sequence    The functor and the Mayer-Vietoris sequence for compact supports  3 Orientation and Integration    Orientation and the integral of a differential form    Stokes' theorem  4 Poincare Lemmas    The Poincare lemma for de Rham cohomology    The Poincare'lemma for compactly supponed cohomology    The degree of a proper map  5 The Mayer-Vietoris Argument    Existence of a good cover    Finite dimensionality of de Rham cohomology    Poincare duality on an orientable manifold    The Kiinneth formula and the Leray-Hirsch theorem    The Poincare dual of a closed oriented submanifold  6 The Thom Isomorphism    Vector bundles and the reduction of structure groups    Operations on vector bundles    Compact cohomology of a vector bundle    Compact vertical cohomology and integration along the fibe    Poincare duality and the Thom class    The global angular form, the Euler class and the Thom class    Relative de Rham theory  7 The Nonorientable Case    The twisted de Rham complex    Integration of densities, Poincare duality and the Thom isomorphismCHAPTERⅡ  The Cech-de Rham Complex  8 The Generalized Mayer-Vietoris Principle    Reformulation of the Mayer-Vietoris sequence    Generalization to countably many open sets and applications  9 More Examples and Applications of the Mayer-Vietoris Principle    Examples: computing the de Rham cohomology from the combinatorics of a good cover    Explicit isomorphisms between the double complex and de Rham and Cech    The tic-tac-toe proof of the Kiinneth formula  10 Presheaves and Cech Cohomology    Presheaves    Cech cohomology  11 Sphere Bundles    Orientability    The Euler class of an oriented sphere bundle    The global angular fonn    Euler number and the isolated singularities of a section    Euler characteristic and the Hopf index theorem  12 The Thom Isomorphism and Poincare Duality Revisited  ……CHAPTERⅢ  Spectral Sequences and ApplicationsCHAPTERⅣ  Characteristic ClassesReferencesList of NotationsIndex

用户评论 (总计3条)

•   入门级教材中的经典。内容丰富但是清晰。必读。
•   内容有难度
•   首先从作者来看L.Bott作为世界著名的拓扑学家其数学能力是毋庸置疑的，读大家的书是能让人感到高屋建瓴的气势。而从写作本身来看，这本书显然做到了讲述清晰易懂非常适合阅读和作为教材。而从内容上看绝对是代数拓扑的必读之作，相比其他代数拓扑而言这本书是比较偏几何的。另外当年陈省身先生讲拓扑学时也曾用这本书，用刘克峰老师的话说读懂这本书就可以开始读文章了。虽然作者尽量对预备知识不做高要求仅要微积分和点集拓扑，但个人觉得学过微分流形和知道一些代数拓扑的基础知识再学会更好。