代数拓扑的微分形式
1999-11
世界图书出版公司
R.BottBottLoringW.Tu
331
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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.
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
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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。