| 词条 | 代数拓扑中微分形式 |
| 释义 | 图书信息出版社: 世界图书出版公司; 第1版 (2009年3月1日) 外文书名: Differenitial Forms In Algebraic Topology 平装: 331页 正文语种: 英语 开本: 32 ISBN: 9787506291903 条形码: 9787506291903 尺寸: 22.2 x 14.8 x 2 cm 重量: 440 g 作者简介作者：(美国)Raoul Bott (美国)Loring W.Tu 内容简介《代数拓扑中微分形式》内容为：The guiding principle in this book is to use differential forms as an aid inexploring some of the less digestible aspects of algebraic topology. Accord-ingly, we move primarily in the realm of smooth manifolds and use thede Rham theory as a prototype of all of cohomology. For applications tohomotopy theory we also discuss by way of analogy cohomoiogy witharbitrary coefficients. Although we have in mind an audience with prior exposure to algebraicor differential topology, for the most part a good knowledge of linearalgebra, advanced calculus, and point-set topology should suffice. Someacquaintance with manifolds, simplicial complexes, singular homology andcohomology, and homotopy groups is helpful, but not really necessary.Within the text itself we have stated with care the more advanced resultsthat are needed, so that a mathematically mature reader who accepts thesebackground materials on faith should be able to read the entire book withthe minimal prerequisites. 目录Introduction CHAPTER Ⅰ De Rham Theory §1 The de Rham Complex on R The de Rham complex Compact supports §2 The Mayer-Vietoris Sequence ThefunctorQ 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 Poincar6 Lemmas The Poincare lemmafordeRham~ohomoiogy The Poincare lemma for compactly supported cohomology The degree of a propermap §5 The Mayer-Vietoris Argument Existence of a good cover Finite dimensionality of de Rham cohomology Poincar6 duality on an orientable manifold The Kiinneth formula and the Leray-Hirsch theorem The Poincar6 dual of a closed oriented submanifold §6 The Thorn 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 fiber Poincar6 duality and the Thorn class The global angular form，the Euler class，and the Thorn class Relative de Rham theory §7 The Nonorientable Case The twisted de Rham COD rplex Integration of densities，Poincard duality，and the Thom isomorphism CHAPTER Ⅱ 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 each The tic—tac-toe proof of the Kfinneth formula §10 Presheaves and Cech Cohomology Presheaves Cech cohomology §11 Sphere Bundles Orientability The Euler class of an oriented sphere bundle The global angular form Euler number and the isolated singularities of a section Euler characteristic and the Hopf index theorem §12 The Thorn Isomorphism and Poincar6 Duality Revisited The Thorn isomorphism Euler class and the zcr0 locus of a section A tic—tac-toe lemma Poincar6 duality §13 Monodromy When is a locally constant presheaf constant? Examples of monodromy CHAPTER Ⅲ Spectral Sequences and Applications §14 The Spectral Sequence of a Filtered Complex Exact Couples The spectral sequence of a filtered complex The spectral sequence of a double complex The spectral sequence of a fiber bundle Some applications PfodUct structures The Gysin sequence Leray’S construction §15 Cohomology with Integer Coefficients Singular homology The cone construction The Mayer-Vietoris sequence for singular chains Singular cohomology The homology spectral sequence §16 The Path Fibration The pathfibration The cohomology of the loop space of a sphere §17 Review of Homotopy Theory Homotopy groups The relative homotopy sequence Some homotopy groups of the spheres Attaching cells Digression on Morse theory The relation between homotopy and homology π3(S2)and the Hopf invariant §18 Applications to Homotopy Theory Eilenberg-MacLane spaces The telescoping construction The cohomology of K(Z，3) Thetransgression Basic tricks of the trade Postnikov approximation Computation ofπ4(S3) The Whitehead tower Computation of π5(S3) §19 Rational Homotopy Theory Minimal modds Examples of Minimal Models The main theorem and applications CHAPTER Ⅳ Characteristic Classes §20 Chern Classes of a Complex Vector Bundle The first Chern class of a complex line bundle The projectivization of a vector bundle Main properties of the Chern classes §21 The Splitting Principle and Flag Manifolds The splitting principle Proof of the Whitney product formula and the equality of the top Chern class and the Euler class Computation of some Chern classes Flag manifolds §22 Pontrjagin Classes Conjugate bundl Realization and complexification The Pontrjagin classes of a real vector bundle Application to the embedding of a manifold in a Euclidean space §23 The Search for the Universal Bund The Grassmannian Digression on the Poincar6 series of a graded algebra The classification of vector bundles The infinite Grassmannian Concluding remarks References List of Notations Index |
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