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Differential forms in algebraic topology 版权信息
- ISBN:9787506291903
- 条形码:9787506291903 ; 978-7-5062-9190-3
- 装帧:一般胶版纸
- 册数:暂无
- 重量:暂无
- 所属分类:>>
Differential forms in algebraic topology 内容简介
《代数拓扑中的微分形式》以微分形式为主要手段,简洁明快地介绍代数拓扑中的许多比较深刻的概念和定理。全书不拘泥于叙述格式,而是强调有关问题的具体背景,从而使读者开阔思路和加深对概念的理解。本书可供拓扑工作者参考,亦可作代数拓扑课研究生教材。 目次:de Rham理论;Cech-de Rham复形;谱序列和应用;示性类。
Differential forms in algebraic topology 目录
Introduction
CHAPTER Ⅰ De Rham Theory
§1 The de Rham Complex on Rn
The de Rham complex
Compact supports
§2 The Mayer-Vietoris Sequence
The functor Ω*
The Mayer-Vietoris sequence
The functor Ω*c and the Mayer-Vietoris sequence for compact supports
§3 Orientation and Integration
Orientation and the integral of a diferential form
Stokes'theorem
§4 Poincare Lemmas
The Poincare lemma for de Rham cohomology
The Poincare lemma for compactly supported 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 Kinneth 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 fiber
Poincare duality and the Thom class
The global angular form, the Euler clas, and the Thom clas
Relative de Rham theory
§7 The Nonorientable Case
The twisted de Rham complex
Integration of densities, Poincare 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 Cech
The tictac-toe proof of the Kinneth 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 Thom Isomorphism and Poincare Duality Revisited
The Thom isomorphism
Euler class and the zero locus of a section
A tic-tac-toe lemma
Poincare 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 fltered complex
The spectral sequence of a double complex
The spectral sequence of a fiber bundle
Some applications
Product 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 path fibration
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)
The transgression
Basic tricks of the trade
Postnikov approximation
Computation of π4(S3)
The Whitehead tower
Computation of π5(S3)
§19 Rational Homotopy Theory
Minimal models
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 bundles
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 Bundle
The Grassmannian
Digression on the Poincare series of a graded algebra
The classification of vector bundles
The infinite Grassmannian
Concluding remarks
References
List of Notations
Index
展开全部
Differential forms in algebraic topology 作者简介
Raoul Bott ,美国哈佛大学剑桥分校(Harvard University Cambridge)数学系教授。
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