代數拓撲中微分形式（簡體書）

《代數拓撲中微分形式》內容為：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.

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