代数拓扑的微分形式（英文版）

    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.

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.

　　本书为英文版。

Introduction
CHAPTERⅠ 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 isomorphism
CHAPTERⅡ 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 Applications
CHAPTERⅣ Characteristic Classes
References
List of Notations
Index