Differential Forms in Algebraic Topology

Developed from a first-year graduate course in algebraic topology, this text is an informal introduction to some of the main ideas of contemporary homotopy and cohomology theory. The materials are structured around four core areas: de Rham theory, the Cech-de Rham complex, spectral sequences, and characteristic classes. By using the de Rham theory of differential forms as a prototype of cohomology, the machineries of algebraic topology are made easier to assimilate. With its stress on concreteness, motivation, and readability, this book is equally suitable for self-study and as a one-semester course in topology.

Table of Contents

I de Rham Theory

1. The de Rham Complex on RAn

2. The Mayer-Vietoris Sequence

3. Orientation and Integration

4. Poincare Lemmas

5. The Mayer-Vietoris Argument

6. The Thom Isomorphism

7. The Nonorientable Case

II The Cech-de Rham Complex

8. The Generalized Mayer-Vietoris Principle

9. More Examples and Applications of the Mayer-Vietoris Principle

10. Presheaves and Cech Cohomology

11. Sphere Bundles

12. Thom Isomorphism and Poincare Duality Revisited

13. Monodromy

III Spectral Sequences and Applications

14. The Spectral Sequence of a Filtered Complex

15. Cohomology with Integer Coefficients

16. The Path Fibration

17. Review of Homotopy Theory

18. Applications to Homoto py Theory

19. Rational Homotopy Theory

IV Characterist ic Classes

20. Chern Classes of a Complex Vector Bundle

21. The Splitting Principle and Flag Manifolds

22. Pont rjagin Classes

23. The Search for the Universal Bundle