Differential forms in algebraic topology / Raoul Bott, Loring W. Tu

This book is an excellent presentation of algebraic topology via differential forms. The first chapter contains the de Rham theory, with stress on computability. Thus, the Mayer-Vietoris technique plays an important role in the exposition. The force of this technique is demonstrated by the fact that the authors at the end of this chapter arrive at a really comprehensive exposition of Poincaré duality, the Euler and Thom classes and the Thom isomorphism.
The second chapter develops and generalizes the Mayer-Vietoris technique to obtain in a very natural way the Čech-de Rham complex and the Čech cohomology for presheaves. The third chapter on spectral sequences is the most difficult one, but also the richest one by the various applications and digressions into other topics of algebraic topology: singular homology and cohomology with integer coefficients and an important part of homotopy theory, including the Hopf invariant, the Postnikov approximation, the Whitehead tower and Serre's theorem on the homotopy of spheres. The last chapter is devoted to a brief and comprehensive description of the Chern and Pontryagin classes.
A book which covers such an interesting and important subject deserves some remarks on the style: On the back cover one can read "With its stress on concreteness, motivation, and readability, Differential forms in algebraic topology should be suitable for self-study.'' This must not be misunderstood in the sense that it is always easy to read the book. The authors invite the reader to understand algebraic topology by completing himself proofs and examples in the exercises. The reader who seriously follows this invitation really learns a lot of algebraic topology and mathematics in general. (MathSciNet)

Bibliogr. p. 307-310. Index