The theory of characteristic classes provides a meeting ground for the various disciplines of differential topology, differential and algebraic geometry, cohomology, and fiber bundle theory. As such, it is a fundamental and an essential tool in the study of differentiable manifolds.

In this volume, the authors provide a thorough introduction to characteristic classes, with detailed studies of Stiefel-Whitney classes, Chern classes, Pontrjagin classes, and the Euler class. Three appendices cover the basics of cohomology theory and the differential forms approach to characteristic classes, and provide an account of Bernoulli numbers.

Based on lecture notes of John Milnor, which first appeared at Princeton University in 1957 and have been widely studied by graduate students of topology ever since, this published version has been completely revised and corrected.

Table of Contents

§1. Smooth Manifo lds

§2. Vector Bund les

§3. Constructing New Vector Bund les Out of Old

§4. Stiefe l-Whitney Classes

§5. Grassmann Manifo lds a nd Universal Bund les

§6. A Cell Structure for Grassmann Manifolds

§7. The Cohomology Ring H*(Gn; Z/ 2 )

§8. Existence of Stiefe l-Whitney Classes

§9. Oriented Bund les a nd the Euler Class

§10. The Thom Isomorphism Theorem

§11. Computatio ns in a Smooth Manifold

§12. Obstructions

§13. Complex Vector Bund les a nd Complex Manifolds

§14. Chern Classes

§15. Pontrjagin Classes

§16. Chern Numbers a nd Pontrjagin Numbers

§17. The OrientedCobordism Ring 0 *

§18. Thom Spaces and Transversality

§19. Multiplicative Sequences a nd the Sig nature Theorem

&20. Combinatorial Pontriaain Classes