John Milnor & James D. Stasheff

• Frontmatter,
• Preface,
• Contents,
• §1. Smooth Manifolds,
• §2. Vector Bundles,
• §3. Constructing New Vector Bundles Out of Old,
• §4. Stiefel-Whitney Classes,
• §5. Grassmann Manifolds and Universal Bundles,
• §6. A Cell Structure for Grassmann Manifolds,
• §7. The Cohomology Ring H*(Gn; Z/2),
• §8. Existence of Stiefel-Whitney Classes,
• §9. Oriented Bundles and the Euler Class,
• §10. The Thom Isomorphism Theorem,
• §11. Computations in a Smooth Manifold,
• §12. Obstructions,
• §13. Complex Vector Bundles and Complex Manifolds,
• §14. Chern Classes,
• §15. Pontrjagin Classes,
• §16. Chern Numbers and Pontrjagin Numbers,
• §17. The Oriented Cobordism Ring O*,
• §18. Thom Spaces and Transversality,
• §19. Multiplicative Sequences and the Signature Theorem,
• §20. Combinatorial Pontrjagin Classes,
• Epilogue,
• Appendix A: Singular Homology and Cohomology,
• Appendix B: Bernoulli Numbers,
• Appendix C: Connections, Curvature, and Characteristic Classes,
• Bibliography,
• Index,

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.