Characteristic Classes/John W.
Milnor and James D. Stasheff. New Delhi, Hindustan Book Agency, 2005, ix,
331 p., (pbk). ISBN 81-85931-52-6. [Texts and Readings in Mathematics, 32]
Characteristic Classes/John W.
Milnor and James D. Stasheff. New Delhi, Hindustan Book Agency, 2005, ix,
331 p., (pbk). ISBN 81-85931-52-6. [Texts and Readings in Mathematics, 32]
Contents: Preface. 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. 18. Thom spaces and transversality. 19. Multiplicative sequences and the signature theorem. 20. Combinatorial Pontrjagin classes. Epilogue. Appendices: 1. Singular homology and cohomology. 2. Bernoulli numbers. 3. Connections, curvature and characteristic classes. Bibliography. Index.
"This book is widely recognised as a classic. It remains the best introduction to the algebraic topology of vector bundles, more specifically to the "Characteristic Classes" associated with the names of Euler, Stiefel and Whitney, Chern, and Pontryagin. The book also contains a lovely treatment of Thom's cobordism theory, and the appendices give brisk accounts of singular homology and cohomology, as well as the Chern-Weil Theory."