From Calculus to Cohomology
Subtitle: De Rham Cohomology and Characteristic Classes
Edition: Reprint
Publisher: Cambridge University Press
Published: 1999
Place of publication: Cambridge
Language: eng
Format: Paperback
Number of pages: 286
ISBN-13: 9780521589567
Copyright date: 1997
Description: De Rham cohomology is the cohomology of differential forms. This book offers a self-contained exposition to this subject and to the theory of characteristic classes from the curvature point of view. It requires no prior knowledge of the concepts of algebraic topology or cohomology.The first 10 chapters study cohomology of open sets in Euclidean space, treat smooth manifolds and their cohomology and end with integration on manifolds. The last 11 chapters cover Morse theory, index of vector fields, Poincare duality, vector bundles, connections and curvature, Chern and Euler classes, and Thom isomorphism, and the book ends with the general Gauss-Bonnet theorem.

The text includes well over 150 exercises, and gives the background necessary for the modern developments in gauge theory and geometry in four dimensions, but it also serves as an introductory course in algebraic topology. It will be invaluable anyone who wishes to know about cohomology, curvature, and their applications.
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Table of contents: title: 1. Introduction; title: 2. The alternating algebra; title: 3. De Rham cohomology; title: 4. Chain complexes and their cohomology; title: 5. The Mayer-Vietoris sequence; title: 6. Homotopy; title: 7. Applications of De Rham cohomology; title: 8. Smooth manifolds; title: 9. Differential forms on smooth manifolds; title: 10. Integration on manifolds; title: 11. Degree, linking numbers and index of vector fields; title: 12. The Poincaré-Hopf theorem; title: 13. Poincaré duality; title: 14. The complex projective space CPn; title: 15. Fiber bundles and vector bundles; title: 16. Operations on vector bundles and their sections; title: 17. Connections and curvature; title: 18. Characteristic classes of complex vector bundles; title: 19. The Euler class; title: 20. Cohomology of projective and Grassmanian bundles; title: 21. Thom isomorphism and the general Gauss-Bonnet formula.