From Calculus to Cohomology
De Rham Cohomology and Characteristic Classes

An introductory textbook on cohomology and curvature with emphasis on applications.

Ib H. Madsen (Author), Jxrgen Tornehave (Author)

9780521589567, Cambridge University Press

Paperback, published 13 March 1997

296 pages
24.7 x 17.4 x 1.9 cm, 0.615 kg

'The book is written in a precise and clear language, it combines well topics from differential geometry, differential topology and global analysis.' European Mathematical Society

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 ten chapters study cohomology of open sets in Euclidean space, treat smooth manifolds and their cohomology and end with integration on manifolds. The last eleven chapters include Morse theory, index of vector fields, Poincaré duality, vector bundles, connections and curvature, and the book ends with the general Gauss-Bonnet theorem. The text includes well over 150 exercises, and gives the background to 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 to anyone studying cohomology, curvature, and their applications.

1. Introduction
2. The alternating algebra
3. De Rham cohomology
4. Chain complexes and their cohomology
5. The Mayer-Vietoris sequence
6. Homotopy
7. Applications of De Rham cohomology
8. Smooth manifolds
9. Differential forms on smooth manifolds
10. Integration on manifolds
11. Degree, linking numbers and index of vector fields
12. The Poincaré-Hopf theorem
13. Poincaré duality
14. The complex projective space CPn
15. Fiber bundles and vector bundles
16. Operations on vector bundles and their sections
17. Connections and curvature
18. Characteristic classes of complex vector bundles
19. The Euler class
20. Cohomology of projective and Grassmanian bundles
21. Thom isomorphism and the general Gauss-Bonnet formula.

Subject Areas: Differential & Riemannian geometry [PBMP]