From Calculus to Cohomology : De Rham Cohomology and Characteristic Classes
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, Poincare 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.
- Online resource
- 05 Sep 2015
- Cambridge University Press (Virtual Publishing)
- Cambridge, United Kingdom
'... a self-contained exposition.' L'Enseignement Mathematique 'This is a very fine book. It treats de Rham cohomology in an intellectually rigourous yet accessible manner which makes it ideal for a beginning graduate student. Moreover, it gets beyond the minimal agenda that many authors have set ... A welcome addition.' Mathematika ' ... a very polished completely self-contained introduction to the theory of differential forms ... the book is very well-written ... I recommend the book as an excellent first reading about curvature, cohomology and algebraic topology to anyone interested in these themes from students to active researchers, and especially to those who deliver lectures concerning the mentioned fields.' Acta. Sci. Math. '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
Table of contents
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 Poincare-Hopf theorem; 13. Poincare 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.