From calculus to cohomology: de Rham cohomology and characteristic classes

Abstract

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.