From Calculus to Cohomology by Madsen

From Calculus to Cohomology: De Rham Cohomology and Characteristic Classes by Ib H. Madsen

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 Poincaré-Hopf theorem
• 13. Poincaré duality
• 14. The complex projective space $CP^n$
• 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.