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 cover Morse theory, index of vector fields, Poincaré duality, vector bundles, connections and curvature, Chern and Euler classes, Thom isomorphism, and the general Gauss-Bonnet theorem. The text includes 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 to anyone who wishes to know about cohomology, curvature, and their applications. 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 To Anyone Who Wishes To Know About Cohomology, Curvature, And Their Applications. Ch. 1. Introduction -- Ch. 2. The Alternating Algebra -- Ch. 3. De Rham Cohomology -- Ch. 4. Chain Complexes And Their Cohomology -- Ch. 5. The Mayer-vietoris Sequence -- Ch. 6. Homotopy -- Ch. 7. Applications Of De Rham Cohomology -- Ch. 8. Smooth Manifolds -- Ch. 9. Differential Forms On Smooth Manifolds -- Ch. 10. Integration On Manifolds -- Ch. 11. Degree, Linking Numbers And Index Of Vector Fields -- Ch. 12. The Poincare-hopf Theorem -- Ch. 13. Poincare Duality -- Ch. 14. The Complex Projective Space Cp[superscript N] -- Ch. 15. Fiber Bundles And Vector Bundles -- Ch. 16. Operations On Vector Bundles And Their Sections -- Ch. 17. Connections And Curvature -- Ch. 18. Characteristic Classes Of Complex Vector Bundles. Ib Madsen And Jørgen Tornehave. Includes Bibliographical References And Index. 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.