From Calculus to Cohomology: De Rham Cohomology and Characteristic ClassesCambridge University Press, 13 mars 1997 - 286 sidor 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. |
Innehåll
Chapter | 15 |
Chapter 4 | 25 |
Chapter 5 | 33 |
Chapter 6 | 39 |
Applications of de Rham Cohomology | 47 |
Chapter 8 | 57 |
Chapter 9 | 65 |
Chapter 10 | 85 |
Chapter 14 | 142 |
Chapter 16 | 160 |
Chapter 17 | 174 |
Chapter 18 | 181 |
Chapter 19 | 197 |
Chapter 21 | 211 |
Appendix | 221 |
Appendix | 227 |
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From Calculus to Cohomology: De Rham Cohomology and Characteristic Classes Ib H. Madsen,Jxrgen Tornehave Ingen förhandsgranskning - 1997 |
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chain complexes chart Chern classes choose Cn+1 cohomology class commutative diagram complex vector bundle connection Construct continuous map coordinate Corollary critical points define Definition deg(f denote diffeomorphism differential disjoint exact sequence Example Exercise exists finite follows formula functor Gauss map given Hence homeomorphic homotopy HP(U induces inner product integral isomorphism Lemma Let f line bundle linear map map f matrix metric Morse function n-dimensional open neighborhood open set open subsets orientation form orthogonal p-form partition of unity polynomial positively oriented Proposition Prove R-linear real vector bundle Rham cohomology Riemannian Rn+1 Rn+k satisfies Show smooth function smooth manifold smooth map smooth submanifold smooth vector bundle subspace tangent bundle topological total space trivial U₁ uniquely V₁ vector field vector space w₁ zero Ω¹