From calculus to cohomology : de Rham cohomology and characteristic classes = 从微积分到上同调 /
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作 者：Ib Madsen and J?rgen Tornehave著.
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ISBN：9787302075639
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简介
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 form 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 set 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 development in gauge theory and geometry in four dimensions ,but it also serves as an introductory course in algebraic topology. It will be invaluable to any one who wishes to know about cohomology, curvature, and their applications.
目录
preface
chapter 1 introduction
chapter 2 the alternating algebra
chapter 3 de rham cohomology
chapter 4 chain complexes and their cohomology
chapter 5 the mayer-vietoris sequence
chapter 6 homotopy
chapter 7 applications of de rham cohomology
chapter 8 smooth manifolds
chapter 9 differential forms on smoth manifolds
chapter 10 integration on manifolds
chapter 11 degree, linking numbers and index of vector fields
chapter 12 the poincar6-hopf theorem
chapter 13 poincare duality
chapter 14 the complex projective space cpn
chapter 15 fiber bundles and vector bundles
chapter 16 operations on vector bundles and their sections
chapter 17 connections and curvature
chapter 18 characteristic classes of complex vector bundles
chapter 19 the euler class
.chapter 20 cohomology of projective and grassmannian bundles
chapter 21 thom isomorphism and the general gauss-bonnet formula
appendix a smooth partition of unity
appendix b invariant polynomials
appendix c proof of lemmas 12.12 and 12.13
appendix d exercises
references
index
chapter 1 introduction
chapter 2 the alternating algebra
chapter 3 de rham cohomology
chapter 4 chain complexes and their cohomology
chapter 5 the mayer-vietoris sequence
chapter 6 homotopy
chapter 7 applications of de rham cohomology
chapter 8 smooth manifolds
chapter 9 differential forms on smoth manifolds
chapter 10 integration on manifolds
chapter 11 degree, linking numbers and index of vector fields
chapter 12 the poincar6-hopf theorem
chapter 13 poincare duality
chapter 14 the complex projective space cpn
chapter 15 fiber bundles and vector bundles
chapter 16 operations on vector bundles and their sections
chapter 17 connections and curvature
chapter 18 characteristic classes of complex vector bundles
chapter 19 the euler class
.chapter 20 cohomology of projective and grassmannian bundles
chapter 21 thom isomorphism and the general gauss-bonnet formula
appendix a smooth partition of unity
appendix b invariant polynomials
appendix c proof of lemmas 12.12 and 12.13
appendix d exercises
references
index
From calculus to cohomology : de Rham cohomology and characteristic classes = 从微积分到上同调 /
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