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Topology Differential Forms in Algebraic Topology by Bott and Tu

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  1. Feb 1, 2013 #1

    micromass

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    Table of Contents:
    Code (Text):

    [LIST]
    [*] Introduction
    [*] De Rham Theory
    [LIST]
    [*] The de Rham Complex on R^n
    [LIST]
    [*] The de Rham complex
    [*] Compact supports
    [/LIST]
    [*] The Mayer-Vietoris Sequence
    [LIST]
    [*] The functor \Omega^*
    [*] The Mayer-Vietoris sequence
    [*] The functor \Omega_c^* and the Mayer-Vietoris sequence for compact supports
    [/LIST]
    [*] Orientation and Integration
    [LIST]
    [*] Orientation and the integral of a differential form
    [*] Stokes' theorem
    [/LIST]
    [*] Poincare Lemma
    [LIST]
    [*] The Poincare lemma for the de Rham cohomology
    [*] The Poincare lemma for compactly supported cohomology
    [*] The degree of a proper map
    [/LIST]
    [*] The Mayer-Vietoris Argument
    [LIST]
    [*] Existence of a good cover
    [*] Finite dimensionality of de Rham cohomology
    [*] Poncare duality on an orientable manifold
    [*] The Kunneth formula and the Leray-Hirsch theorem
    [*] The Poincare dual of a closed oriented submanifold
    [/LIST]
    [*] The Thom Isomorphism
    [LIST]
    [*] Vector bundles and the reduction of structure groups
    [*] Operations on vector bundles
    [*] Compact cohomology of a vector bundle
    [*] Compact vertical cohomology and integration along the fiber
    [*] Poincare duality and the Thom class
    [*] The global singular form, the Euler class, and the Thom class
    [*] Relative de Rham theory
    [/LIST]
    [*] The Nonorientable Case
    [LIST]
    [*] The twisted de Rham complex
    [*] Integration of densities, Poincare duality, and the Thom isomorphism
    [/LIST]
    [/LIST]
    [*] The Cech-de Rham Complex
    [LIST]
    [*] The Generalized Mayer-Vietoris Principle
    [LIST]
    [*] Reformulation of the Mayer-Vietoris Sequence
    [*] Generalization to countably many open sets and applications
    [/LIST]
    [*] More Examples and Applications of the Mayer-Vietoris Principle
    [LIST]
    [*] Examples: computing the de Rham cohomology from the combinatorics of a good cover
    [*] Explicit isomorphisms between the double complex and de Rham and Cech
    [*] The tic-tac-toe proof of the Kunneth formula
    [/LIST]
    [*] Presheaves and Cech Cohomology
    [LIST]
    [*] Presheaves
    [*] Cech cohomology
    [/LIST]
    [*] Sphere Bundles
    [LIST]
    [*] Orientability
    [*] The Euler class of an oriented sphere bundle
    [*] The global angular form
    [*] Euler number and the isolated singularities of a section
    [*] Euler characteristic and the Hopf index theorem
    [/LIST]
    [*] The Thom Isomorphism and Poincare Duality Revisited
    [LIST]
    [*] The Thom isomorphism
    [*] Euler class and the zero locus of a section
    [*] A tic-tac-toe lemma
    [*] Poincare duality
    [/LIST]
    [/LIST]
    [*] Spectral Sequence and Applications
    [LIST]
    [*] The Spectral Sequence of a Filtered Complex
    [LIST]
    [*] Exact couples
    [*] The spectral sequence of a filtered complex
    [*] The spectral sequence of a double complex
    [*] The spectral sequence of a fiber bundle
    [*] Some applications
    [*] Product structures
    [*] The Gysin sequence
    [*] Leray's construction
    [/LIST]
    [*] Cohomology with Integer Coefficients
    [LIST]
    [*] Singular homology
    [*] The cone construction
    [*] The Mayer-Vietoris sequence for singular chains
    [*] Singular cohomology
    [*] The homology spectral sequence
    [/LIST]
    [*] The Path Fibration
    [LIST]
    [*] The path fibration
    [*] The cohomology of the loop space of a sphere
    [/LIST]
    [*] Review of Homotopy Theory
    [LIST]
    [*] Homotopy groups
    [*] The relative homotopy sequence
    [*] Some homotopy groups of the spheres
    [*] Attaching cells
    [*] Digression on Morse theory
    [*] The relation between homotopy and homology
    [*] \pi_3(S^2) and the Hopf invariant
    [/LIST]
    [*] Applications to Homotopy Theory
    [LIST]
    [*] Eilenberg-MacLane spaces
    [*] The telescoping construction
    [*] The cohomology of K(Z,3)
    [*] The transgression
    [*] Basic tricks of the trade
    [*] Postnikov approximation
    [*] Computation of \pi_4(S^3)
    [*] The Whitehead tower
    [*] Computation of \pi_5(S^3)
    [/LIST]
    [*] Rational Homotopy Theory
    [LIST]
    [*] Minimal models
    [*] Examples of Minimal Models
    [*] The main theorem and applications
    [/LIST]
    [/LIST]
    [*] Characteristic Classes
    [LIST]
    [*] Chern Classes of a Complex Vector Bundle
    [LIST]
    [*] The first Chern class of a complex line bundle
    [*] The projectivization of a vector bundle
    [*] Main properties of the Chern classes
    [/LIST]
    [*] The SPlitting Principle and Flag Manifolds
    [LIST]
    [*] The splitting principle
    [*] Proof of the Whitney product formula and the equality of the top Chern class and the Euler class
    [*] Computation of some Chern classes
    [*] Flag manifolds
    [/LIST]
    [*] Pontrjagin Classes
    [LIST]
    [*] Conjugate bundles
    [*] Realization and complexification
    [*] The Pontrjagin classes of a real vector bundle
    [*] Application to the embedding of a manifold in a Euclidean space
    [/LIST]
    [*] The Search for the Universal Bundle
    [LIST]
    [*] The Grassmannian
    [*] Digression on the Poincare series of a graded algebra
    [*] The classification of vector bundles
    [*] The infinite Grassmannian
    [*] Concluding remarks
    [/LIST]
    [/LIST]
    [*] References
    [*] List of Notations
    [*] Index
    [/LIST]
     
     
    Last edited by a moderator: May 6, 2017
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  3. Feb 2, 2013 #2

    mathwonk

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    a superb book, authoritative, deep and crystal clear.
     
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