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Main Question or Discussion Point
- Author: Raoul Bott, Loring Tu
- Title: Differential Forms in Algebraic Topology
- Amazon Link: https://www.amazon.com/dp/1441928154/?tag=pfamazon01-20&tag=pfamazon01-20
- Prerequisities: Differential Geometry, Algebraic Topology
- Level: Grad
Table of Contents:
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[*] Introduction
[*] De Rham Theory
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[*] The de Rham Complex on R^n
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[*] The de Rham complex
[*] Compact supports
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[*] The Mayer-Vietoris Sequence
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[*] The functor \Omega^*
[*] The Mayer-Vietoris sequence
[*] The functor \Omega_c^* and the Mayer-Vietoris sequence for compact supports
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[*] Orientation and Integration
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[*] Orientation and the integral of a differential form
[*] Stokes' theorem
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[*] Poincare Lemma
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[*] The Poincare lemma for the de Rham cohomology
[*] The Poincare lemma for compactly supported cohomology
[*] The degree of a proper map
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[*] The Mayer-Vietoris Argument
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[*] 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
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[*] The Thom Isomorphism
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[*] 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
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[*] The Nonorientable Case
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[*] The twisted de Rham complex
[*] Integration of densities, Poincare duality, and the Thom isomorphism
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[*] The Cech-de Rham Complex
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[*] The Generalized Mayer-Vietoris Principle
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[*] Reformulation of the Mayer-Vietoris Sequence
[*] Generalization to countably many open sets and applications
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[*] More Examples and Applications of the Mayer-Vietoris Principle
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[*] 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
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[*] Presheaves and Cech Cohomology
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[*] Presheaves
[*] Cech cohomology
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[*] Sphere Bundles
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[*] 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
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[*] The Thom Isomorphism and Poincare Duality Revisited
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[*] The Thom isomorphism
[*] Euler class and the zero locus of a section
[*] A tic-tac-toe lemma
[*] Poincare duality
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[*] Spectral Sequence and Applications
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[*] The Spectral Sequence of a Filtered Complex
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[*] 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
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[*] Cohomology with Integer Coefficients
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[*] Singular homology
[*] The cone construction
[*] The Mayer-Vietoris sequence for singular chains
[*] Singular cohomology
[*] The homology spectral sequence
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[*] The Path Fibration
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[*] The path fibration
[*] The cohomology of the loop space of a sphere
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[*] Review of Homotopy Theory
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[*] 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
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[*] Applications to Homotopy Theory
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[*] 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)
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[*] Rational Homotopy Theory
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[*] Minimal models
[*] Examples of Minimal Models
[*] The main theorem and applications
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[*] Characteristic Classes
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[*] Chern Classes of a Complex Vector Bundle
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[*] The first Chern class of a complex line bundle
[*] The projectivization of a vector bundle
[*] Main properties of the Chern classes
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[*] The SPlitting Principle and Flag Manifolds
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[*] 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
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[*] Pontrjagin Classes
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[*] Conjugate bundles
[*] Realization and complexification
[*] The Pontrjagin classes of a real vector bundle
[*] Application to the embedding of a manifold in a Euclidean space
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[*] The Search for the Universal Bundle
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[*] The Grassmannian
[*] Digression on the Poincare series of a graded algebra
[*] The classification of vector bundles
[*] The infinite Grassmannian
[*] Concluding remarks
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[*] References
[*] List of Notations
[*] Index
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