Differential Forms in Algebraic Topology by Bott and Tu

[micromass]

• Author: Raoul Bott, Loring Tu
• Title: Differential Forms in Algebraic Topology
• Amazon Link: https://www.amazon.com/dp/1441928154/?tag=pfamazon01-20
• Prerequisities: Differential Geometry, Algebraic Topology
• Level: Grad

Table of Contents:

[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]

 

[mathwonk]

a superb book, authoritative, deep and crystal clear.