nLab Differential Forms in Algebraic Topology

This entry collects material related to the book

• Raoul Bott, Loring Tu:

  
  

  Differential Forms in Algebraic Topology

  
  

  Graduate Texts in Mathematics 82

  Springer (1982)

  doi:10.1007/978-1-4757-3951-0

on applications of differential forms, or rather of de Rham cohomology, in algebraic topology.

Related references:

• Victor Guillemin, Peter Haine, Differential Forms, World Scientific (2019) (doi:10.1142/11058)

1 The de Rham complex on $\mathbb{R}^n$

• differential forms

• wedge product

• exterior differential

• de Rham complex

2 The Mayer-Vietoris sequence

3 Orientation and integration

• volume form

• integration of differential forms

• Stokes theorem

4 Poincaré lemma

• Poincare lemma

• Gysin sequence

5 The Mayer-Vietoris argument

6 The Thom isomorphism

• vector bundles

• fiber integration, projection formula

• Thom isomorphism

7 The Nonorientable case

8 The Generalized Mayer-Vietoris Principle

9 More Examples and Applications of the Mayer-Vietoris Principle

10 Presheaves and Cech cohomology

• presheaves

• Cech cohomology

11 Sphere bundles

• spherical fibration

• Euler form

• Euler characteristic

• Poincaré–Hopf theorem

• Hopf degree theorem

13 Monodromy

• monodromy

14 The spectral sequence of a filtered complex

• spectral sequence

• spectral sequence of a filtered complex

17 Review of homotopy theory

• homotopy theory

• homotopy groups of spheres

• cell attachment

• Morse theory

• Hopf invariant

19 Rational homotopy theory

• rational homotopy theory

20 Chern Classes of a Complex Vector Bundle

• complex vector bundle

• Chern class

21 The Splitting Principle and Flag manifolds

• splitting principle

22 Pontrjagin classes

• Pontrjagin classes

23 The Search for the Universal Principal Bundle

• classifying space

• universal principal bundle

category: reference