de Rham cohomology


Let X be a paracompact π’žβˆž differential manifold. Let

Ω⁒X=βŠ•i=0∞Ωi⁒X

denote the graded-commutative ℝ-algebraPlanetmathPlanetmath of differential formsMathworldPlanetmath on X. Together with the exterior derivative

di:Ξ©iXβ†’Ξ©i+1X (i=0,1,…),

Ω⁒X forms a chain complexMathworldPlanetmath (Ω⁒X,d) of ℝ-vector spaces. The HdRi⁒X of X are defined as the homology groups of this complex, that is to say

HdRiX:=(kerdi)/(imdi-1) (i=0,1,…),

where Ξ©-1⁒X is taken to be 0, so d-1:0β†’Ξ©0⁒X is the zero map. The wedge product in Ω⁒X induces the structure of a graded-commutative ℝ-algebra on

HdR⁒X:=βŠ•i=0∞HdRi⁒X.

If X and Y are both paracompact π’žβˆž manifolds and f:Xβ†’Y is a differentiable map, there is an induced map

f*:HdR⁒Yβ†’HdR⁒X,

defined by

f*⁒[Ο‰]:=[f*⁒ω] forΒ Ο‰βˆˆker⁑d.

Here [Ο‰] denotes the class of Ο‰ modulo imd, and the second f* is the map Ω⁒Y→Ω⁒X induced by the functorMathworldPlanetmath Ξ©. This action on differentiable maps makes the de Rham cohomologyMathworldPlanetmath into a contravariant functor from the categoryMathworldPlanetmath of paracompact π’žβˆž manifolds to the category of graded-commutative ℝ-algebras. It turns out to be homotopy invariant; this implies that homotopy equivalent manifolds have isomorphic de Rham cohomology.

Title de Rham cohomology
Canonical name DeRhamCohomology
Date of creation 2013-03-22 14:24:40
Last modified on 2013-03-22 14:24:40
Owner pbruin (1001)
Last modified by pbruin (1001)
Numerical id 9
Author pbruin (1001)
Entry type Definition
Classification msc 55N05
Classification msc 58A12
Defines de Rham cohomology group