# de Rham cohomology

Let $X$ be a paracompact ${\cal C}^{\infty}$ differential manifold. Let

 $\Omega X=\bigoplus_{i=0}^{\infty}\Omega^{i}X$

denote the graded-commutative $\mathbb{R}$-algebra of differential forms on $X$. Together with the exterior derivative

 $d^{i}\colon\Omega^{i}X\to\Omega^{i+1}X\quad(i=0,1,\ldots),$

$\Omega X$ forms a chain complex $(\Omega X,d)$ of $\mathbb{R}$-vector spaces. The ${\rm H}_{\rm dR}^{i}X$ of $X$ are defined as the homology groups of this complex, that is to say

 ${\rm H}_{\rm dR}^{i}X:=(\ker d^{i})/(\mathop{\mathrm{im}}d^{i-1})\quad(i=0,1,% \ldots),$

where $\Omega^{-1}X$ is taken to be 0, so $d^{-1}\colon 0\to\Omega^{0}X$ is the zero map. The wedge product in $\Omega X$ induces the structure of a graded-commutative $\mathbb{R}$-algebra on

 ${\rm H}_{\rm dR}X:=\bigoplus_{i=0}^{\infty}{\rm H}_{\rm dR}^{i}X.$

If $X$ and $Y$ are both paracompact ${\cal C}^{\infty}$ manifolds and $f\colon X\to Y$ is a differentiable map, there is an induced map

 $f^{*}\colon{\rm H}_{\rm dR}Y\to{\rm H}_{\rm dR}X,$

defined by

 $f^{*}[\omega]:=[f^{*}\omega]\quad\hbox{for \omega\in\ker d}.$

Here $[\omega]$ denotes the class of $\omega$ modulo $\mathop{\mathrm{im}}d$, and the second $f^{*}$ is the map $\Omega Y\to\Omega X$ induced by the functor $\Omega$. This action on differentiable maps makes the de Rham cohomology into a contravariant functor from the category of paracompact ${\cal C}^{\infty}$ manifolds to the category of graded-commutative $\mathbb{R}$-algebras. It turns out to be homotopy invariant; this implies that homotopy equivalent manifolds have isomorphic de Rham cohomology.

Title de Rham cohomology DeRhamCohomology 2013-03-22 14:24:40 2013-03-22 14:24:40 pbruin (1001) pbruin (1001) 9 pbruin (1001) Definition msc 55N05 msc 58A12 de Rham cohomology group