PlanetMath: de Rham cohomology

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de Rham cohomology

(Definition)

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

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

denote the graded-commutative $\mathbb{R}$ -algebra of differential forms on

. Together with the exterior derivative

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

$\Omega X$ forms a chain complex $(\Omega X,d)$ of $\mathbb{R}$ -vector spaces. The de Rham cohomology groups ${\rm H}_{\rm dR}^i X$ of

are defined as the homology groups of this complex, that is to say

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

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

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

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

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

defined by

$\displaystyle 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

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.

"de Rham cohomology" is owned by pbruin.

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Also defines:

de Rham cohomology group

Attachments:: example of de Rham cohomology (Example) by pbruin

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Cross-references: isomorphic, homotopy equivalent, implies, homotopy invariant, category, action, functor, class, map, induced, differentiable map, structure, induces, wedge product, zero map, complex, homology groups, chain complex, exterior derivative, differential forms, graded-commutative, differential manifold, paracompact
There are 15 references to this entry.

This is version 6 of de Rham cohomology, born on 2004-06-12, modified 2007-10-09.
Object id is 5913, canonical name is DeRhamCohomology.
Accessed 5519 times total.

Classification:

AMS MSC:	55N05 (Algebraic topology :: Homology and cohomology theories :: Cech types)
	58A12 (Global analysis, analysis on manifolds :: General theory of differentiable manifolds :: de Rham theory)

Pending Errata and Addenda

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