# De Rham theorem

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A theorem expressing the real cohomology groups of a differentiable manifold $M$ in terms of the complex of differential forms (cf. Differential form) on $M$. If $E ^ {*} ( M) = \sum _ {p = 0 } ^ {n} E ^ {p} ( M)$ is the de Rham complex of $M$, where $E ^ {p} ( M)$ is the space of all infinitely-differentiable $p$- forms on $M$ equipped with the exterior differentiation, then de Rham's theorem establishes an isomorphism between the graded cohomology algebra $H ^ {*} ( E ^ {*} ( M))$ of the complex $E ^ {*} ( M)$ and the cohomology algebra $H ^ {*} ( M, \mathbf R )$ of $M$ with values in $\mathbf R$. An explicit interpretation of this isomorphism is that to each closed $p$- form $\omega$ there is associated a linear form $\gamma \rightarrow \int _ \gamma \omega$ on the space of $p$- dimensional singular cycles $\gamma$ in $M$.

The theorem was first established by G. de Rham [1], although the idea of a connection between cohomology and differential forms goes back to H. Poincaré.

There are various versions of de Rham's theorem. For example, the cohomology algebra $H ^ {*} ( E _ {c} ^ {*} ( M))$ of the complex $E _ {c} ^ {*} ( M)$ of forms with compact supports is isomorphic to the real cohomology algebra $H _ {c} ^ {*} ( M, \mathbf R )$ of the manifold $M$ with compact supports. The cohomology of $M$ with values in a locally constant sheaf of vector spaces is isomorphic to the cohomology of the complex of differential forms with values in the corresponding flat vector bundle [3]. The cohomology of a simplicial set $S$ with values in any field $k$ of characteristic 0 is isomorphic to the cohomology of the corresponding de Rham polynomial complex over $k$. In the case when $S$ is the singular complex of an arbitrary topological space $X$ one obtains in this way a graded-commutative differential graded $k$- algebra $A _ { \mathop{\rm dR} } ( X)$ with cohomology algebra $H ^ {*} ( A _ { \mathop{\rm dR} } ( X))$ isomorphic to the singular cohomology algebra $H ^ {*} ( X, k)$( see [4]). If $X$ is a smooth affine algebraic variety over $\mathbf C$, then the cohomology algebra $H ^ {*} ( X, \mathbf C )$ is isomorphic to the cohomology algebra of the complex of regular differential forms on $M$( see de Rham cohomology).

#### References

 [1] G. de Rham, "Sur l'analysis situs des variétés à $n$ dimensions" J. Math. Pures Appl. Sér. 9 , 10 (1931) pp. 115–200 [2] G. de Rham, "Differentiable manifolds" , Springer (1984) (Translated from French) [3] M.S. Raghunathan, "Discrete subgroups of Lie groups" , Springer (1972) [4] D. Lehmann, "Théorie homotopique des forms différentiélles (d'après D. Sullivan)" Astérisque , 45 (1977)
How to Cite This Entry:
De Rham theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=De_Rham_theorem&oldid=50392
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article