# De Rham theorem

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 $M$.
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 . 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 ). 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).