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 $.

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 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)