De Rham theorem

A theorem expressing the real cohomology groups of a differentiable manifold in terms of the complex of differential forms (cf. Differential form) on . If is the de Rham complex of , where is the space of all infinitely-differentiable -forms on equipped with the exterior differentiation, then de Rham's theorem establishes an isomorphism between the graded cohomology algebra of the complex and the cohomology algebra of with values in . An explicit interpretation of this isomorphism is that to each closed -form there is associated a linear form on the space of -dimensional singular cycles in .

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 of the complex of forms with compact supports is isomorphic to the real cohomology algebra of the manifold with compact supports. The cohomology of 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 with values in any field of characteristic 0 is isomorphic to the cohomology of the corresponding de Rham polynomial complex over . In the case when is the singular complex of an arbitrary topological space one obtains in this way a graded-commutative differential graded -algebra with cohomology algebra isomorphic to the singular cohomology algebra (see [4]). If is a smooth affine algebraic variety over , then the cohomology algebra is isomorphic to the cohomology algebra of the complex of regular differential forms on (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)