De Rham cohomology

In differential geometry, differential forms on a smooth manifold which are exterior derivatives are called exact; and forms whose exterior derivatives are 0 are called closed (see closed and exact differential forms).

Exact forms are closed, so the vector spaces of k-forms along with the exterior derivative are a cochain complex. The vector spaces of closed forms modulo exact forms are called the de Rham cohomology groups. The wedge product endows the direct sum of these groups with a ring structure.

De Rham's theorem, proved by Georges de Rham in 1931, states that for a compact oriented smooth manifold M, these groups are isomorphic as real vector spaces with the singular cohomology groups H^p(M;R). Further, the two cohomology rings are isomorphic (as graded rings).

The general Stokes' theorem is an expression of duality between de Rham cohomology and the homology of chains.