De Rham Cohomology - de Rham's Theorem

De Rham's Theorem

Stokes' theorem is an expression of duality between de Rham cohomology and the homology of chains. It says that the pairing of differential forms and chains, via integration, gives a homomorphism from de Rham cohomology to singular cohomology groups Hk(M; R). De Rham's theorem, proved by Georges de Rham in 1931, states that for a smooth manifold M, this map is in fact an isomorphism.

The wedge product endows the direct sum of these groups with a ring structure. A further result of the theorem is that the two cohomology rings are isomorphic (as graded rings), where the analogous product on singular cohomology is the cup product.