The de Rham Theorem

Abstract

The topological invariance of the de Rham groups suggests that there should be some purely topological way of computing them. There is indeed: in this chapter, we prove the de Rham theorem, which says that the de Rham groups of a smooth manifold are naturally isomorphic to its singular cohomology groups, which are algebraic-topological invariants that measure “holes” in a rather direct sense. First we give a very brief introduction to singular homology and cohomology, and prove that they can be computed by restricting attention only to smooth simplices. In the final section of the chapter, we prove the de Rham theorem by showing that integration of differential forms over smooth simplices induces isomorphisms between the de Rham groups and the singular cohomology groups.