Sheaves, Cohomology, and the de Rham Theorem

Abstract

The principal objective in this chapter is a proof of the de Rham theorem, one version of which we have stated in 4.17. In its most complete form it asserts that the homomorphism from the de Rham cohomology ring to the differentiable singular cohomology ring given by integration of closed forms over differentiable singular cycles is a ring isomorphism. The approach will be to exhibit both the de Rham cohomology and the differentiable singular cohomology as special cases of sheaf cohomology and to use a basic unique­ness theorem for homomorphisms of sheaf cohomology theories to prove that the natural homomorphism between the de Rham and differentiable singular theories is an isomorphism. As an added dividend of this approach we shall also obtain the existence of canonical isomorphisms of the de Rham and differentiable singular cohomology theories with the continuous singular theory, the Alexander-Spanier theory, and the Čech cohomology theory for differentiable manifolds. From these isomorphisms we shall conclude that the de Rham cohomology theory is a topological invariant of a differentiable manifold.