The de Rham Theorem

We describe the deep connection between the two fundamental notions of de Rham cohomology and differentiable singular cohomology. We develop relevant sheaf theory and notions of cohomology, and show the existence of an axiomatic sheaf cohomology theory via sheaf resolutions. The strength of sheaf cohomology theory lies in that we can prove that sheaf cohomology theories are unique. Furthermore, we determine sheaf resolutions which give rise to sheaf cohomologies that are isomorphic to the classical differentiable singular cohomology and de Rham cohomology respectively. By uniqueness we assert that the cohomologies are isomorphic, with the principal ideal domain set to the field of real numbers. The de Rham theorem determines the isomorphism, and we conclude that, though they are concerned with differential forms, the de Rham cohomology groups are topological invariants.