De Rham Cohomology

Abstract

In Chapter 14, we defined closed and exact differential forms, and showed that every exact form is closed. In this chapter, we explore the converse question: Is every closed form exact? The answer is locally yes, but globally no. The question of which closed forms are exact depends on subtle topological properties of the manifold, connected with the existence of “holes” of various dimensions. Making this dependence quantitative leads to a new set of invariants of smooth manifolds, called the de Rham cohomology groups, which are the subject of this chapter. They are easily shown to be diffeomorphism invariants, but surprisingly they turn out also to be topological invariants. We prove a general theorem, called the Mayer–Vietoris theorem, that expresses the de Rham groups of a manifold in terms of those of its open subsets. Using it, we compute the de Rham groups of spheres and the top-degree groups of compact manifolds, and give a brief introduction to degree theory for maps between compact manifolds of the same dimension.