Chapter 8 - de Rham Cohomology

Abstract

We begin this chapter with some algebraic constructions that we will need later: (co) chain complexes and related matters. Then we introduce the de Rham cohomology of a smooth manifold with boundary M, which measures in a precise way the difference between closed and exact differential forms on M. Actually, we introduce the notion of a smooth pair (M,A) and consider the de Rham cohomology of a smooth pair. We develop the basic properties of de Rham cohomology, including smooth homotopy invariance, the long exact sequence of a smooth pair, and the Mayer-Vietoris sequence. We see how to reinterpret some of our previous work in terms of de Rham cohomology. We use the properties of de Rham cohomology to compute the de Rham cohomology of the n-sphere, for every n. We also consider a variant of de Rham cohomology: de Rham cohomology with compact supports. We develop its (analogous) basic properties and do some computations here, as well.