De Rham Cohomology

Abstract

We turn now from classical vector analysis to a completely different aspect of the calculus of differential forms. Consider the de Rham complex

$$0 \to {\Omega ^0}M{\Omega ^1}M \cdots $$

of a manifold M. The property d ο d = 0 means that

$$im(d:{\Omega ^{k - 1}}M \to {\Omega ^k}M) \subset \ker (d:{\Omega ^k}M \to {\Omega ^{k + 1}}M)$$

for every k, so we can take the quotient of these two vector spaces.