de Rham cohomology

Summary

De Rham cohomology is a cohomology theory of smooth manifolds built from differential forms. Its groups measure how far closed forms, those with $d\omega = 0$, are from being exact, of the form $\omega = d\eta$. They are real vector spaces that are diffeomorphism invariants, and by the de Rham theorem they agree with singular cohomology with real coefficients. This links calculus on a manifold to its topology.