De Rham and Cech cohomology of smooth manifolds
Using the exterior differential on smooth differential forms and the fact that , we define the de Rham cohomology with real coefficients as the quotient space of closed -forms by exact -forms. This is a global invariant of a manifold (for non-compact manifolds we may also consider a version for compactly supported forms, which yields different results).
De Rham cohomology can be computed using the Poincare lemma. If is an open covering of such that all opens sets and all their non-empty finite intersections are topologically trivial (homeomorphic to open balls), then for any closed form one can construct its primitives such that in . The th forms may disagree on the intersections , but one can attempt to twist them by suitable closed forms . The corresponding system of -forms satisfies certain linear conditions on pairwise intersections; to satisfy these conditions one has to look for forms on triple intersections etc.
This construction gives rise to the notion of the Cech cohomology defined via systems of linear algebraic equations and reduces computation of the de Rham cohomology to a problem from linear algebra, determined by the combinatorics of the pairwise and multiple intersections of the sets . In particular, one can conclude that the de Rham cohomology of compact manifolds is finite-dimensional.
NB. The class was shorter than usual because of the Hanukka lighting ceremony.