de Rham Cohomology

de Rham Complex

de Rham Complex

Let be a smooth manifold. The de Rham complex is the sequence of differential forms on : where is the space of smooth differential -forms on , and is the exterior derivative.

de Rham Cohomology Group

The th de Rham cohomology group of a smooth manifold is defined as the quotient That is, it’s the space of closed -forms modulo the space of exact -forms.

de Rham’s theorem

de Rham's Theorem

Let be a smooth manifold. Then the de Rham cohomology groups are isomorphic to the singular cohomology groups with real coefficients. In particular, they are independent of the choice of differential forms used to define them.