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 space 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.