De Rham Cohomology - Definition

Definition

The de Rham complex is the cochain complex of exterior differential forms on some smooth manifold M, with the exterior derivative as the differential.

where Ω0(M) is the space of smooth functions on M, Ω1(M) is the space of 1-forms, and so forth. Forms which are the image of other forms under the exterior derivative, plus the constant 0 function in are called exact and forms whose exterior derivative is 0 are called closed (see closed and exact differential forms); the relationship then says that exact forms are closed.

The converse, however, is not in general true; closed forms need not be exact. A simple but significant case is the 1-form of angle measure on the unit circle, written conventionally as dθ (described at closed and exact differential forms). There is no actual function θ defined on the whole circle of which dθ is the derivative; the increment of 2π in going once round the circle in the positive direction means that we can't take a single-valued θ. We can, however, change the topology by removing just one point.

The idea of de Rham cohomology is to classify the different types of closed forms on a manifold. One performs this classification by saying that two closed forms α and β in are cohomologous if they differ by an exact form, that is, if is exact. This classification induces an equivalence relation on the space of closed forms in . One then defines the -th de Rham cohomology group to be the set of equivalence classes, that is, the set of closed forms in modulo the exact forms.

Note that, for any manifold M with n connected components

This follows from the fact that any smooth function on M with zero derivative (i.e. locally constant) is constant on each of the connected components of M.