In mathematics, de Rham cohomology (after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. It is a cohomology theory based on the existence of differential forms with prescribed properties.
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 Ω0(M) are called exact and forms whose exterior derivative is 0 are called closed (see closed and exact differential forms); the relationship d 2 = 0 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.