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Concept# De Rham cohomology

Summary

In mathematics, de Rham cohomology (named 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.
On any smooth manifold, every exact form is closed, but the converse may fail to hold. Roughly speaking, this failure is related to the possible existence of "holes" in the manifold, and the de Rham cohomology groups comprise a set of topological invariants of smooth manifolds that precisely quantify this relationship.
The integration on forms concept is of fundamental importance in differential topology, geometry, and physics, and also yields one of the most important examples of cohomology, namely de Rham cohomology, which (roughly speaking) measures precisely the extent to which the fundamental theorem of calculus fails in higher dimensions and on general manifolds.
The de Rham complex is the cochain complex of 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 that 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.
In contrast, closed forms are not necessarily exact. An illustrative case is a circle as a manifold, and the 1-form corresponding to the derivative of angle from a reference point at its centre, typically written as dθ (described at Closed and exact differential forms). There is no function θ defined on the whole circle such that dθ is its derivative; the increase of 2π in going once around the circle in the positive direction implies a multivalued function θ.

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