De Rham cohomology

For Grothendieck's de Rham cohomology of varieties, see algebraic de Rham cohomology.
Vector field corresponding to a differential form on the punctured plane that is closed but not exact, showing that the de Rham cohomology of this space is non-trivial.

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.[1]

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.
— Terence Tao, Differential Forms and Integration[2]
  1. ^ Lee 2013, p. 440.
  2. ^ Tao, Terence (2007) "Differential Forms and Integration" Princeton Companion to Mathematics 2008. Timothy Gowers, ed.