Closed and Exact Differential Forms - Formulation As Cohomology

Formulation As Cohomology

When the difference of two closed forms is an exact form, they are said to be cohomologous to each other. That is, if ζ and η are closed forms, and one can find some β such that

then one says that ζ and η are cohomologous to each other. Exact forms are sometimes said to be cohomologous to zero. The set of all forms cohomologous to a given form (and thus to each other) is called a de Rham cohomology class; the general study of such classes is known as cohomology. It makes no real sense to ask whether a 0-form (smooth function) is exact, since d increases degree by 1; but the clues from topology suggest that only the zero function should be called "exact". The cohomology classes are identified with locally constant functions.

A corollary of the Poincaré lemma is that de Rham cohomology is homotopy-invariant. Non-contractible spaces need not have trivial de Rham cohomology. For instance, on the circle S1, parametrized by t in, the closed 1-form dt is not exact.