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de Rham cohomology

The measurement of exactly how many closed forms on a region fail to be exact -- a number that detects holes.

De Rham cohomology measures the gap between closed and exact forms on a region -- zero when every closed form is exact (a simply connected region, by the Poincar'e lemma), nonzero exactly when the region has a topological hole a closed-but-not-exact form can detect, like the punctured plane's winding number.

Examples

  • The punctured plane's first de Rham cohomology group is one-dimensional, generated by the winding-number 1-form.

See also

Closed form Simply connected region
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