proof of Poincaré lemma

Let X be a smooth manifoldMathworldPlanetmath, and let ω be a closed differential form of degree k>0 on X. For any xX, there exists a contractible neighbourhood UX of x (i.e. U is homotopy equivalent to a single point), with inclusion map


To construct such a neighbourhood, take for example an open ball in a coordinate chart around x. Because of the homotopy invariance of de Rham cohomologyMathworldPlanetmath, the kth de Rham cohomology group Hk(U) is isomorphic to that of a point; in particular,

Hk(U)=0for all k>0.

Since d(ι*ω)=ι*(dω)=0, this implies that there exists a (k-1)-form η on U such that dη=ι*ω. In the case where X is a contractible manifold, such an η exists globally since we can choose U=X above.

Title proof of Poincaré lemma
Canonical name ProofOfPoincareLemma
Date of creation 2013-03-22 14:24:36
Last modified on 2013-03-22 14:24:36
Owner pbruin (1001)
Last modified by pbruin (1001)
Numerical id 4
Author pbruin (1001)
Entry type Proof
Classification msc 53-00
Classification msc 55N05