Poincaré lemma - Imported Knowlpedia Content

Statement

The Poincaré lemma states that every closed differential form of degree $k>0$ is locally exact. Explicitly, if $M$ is a smooth manifold, $p\in M$ , and $\omega\in\Omega^k(M)$ satisfies $d\omega=0$ , then some open neighborhood $W$ of $p$ admits $\eta\in\Omega^{k-1}(W)$ with $\omega|_W=d\eta$ . Equivalently, on every star-shaped open set $U\subseteq\mathbb R^n$ , each closed form of positive degree is exact. In degree zero, the corresponding statement is that a function with zero differential is locally constant.

Homotopy operator

For a star-shaped $U$ , radial contraction to the center supplies an operator $K:\Omega^k(U)\to\Omega^{k-1}(U)$ satisfying

dK+Kd=\operatorname{id}-c^*,

where $c$ is the constant map to the center. The operator is obtained by contracting the pulled-back form with the interval direction and integrating in that direction. For $k>0$ , $c^*=0$ , so a closed $\omega$ obeys $\omega=d(K\omega)$ .

Local rather than global exactness

The lemma does not say that every closed form on a manifold is globally exact. The angular one-form on $\mathbb R^2\setminus\{0\}$ is locally exact but represents a nonzero de Rham cohomology class. Global failure to patch local primitives is precisely the information that positive-degree de Rham cohomology records.

Role in de Rham theory

The lemma says that the de Rham complex is locally exact in positive degrees. Together with smooth partitions of unity, this local statement is the key analytic input in the de Rham theorem; see Bott and Tu, Chapter I.

References

Raoul Bott and Loring W. Tu, Differential Forms in Algebraic Topology, Springer, 1982. DOI record. Relevant: Chapter I, the Poincaré lemma and de Rham theory.
Loring W. Tu, An Introduction to Manifolds, 2nd ed., Springer, 2011. DOI record. Relevant: the chapters on differential forms and de Rham theory.