# proof of Poincaré lemma

Let $X$ be a smooth manifold, and let $\omega$ be a closed differential form of degree $k>0$ on $X$. For any $x\in X$, there exists a contractible neighbourhood $U\subset X$ of $x$ (i.e. $U$ is homotopy equivalent to a single point), with inclusion map

 $\iota\colon U\hookrightarrow X.$

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 cohomology, the $k$th de Rham cohomology group ${\rm H}^{k}(U)$ is isomorphic to that of a point; in particular,

 ${\rm H}^{k}(U)=0\quad\hbox{for all k>0}.$

Since $d(\iota^{*}\omega)=\iota^{*}(d\omega)=0$, this implies that there exists a $(k-1)$-form $\eta$ on $U$ such that $d\eta=\iota^{*}\omega$. In the case where $X$ is a contractible manifold, such an $\eta$ exists globally since we can choose $U=X$ above.

Title proof of Poincaré lemma ProofOfPoincareLemma 2013-03-22 14:24:36 2013-03-22 14:24:36 pbruin (1001) pbruin (1001) 4 pbruin (1001) Proof msc 53-00 msc 55N05