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

$$\iota :U\hookrightarrow X.$$

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 ${\mathrm{H}}^k(U)$ is isomorphic to that of a point; in particular,

$${\mathrm{H}}^k(U)=0\mathit{\hspace{1em}}\text{for all }k>0.$$

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.