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.