I have learned that, if we have a connected, oriented and compact n-dimensional manifold, the top de Rham cohomology is isomorphic to $\mathbb{R}$, i.e $$ H_{\text{dR}}^n(M) \cong \mathbb{R}.$$ However, Poincaré's lemma states that for any star-shaped subset of $ \mathcal{U} \subset \mathbb{R}^n$, it holds that $$ {H_{\text{dR}}^k(\mathcal{U}) = 0} \qquad \text{for all }k \neq 0.$$
Consider now the ball of radius $1$, this is evidently star shaped. Furthermore, it seems to me that it also suffices the properties in the first result. What is wrong in this reasoning?
I know the statement in the form: "top de-Rham cohomology of compact, connected, oriented manifold without boundary is isomorphic to $\mathbb{R}$".