Differential forms on a point

Question

For the proof of Poincaré lemma, it's essential to evaluate $\Omega^p(*)$ where $*$ is zero dimensional manifold and $\Omega^p$ is a collection of all $p$-forms on given manifold. Clearly, $\Omega^0 (*) =\mathbb{R}$. But, I want to show that $\Omega^p(*)=0$ where $p>0$.

Answer 1

If $M$ is a manifold of dimension $n$, then for all $p > n$, $\Omega^p(M) = \{0\}$. In particular, as a point is a zero-dimensional manifold, $\Omega^p(\ast) = \{0\}$ for all $p > 0$.