The Poincaré Lemma (Amann-Escher XI.3.11) states
If $X$ is contractible, then every closed differential form on $X$ is exact,
where contractible means that the identity on $X$ is null-homotic. Amann-Escher however requires such a homotpy to be smooth ($C^\infty$).
So my question is whether the homotopy really must be smooth for the lemma to hold. In particular
I guess there are examples of manifolds (or maybe subsets of $\mathbb{R}^n$) which are "topologically contractible" but not "smoothly contractible"?
should "topologically contractible" not be strong enough for the theorem to hold, would something like a "$C^2$-contractible" be enough?
The answer to this question provides a reference for the fact that homotopic smooth maps are always smooth-homotopic, so in particular $\text{Id}_M$ and $constant$ are homotopic by a $C^\infty$-map.
A related standard fact you may know is that the singular (“continuous”) cohomology groups and the de Rham cohomology groups are isomorphic for smooth manifolds. This is “de Rham’s theorem”, I believe.