There is a lot known about $\pi_0(Diff(D^n), \partial)$, diffeomorphisms of the closed n-disk that act trivially on the boundary. I was wondering what is known about $\pi_0(Diff(D^n))$, where diffeomorphisms can act non-trivially on the boundary. Here I mean $C^\infty$ diffeomorphisms but am not exactly sure what topology to take on the space of $C^\infty$ diffeomorphisms.
I should mention that $C^1$-maps are studied by Labach in his paper: "Diffeomorphisms of n-disk" https://www.jstage.jst.go.jp/article/pjab1945/43/6/43_6_448/_article/-char/en He shows that the space of orientation preserving $C^1$-diffeomorphisms are homotopy equivalent to $SO(n)$. I think the proof is to take a diffeomorphism of $D^n$ (which we can assume fixes the origin) and "blow it up" from the interior by pushing everything away from the origin. This is a family of $C^\infty$ diffeomorphisms but I don't think it defines a $C^\infty$ map $D^n \times D^1 \rightarrow D^n$.
In dimension $n \geq 6$, the map $$\pi_0\,\mathrm{O}(n+1) \to \pi_0\,\mathrm{Diff}(D^n)$$ induced by rotation is an isomorphism by a result of Cerf (Corollaire 2 of La stratification naturelle..., note that for Cerf diffeomorphisms need to preserve orientations). The same is true for $n \leq 3$ by techniques particular to those dimensions, and I believe the cases $n=4,5$ are open questions.