Short version of question:
Does $\operatorname{dif}(S^n)$ have more than two connected components?
Reading this article on Smale's conjecture and the resolution in higher dimensions, I had a question which wasn't answered in the article.
If $\operatorname{dif}(M)$ is the group of diffeomorphisms $M\to M,$ then the general question is whether $\operatorname{dif}(S^n)$ is homotopy-equivalent to the subgroup $O(n+1)$ of rotations and reflections of $S^n.$
Turns out, the answer is "No" for $n\geq 4.$ And the article indicates there are classes of diffeomorphisms that present problems.
Naively, that would seem to imply that there are diffeomorphisms that are not homotopic to a rotation or reflection – that is, there are more than two path-connected components of $\operatorname{dif}(S^n).$ But that's a much stronger result than the spaces not being homotopy equivalent.
Given any $h:S^n\to S^n$ which is a homeomorphism, we get it must be homotopic to either the identity or reflection, by considering it as either $1$ or $-1$ in $\pi_n(S^n)\cong \mathbb Z.$ But that homotopy might not be able to stay in $\operatorname{dif}(M).$
So, are the "problem diffeomorphisms" for $n>4$ path-disconnected from $O(n+1),$ or are they just like how a circle is not homotopy equivalent to an arc, with no individual points on the rest of the circle really being "the problem point?"
(I restrict to $n>4$ only because the article indicates they don't yet know much about the problem diffeomorphisms for $n=4.$)
Best Answer
I'll use the notation $\operatorname{Diff}_\partial$ to denote $\operatorname{Diff}(D^n\text{ rel } \partial D^n) = \{f:D^n\rightarrow D^n| f\text{ is a diffeomorphism and } f|_{\partial D^n} = Id_{\partial D^n}\}$.
From this MO question one finds a proof that $\operatorname{Diff}(S^n)$ has the homotopy type of $O(n+1)\times \operatorname{Diff}_\partial$. So, there are diffeomorphisms of $S^n$ which are not path connected to $O(n+1)$ precisely when $\operatorname{Diff}_\partial$ has multiple path components.
From the "Historical Remarks" portion of this preprint, one finds the following facts: For $n=1,2,3$, $\operatorname{Diff}_\partial$ has the homotopy type of a point. For $n=1$, this is "classical", for $n=2$ it's a result of Smale, and for $n=3$ it's a result of Hatcher. Thus, in these dimensions, all diffeomorphisms of $S^n$ are path connected in $\operatorname{Diff}(S^n)$ to $O(n+1)$.
For $n\geq 5$, there is an isomorphism between $\pi_0(\operatorname{Diff}_\partial)$ and $\Theta_{n+1}$, the group of exotic $(n+1)$-spheres. Thus, whenever $|\Theta_{n+1}|\neq 1$, there are diffeomorphisms of $S^n$ which are not path connected to $O(n+1)$. This case is typical. In fact, the only known examples with $n\geq 5$, and $|\Theta_{n+1}| = 1$ are when $n+1\in \{6,7,13,55,62\}$ (see this preprint. Moreover, from this paper, the set I just listed is complete for even entries.
All this leaves the case $n=4$. But in this preprint, one finds a proof that there are diffeomorphisms of $S^4$ which are not path connected to $O(5)$.