When is a bi-Lipschitz homeomorphism smoothable? Suppose I have a smooth Riemannian manifold $X$ with induced distance function $d$, and a bi-Lipschitz (with respect to $d$) homeomorphism
$$\phi: X \to X.$$
Under what circumstances could $\phi$ be smoothable to a diffeomorphism? By "smoothable" in this case I mean "homotopic to a diffeomorphism through bi-Lipschitz homeomorphisms" (this might not be standard, I suppose). Are there any clear obstructions?
 A: There is some interest in a related question in non-linear elasticity, specifically people there would consider a function "smoothable" if there is a close-by (in some norm applying both to function and its inverse) diffeomorphism.
In 2D there are some results with regards to this, see e.g. Smooth approximation of bi-Lipschitz orientation-preserving homeomorphisms by Danieri & Pratelli, who prove that there is a close diffeomorphism in some bi-Sobolev norm for domains in the plane (which if I am not mistaken should imply the same result at least for compact manifolds). But the proof uses a bi-Lipschitz extension theorem, so I am not sure if one can construct homotopies from that result easily and there seems to be no way to extend this to higher dimensions.
A: Any self-homeomorphism of a manifold of dimension $\neq 4$ is topologically isotopic to a bi-Lipschitz homeomorphism, see lemma 2.4 in Lipschitz and quasiconformal approximation of homeomorphism pairs
by Jouni Luukkainen.
There are exotic spheres (e.g. in dimension $7$) that admit a self-homeomorphism that is not homotopic to a diffeomorphism, see here. This gives a bi-Lipschitz homeomorphism that is not homotopic to a diffeomorphism.
