In 1926 Kneser showed that homeomorphisms of $\mathbf{S}^2$ admit a retraction into the orthogonal group $O(3)$. Smale extended this result to Diffeomorphisms of $\mathbf{S}^2$ in 1958; however, in that paper he assumes that the diffeomorphisms are of class $C^k$ for $k\geq 2$. So my question is: Has anyone established Smale's theorem for $C^1$ diffeomorphisms?
Let $M$ be a closed $C^{\infty}$manifold, then for any $k>0$ the canonical inclusion: $$Diff^{C^{\infty}}(M)\subset Diff^{C^{k}}(M)$$ is a homotopy equivalence. Embed $M$ in an euclidean space $\mathbb{R}^n$ as a $C^{\infty}$submanifold and build a smoothing operator by using the convolution product.

2$\begingroup$ You seem to be saying that first retract the space of $C^1$ diffeomorphisms into the space of $C^\infty$ diffeomorphisms. This would involve deforming a $C^1$ diffeomorphism into a $C^\infty$ one in a canonical way. Is it obvious that one can do that? (I know how to smoothen a $C^1$ diffeomorphism, but do not immediately see how to do it canonically) $\endgroup$ – Mohammad Ghomi Nov 29 '17 at 15:14

1$\begingroup$ OK, maybe I am beginning to see how this should work: for every point $p\in S^2$, let $\theta=\theta_{p,t}$ be a smooth convolution kernel function with support in a ball of radius $t$ centered at $p$, and total integral $1$. So as $t\to 0$, $\theta$ converges to the Dirac delta function at $p$. Now for some $\epsilon>0$, take the convolution of the $C^1$ diffeomorphism $f\colon S^2\to S^2\subset R^3$ with $\theta$ for all $t<\epsilon$, and then project the resulting map back to $S^2$, i.e., let $f_t(p)$ be the projection of $f\circ\theta_{p,t}$ into $S^2$. $\endgroup$ – Mohammad Ghomi Nov 29 '17 at 16:52

2$\begingroup$ @Mohammed: there is no canonical smoothing that I know of, but there are certainly "fairly canonical" families of smoothings. It's enough for this theorem. $\endgroup$ – Ryan Budney Nov 30 '17 at 3:21
I think it should be covered by M.T. Wang's papers on deformation of graphs of diffeomorphisms, in particular a combination of the main results in https://intlpress.com/site/pub/pages/journals/items/cag/content/vols/0012/0003/a004/ and https://intlpress.com/site/pub/pages/journals/items/mrl/content/vols/0008/0005/a007/.
I am quite sure it follows from:
Bloch, Ethan D.; Connelly, Robert; Henderson, David W., The space of simplexwise linear homeomorphisms of a convex 2disk, Topology 23, 161175 (1984). ZBL0547.57016.
(using an idea of Nico Kuiper, which can be found in the references).