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?



Smale's theorem for $C^1$ diffeomorphisms of the sphere
Let $M$ be a closed $C^{\infty}$-manifold, then for any $k>0$ the canonical inclusion:
$$\mathrm{Diff}^{C^{\infty}}(M)\subset \mathrm{Diff}^{C^{k}}(M)$$
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.


