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.

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.