Smooth approximations and their applications to homotopy types (2008.11991v3)

Abstract: Let $M, N$ the be smooth manifolds, $\mathcal{C}^{r}(M,N)$ the space of ${C}^{r}$ maps endowed with weak $C^{r}$ Whitney topology, and $\mathcal{B} \subset \mathcal{C}^{r}(M,N)$ an open subset. It is proved that for $0\leq r<s\leq\infty$ the inclusion $\mathcal{B} \cap \mathcal{C}^{s}(M,N) \subset \mathcal{B}$ is a weak homotopy equivalence. It is also established a parametrized variant of such a result. In particular, it is shown that for a compact manifold $M$, the inclusion of the space of $\mathcal{C}^{s}$ isotopies $[0,1]\times M \to M$ fixed near ${0,1}\times M$ into the space of loops $\Omega(\mathcal{D}^{r}(M), \mathrm{id}{M})$ of the group of $\mathcal{C}^{r}$ diffeomorphisms of $M$ at $\mathrm{id}{M}$ is a weak homotopy equivalence.