Smooth approximations and their applications to homotopy types

Abstract

Let $M, N$ the be smooth manifolds, $mathcal{C}^{r}(M,N)$ the space of ${C}^{r}$ maps endowed with the corresponding weak Whitney topology, and $mathcal{B} subset mathcal{C}^{r}(M,N)$ an open subset.It is proved that for $0<r<sleqinfty$ 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 $eta:[0,1]times M to M$ 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.