Smooth approximations and their applications to homotopy types

TL;DR

This paper proves that certain inclusions of smooth map spaces between manifolds are weak homotopy equivalences, extending to parametrized cases, with applications to isotopies and diffeomorphism groups.

Contribution

It establishes that inclusions of spaces of higher regularity maps into lower regularity spaces are weak homotopy equivalences, including parametrized variants and applications to diffeomorphism groups.

Findings

Inclusion of C^s maps into C^r maps is a weak homotopy equivalence.

Parametrized versions of the homotopy equivalence are proved.

Applications to isotopies and diffeomorphism groups of manifolds.

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.