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 weak $C^r$ Whitney topology, and $\mathcal{B}\subset {\mathcal{C}}^r(M,N)$ an open subset. It is proved that for $0\le r<s\le \infty$ 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.

31 pages, improved exposition, fixed some misprints, added few references

References in corpus (1)

• Actions of groups of foliated homeomorphisms on spaces of leaves

Cited by in corpus (5)

• Homotopy types of diffeomorphism groups of polar Morse-Bott foliations on lens spaces, 1
• Homotopy types of diffeomorphisms groups of simplest Morse-Bott foliations on lens spaces, 2
• Homeotopy groups of leaf spaces of one-dimensional foliations on non-compact surfaces with non-compact leaves
• Diffeomorphism groups of Morse-Bott foliation on the solid Klein bottle by Klein bottles parallel to the boundary
• Vector bundle automorphisms preserving Morse-Bott foliations