The orientation-preserving diffeomorphism group of S^2 deforms to SO(3) smoothly

TL;DR

This paper constructs a diffeologically smooth strong deformation retraction of the orientation-preserving diffeomorphism group of S^2 onto SO(3), extending Smale's classical result with smooth structure considerations.

Contribution

It provides a diffeologically smooth deformation retraction of the diffeomorphism group of S^2 onto SO(3), enhancing Smale's original topological result with smoothness properties.

Findings

Established a diffeologically smooth deformation retraction

Extended Smale's result with smooth structure

Demonstrated smooth deformation in diffeological sense

Abstract

Smale proved that the orientation-preserving diffeomorphism group of S^2 has a continuous strong deformation retraction to SO(3). In this paper, we construct such a strong deformation retraction which is diffeologically smooth.