Isometry groups and mapping class groups of spherical 3-orbifolds

TL;DR

This paper classifies the isometry groups of spherical 3-orbifolds and proves a version of the Smale Conjecture for these orbifolds, linking their isometry and diffeomorphism groups.

Contribution

It determines the isometry groups of spherical 3-orbifolds and establishes an isomorphism between their isometry and diffeomorphism group components, extending the Smale Conjecture.

Findings

Isometry groups of spherical 3-orbifolds are classified.

The inclusion of isometry into diffeomorphism groups induces an isomorphism on $oldsymbol{\pi_0 ext{-}groups}$.

Proves the $oldsymbol{ ext{Smale Conjecture}}$ for spherical 3-orbifolds.

Abstract

We study the isometry groups of compact spherical orientable $3$-orbifolds $S^3/G$, where $G$ is a finite subgroup of $\mathrm{SO}(4)$, by determining their isomorphism type. Moreover, we prove that the inclusion of $\mbox{Isom}(S^3/G)$ into $\mbox{Diff}(S^3/G)$ induces an isomorphism of the $\pi_0$ groups, thus proving the $\pi_0$-part of the natural generalization of the Smale Conjecture to spherical $3$-orbifolds.