Isometry groups and mapping class groups of spherical 3-orbifolds
The authors consider compact and orientable spherical 3-orbifolds, i.e. 3-orbifolds of the form \(\mathcal{O} = S^3/G\) where \(G\) denotes a finite subgroup of the orthogonal group SO(4). On the basis of the classification of the finite subgroups of SO(4), the two main results of the present paper are the following. First, an explicit algebraic computation of the isometry groups of all spherical 3-orbifolds \(S^3/G\), first of the orientation-preserving isometry groups (isomorphic to the quotients of the normalizers of the groups \(G\) in SO(4) by \(G\)), then also of the full isometry groups. Secondly, the following ``version on the \(\pi_0\)-level'' of the generalized Smale conjecture for spherical 3-orbifolds: The inclusion \(\mathrm{Isom}(\mathcal O) \hookrightarrow \mathrm{Diff}(\mathcal O)\) induces an isomorphism between \(\pi_0 \, \mathrm{Isom}(\mathcal O)\) and the orbifold mapping class group \(\pi_0 \, \mathrm{Diff}(\mathcal O)\); in particular, together with the first result this gives an explicit computation of the mapping class groups of spherical 3-orbifolds. For the case of spherical 3-manifolds instead (the case of free actions of \(G\) on \(S^3\)), these results have been proved in a paper by \textit{D. McCullough} [J. Lond. Math. Soc., II. Ser. 65, No. 1, 167--182 (2002; Zbl 1012.57023)], so in the present paper the authors can restrict to the cases where \(\mathcal O\) has nonempty singular set; they use geometric structures on 3-manifolds (hyperbolic and Seifert fibered), the geometrization of finite group-actions on 3-manifolds and the analysis of Seifert fibrations of spherical 3-orbifolds given in a previous paper by the two present authors [Rev. Mat. Iberoam. 31, No. 3, 811--840 (2015; Zbl 1328.57023)]. The generalized Smale conjecture for spherical 3-orbifolds states that the inclusion \(\mathrm{Isom}(\mathcal O) \hookrightarrow \mathrm{Diff}(\mathcal O)\) is a homotopy equivalence; for the case of spherical 3-manifolds, this is known for the 3-sphere (the original Smale conjecture proved by Hatcher) and various other spherical 3-manifolds (see [\textit{S. Hong} et al., Diffeomorphisms of elliptic 3-manifolds. Berlin: Springer (2012; Zbl 1262.57001)], and a recent preprint by \textit{R. H. Bamler} and \textit{B. Kleiner} [``Ricci flow and diffeomorphism groups of 3-manifolds'', \url{arXiv:1712.06197}]).
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