Isometry groups and mapping class groups of spherical 3-orbifolds
type: Item
instance of: scholarly article
title: Isometry groups and mapping class groups of spherical 3-orbifolds
zbMATH Open document ID: 1422.57046
author: Mattia Mecchia
author: Andrea Seppi
published in: Mathematische Zeitschrift
publication date: 2019-07-18T00:00:00Z
full work available at URL: https://arxiv.org/abs/1607.06281
review text: 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 …
Mathematics Subject Classification ID: 57M50
zbMATH DE Number: 7081700
zbMATH Keywords: spherical 3-orbifold
zbMATH Keywords: isometry group
zbMATH Keywords: mapping class group
zbMATH Keywords: generalized Smale conjecture
reviewed by: Bruno Zimmermann
MaRDI profile type: Publication
OpenAlex ID: W2963351762
arXiv ID: 1607.06281
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DOI: 10.1007/S00209-018-2166-2
description: scientific article