On finite simple groups acting on homology 3-spheres
type: Item
instance of: scholarly article
title: On finite simple groups acting on homology 3-spheres
zbMATH Open document ID: 1012.57028
DOI: 10.1016/S0166-8641(01)00271-1
published in: Topology and its Applications
publication date: 2003-03-10T00:00:00Z
review text: The main theorem is: The only nonabelian finite simple group which acts faithfully by diffeomorphisms on a homology 3-sphere is the alternating or dodecahedral group \(\mathbf A_5 \cong \text{PSL}(2,5) \). \textit{M. Reni} [J. Lond. Math. Soc., II. Ser. 63, No. 1, 226-246 (2001; Zbl 1012.57025)] and \textit{B. Zimmermann} [Monatsh. Math. 135, No. 3, 253-258 (2002; Zbl 0994.57019)] showed that a finite group \(G\) of orientation-preserving diffeomorphisms of a homology 3-sphere is either solvable or it has a unique composition factor which is isomorphic to \(\text{PSL}(2,q)\) for \(q\) an odd prime. In this paper the author shows: For a prime \(p\geq 5\), the group \(G=\text{PSL}(2,p)\) acts as a group of diffeomorphisms of homology 3-spheres if and only if \(p=5\), i.e. \(G\) is isomorphic to the dodecahedral group \(\mathbf A_5\).
Mathematics Subject Classification ID: 57M60
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 57S17
zbMATH DE Number: 1879119
zbMATH Keywords: homology 3-sphere
zbMATH Keywords: finite group action
zbMATH Keywords: simple group
author: Bruno Zimmermann
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1879119