On finite groups acting on \(\mathbb Z_2\)-homology 3-spheres
type: Item
instance of: scholarly article
title: On finite groups acting on \(\mathbb Z_2\)-homology 3-spheres
zbMATH Open document ID: 1063.57017
author: Mattia Mecchia
author: Bruno Zimmermann
published in: Mathematische Zeitschrift
publication date: 2005-02-02T00:00:00Z
review text: The authors obtain information about orientation-preserving actions of finite groups on integral homology \(3\)-spheres and \(\mathbb{Z}/2\)-homology \(3\)-spheres. The latter is an intermediate case between rational homology \(3\)-spheres, for which any finite group acts on some such manifold, and integral homology \(3\)-spheres, on which an extremely restricted class can act. The first main result says that if \(G\) acts on a \(\mathbb{Z}/2\)-homology \(3\)-sphere, then \(G\) can be obtained as follows: let \(\text{O}(G)\) be the maximal normal subgroup of \(G\) of odd order, then either \(G\) is solvable, or \(G/\text{O}(G)\) has a normal subgroup, with solvable factor group, isomorphic to one of \(\text{PSL}(2,q)\), \(\text{PSL}(2,q)\times\mathbb{Z}/2\), \(\text{SL}(2,q)\times_{\mathbb{Z}/2} C\), \(\text{SL}(2,q)\times_{\mathbb{Z}/2}\text{SL}(2,q')\), or \(\widehat{A}_7\), where \(C\) is solvable with a unique involution, \(q\) and \(q'\) are odd powers greater than four, and \(\widehat{A}_7\) is the unique perfect extension of the alternating group \(A_7\). Specializing the arguments for the first main result gives the second: a nonsolvable group acting on an integral homology \(3\)-sphere must be either \(A_5\), \(A_5\times \mathbb{Z}/2\), \(A_5^*\times_{\mathbb{Z}/2} C\), or \(A_5^*\times_{\mathbb{Z}/2} A_5^*\), where \(C\) is as before. There are additional restrictions, for example in the latter case the groups \(C\) must act freely, so must have periodic cohomology of period~\(4\) and appear on J. Milnor's well-known list of candidates to act freely. In neither case is it known that all groups of these forms do act. The authors have the impression that every group \(\text{SL}(2,q)\) and \(\text{PSL}(2,q)\) acts on some \(\mathbb{Z}/2\)-homology \(3\)-sphere, and have produced numerous examples of such actions. The proof of these results proceeds by using topological information to obtain various group-theoretic restrictions on \(G\), and then utilizing an …
reviewed by: Darryl McCullough
Mathematics Subject Classification ID: 57M60
zbMATH DE Number: 2131999
zbMATH Keywords: group
zbMATH Keywords: action
zbMATH Keywords: 3-manifold
zbMATH Keywords: sphere
zbMATH Keywords: homology sphere
zbMATH Keywords: rational homology sphere
zbMATH Keywords: \(\mathbb Z/2\) homology sphere
MaRDI profile type: Publication
full work available at URL: s00209-004-0672-x
OpenAlex ID: W2105872670
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DOI: 10.1007/S00209-004-0672-X
description: scientific article; zbMATH DE number 2131999