On the classification of finite groups acting on homology 2-spheres
type: Item
instance of: scholarly article
title: On the classification of finite groups acting on homology 2-spheres
zbMATH Open document ID: 1068.57013
published in: Pacific Journal of Mathematics
publication date: 2005-05-03T00:00:00Z
review text: In this paper, the author obtains information about orientation-preserving actions of nonsolvable finite groups on integral homology \(3\)-spheres. The main result is that they must be isomorphic to one of the groups \(\mathbb{A}_5\), \(\mathbb{A}_5\times \mathbb{Z}_2\), \(\mathbb{A}_5^*\times_Z \mathbb{A}_5^*\), \(\mathbb{A}_5^*\times_Z \mathbb{S}_4^*\), \(\mathbb{A}_5^*\times_Z \mathbb{A}_4^*\), \(\mathbb{A}_5^*\times_Z \mathbb{D}_{4n}^*\), \(\mathbb{A}_5^*\times_Z \mathbb{Z}_{2n}^*\), \(\mathbb{A}_5^*\times_Z Q(8n,k,l)\), or \(\mathbb{A}_5^*\times_Z (\mathbb{D}_{4n}^*\times \mathbb{Z}_k)\) (with restrictions on \(n\), \(k\), and \(\ell\) in some of the cases). Here, \(G\times_ZH\) means the central product obtained by identifying the centers of the two groups. In all cases \(Z\cong \mathbb{Z}_2\), and except for the first two cases, \(G\) and \(H\) appear in Milnor's well-known list of groups with periodic cohomology of period four and satisfying certain other necessary conditions to act freely on a homology \(3\)-sphere. In most but not all cases of the main result, it is known that the group does act on some homology \(3\)-sphere. The proof proceeds by using topological information to obtain various group-theoretic restrictions on \(G\), and then utilizing the theory of finite groups to find all groups satisfying them. The paper continues earlier work of \textit{M. Reni} [J. London Math. Soc. (2) 63, No.1, 226--246 (2001; Zbl 1012.57025)]. In further work with M. Mecchia, [\textit{M. Mecchia} and \textit{B. Zimmermann}, Math. Z. 248, No. 4, 675--693 (2004; Zbl 1063.57017)], the author has obtained similar results for actions on \(\mathbb{Z}_2\)-homology \(3\)-spheres. For that more general case, the list of groups that can act is quite a bit longer.
reviewed by: Darryl McCullough
Mathematics Subject Classification ID: 57M60
Mathematics Subject Classification ID: 57S25
zbMATH DE Number: 2164521
zbMATH Keywords: group
zbMATH Keywords: simple
zbMATH Keywords: non-solvable
zbMATH Keywords: action
zbMATH Keywords: 3-manifold
zbMATH Keywords: sphere
zbMATH Keywords: homology sphere
author: Bruno Zimmermann
MaRDI profile type: Publication
full work available at URL: pjm.2004.217.387
OpenAlex ID: W1987552857
DOI: 10.2140/PJM.2004.217.387
Recommended article: On finite groups acting on \(\mathbb Z_2\)-homology 3-spheres
Recommended article: On finite simple and nonsolvable groups acting on homology 4-spheres
Recommended article: Some results and conjectures on finite groups acting on homology spheres
Recommended article: Finite groups acting on homology 3-spheres: On a result of M. Reni
Recommended article: Q5387274
Recommended article: On finite groups acting on homology 4-spheres and finite subgroups of SO(5)
Recommended article: Q3334647
Recommended article: FREE FINITE GROUP ACTIONS ON RATIONAL HOMOLOGY 3-SPHERES
Recommended article: On finite simple groups acting on integer and mod 2 homology 3-spheres
Recommended article: On finite simple groups acting on homology spheres with small fixed point sets
description: scientific article; zbMATH DE number 2164521