On the classification of finite groups acting on homology 2-spheres
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.
scientific article; zbMATH DE number 2164521
scientific article