On finite groups acting on acyclic low-dimensional manifolds

TL;DR

This paper classifies finite groups acting smoothly on acyclic manifolds of dimensions three to five, showing they are closely related to classical orthogonal groups and identifying specific simple groups involved in five-dimensional cases.

Contribution

It proves that finite groups acting on acyclic 3- and 4-manifolds are subgroups of orthogonal groups, and identifies the only nonabelian simple groups acting on acyclic 5-manifolds.

Findings

Finite groups on acyclic 3- and 4-manifolds are subgroups of O(3) and O(4).

Only A_5 and A_6 act smoothly on acyclic 5-manifolds among nonabelian simple groups.

Candidates for orientation-preserving actions on acyclic 5-manifolds are closely related to subgroups of SO(5).

Abstract

We consider finite groups which admit a faithful, smooth action on an acyclic manifold of dimension three, four or five (e.g. euclidean space). Our first main result states that a finite group acting on an acyclic 3- or 4-manifold is isomorphic to a subgroup of the orthogonal group O(3) or O(4), respectively. The analogue remains open in dimension five (where it is not true for arbitrary continuous actions, however). We prove that the only finite nonabelian simple groups admitting a smooth action on an acyclic 5-manifold are the alternating groups A_5 and A_6, and deduce from this a short list of finite groups, closely related to the finite subgroups of SO(5), which are the candidates for orientation-preserving actions on acyclic 5-manifolds.