On finite simple and nonsolvable groups acting on homology 4-spheres

Abstract

The only finite nonabelian simple group acting on a homology 3-spherenecessarily non-freely -is the dodecahedral group A 5 ∼ = PSL(2, 5) (in analogy, the only finite perfect group acting freely on a homology 3-sphere is the binary dodecahedral group A * 5 ∼ = SL(2, 5)). In the present paper we show that the only finite simple groups acting on a homology 4-sphere, and in particular on the 4-sphere, are the alternating or linear fractional groups groups A 5 ∼ = PSL(2, 5) and A 6 ∼ = PSL (2, 9). From this we deduce a short list of groups which contains all finite nonsolvable groups admitting an action on a homology 4-spheres.1991 Mathematics Subject Classification. 57M60, 57S17, 57S25

On finite groups acting on acyclic low-dimensional manifolds

“…By Lemma 1, every 2-subgroup of G is in particular a subgroup of SO (5) and it is therefore generated by at most 4 elements, (e.g. by [MeZ2,Proposition 3.1. ] Proof.…”

“…The subgroup A 4 × Q 2 contains a normal elementary 2-group of rank two; by Lemma 3 the group A 4 × Q 2 should be a subgroup of SO (5) which is not the case (e.g. by [MeZ2]). …”

“…the Smith conjecture does not remain true for the 4-sphere, that is the fixed point set of a periodic diffeomorphism of S 4 may be a knotted 2-sphere). However it has been shown in [MeZ2] and [MeZ3] that a finite group which admits an orientation-preserving action on the 4-sphere, and more generally on any homology 4-sphere, is isomorphic to a subgroup of the orthogonal group SO(5) (up to 2-fold extensions in the case of solvable groups).…”

Finite isometry groups of 4-manifolds with positive sectional curvature

“…By [16] a finite group which acts pseudofreely on S 2n is either cyclic or a dihedral group. In [21] non-solvable groups acting on homology 4-spheres are listed. But it seems a quite difficult problem to classify all finite groups acting smoothly on S 4 , in particular, to study whether the finite groups are subgroups of O(5).…”

“…By [21] G must be a solvable group since |G| is odd. Hence G is obtained from {1} by a finite number of extensions with abelian kernels.…”