On finite simple groups acting on integer and mod 2 homology 3-spheres

Abstract

We prove that the only finite non-abelian simple groups G which possibly admit an action on a Z 2 -homology 3-sphere are the linear fractional groups PSL ( 2 , q ) , for an odd prime power q (and the dodecahedral group A 5 ≅ PSL ( 2 , 5 ) in the case of an integer homology 3-sphere), by showing that G has dihedral Sylow 2-subgroups and applying the Gorenstein–Walter classification of such groups. We also discuss the minimal dimension of a homology sphere on which a linear fractional group PSL ( 2 , q ) acts.