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

The only finite non-Abelian simple group acting on a homology 3-sphere—necessarily 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 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-sphere.