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

摘要

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.