On finite groups acting on $Bbb{Z}_2$ - homology 3-spheres.

We consider finite group G admitting orientation-preserving actions on homology 3-spheres (arbitrary, i.e. not necessarily free actions), concentrating on the case of nonsolvable groups. It is known that every finite group G admits actions on rational homology 3-spheres (and even free actions). On the other hand, the class of groups admitting actions on integer homology 3-spheres is very restricted (and close to the class of finite subgroups of the orthogonal group SO(4), acting on the 3-sphere). In the present paper, we consider the intermediate case of Z_2-homology 3-spheres (i.e., with the Z_2-homology of the 3-sphere where Z_2 denote the integers mod two; we note that these occur much more frequently in 3-dimensional topology than the integer ones). Our main result is a list of finite nonsolvable groups G which are the candidates for orientation-preserving actions on Z_2-homology 3- spheres. From this we deduce a corresponding list for the case of integer homology 3-spheres. In the integer case, the groups of the list are closely related to the dodecahedral group A_5 isomorphic to PSL(2, 5) or the binary dodecahedral group SL(2, 5); most of these groups are subgroups of the orthogonal group SO(4) and hence admit actions on S^3. Roughly, in the case of Z_2-homology 3-spheres the groups PSL(2, 5) and SL(2, 5) get replaced by the groups PSL(2, q) and SL(2, q), for an arbitrary odd prime power q. We have many examples of actions of the groups PSL(2, q) and SL(2, q) on Z_2-homology 3-spheres, for various small values of q (constructed as regular coverings of suitable hyperbolic 3-orbifolds and 3-manifolds, using computer- supported methods to calculate the homology of the coverings).We think that all of them occur but have no method to prove this at present (in particular, the exact classification of the finite nonsolvable groups admitting actions on Z_2-homology 3-spheres remains still open).