Group actions on homology spheres

This can be stated in a more symmetric manner. Let r be any positive integer not equal to 3. Then n acts freely and homologically trivially on Z r i ff n acts freely and homologically trivially on SL In fact, there is a one-to-one correspondence between such actions on U and such actions on S r. (The classification of such actions is discussed in w In addition the actions constructed have the property that every group element is isotopic to the identity. Theorem A holds in the topological or piecewise-linear categories. Well-known counterexamples exist in the smooth category even for homotopy spheres. Theorem A would be false without the assumption of homological triviality or without the assumption of simple-connectivity (see remark 8.8). In fact, there exist groups n and simply-connected Z/In[ homology spheres Z r, such that n acts freely on U, but not on SL Also there are groups n which act freely on S r, but do not act freely on certain lens spaces Lp, with p prime to In[. If the group n is cyclic, a result similar to our main theorem was first proved by L~Sffler [L] by different methods. The second author independently established the cyclic case in [Wel ] , by a method that extends to cyclic group actions on more general manifolds (see [C-W1]), and, via this paper, to general finite groups. However the difficulties in making this extension have application elsewhere. The main theorem is motivated by the philosophy that all of the geometry of a group action is present at the order of the group. Thus, it is only natural to expect that any manifold "homologically resembling" a sphere admitting a