Free Finite Group Actions on S by Ronnie Lee and Charles Thomas

In this paper we describe the first stages of a theory of 3-manifolds with finite fundamental group. The strong conjecture that any free finite group action on S is conjugate to a linear action is known for some cyclic groups, see [3], [4], and is supported by recent work of one of us on fundamental groups [2]. Here we concern ourselves with the weaker conjecture that any compact 3-manifold with finite fundamental group is homotopy equivalent to a Clifford-Klein form. (Note that both conjectures are phrased to avoid problems with homotopy 3-spheres.) It is known, see for example [6], that the homotopy type of such a manifold is determined by the fundamental group and the first /c-invariant. By exploiting the link between /c-invariant and finiteness obstruction we are able to decide which homotopy types correspond to finite Poincaré complexes, and thus restrict the possible homotopy types for manifolds. There are nonstandard types for some groups, and a corollary of our argument is the existence in dimensions An — 1, n ^ 2, of free actions homotopically distinct from orthogonal ones. When n = 1, we can only produce such an action on a homology sphere, and it would be most interesting to know the fundamental group.