un 1 99 8 S 3 × S 3 acts freely on S 3 × S 3

A free action of the direct product of two copies of the symmetric group on 3 elements on the cartesian product of two copies of the 3-sphere is constructed. This nonlinear action is constructed using surgery. The action provides a counterexample to a conjecture of Lewis made in 1968. Results of P. A. Smith [10] and J. Milnor [5] show that a dihedral group cannot act freely on a sphere. This implies that if a group G acts freely on a sphere, then any element of order 2 must be central. Based on this G. Lewis [3] conjectured that if a group G acts freely on S × S, then any subgroup isomorphic to the Klein 4-group (Z2×Z2) must intersect the center non-trivially. (This condition must hold true for a free, linear action.) We will use standard techniques from surgery theory to prove the assertion in the title of this note and thus give a counterexample to Lewis’ conjecture. R. G. Swan [11] constructed a finite CW -complex X whose fundamental group is the symmetric group S3 and whose universal cover X̃ has the homotopy type of the 3-sphere S. Milnor’s result shows that X does not have the homotopy type of a closed manifold. However, there is the following well-known result, of which we sketch a proof. ∗Partially support by an NSF grant. I thank Alejandro Adem bringing this conjecture to my attention and Peter May for providing background for the proof.