Any finite group acts freely and homologically trivially on a product of   spheres
  The main theorem is that if K is a finite CW complex with finite fundamental group G and universal cover homotopy equivalent to a product of spheres X, then G acts smoothly and freely on X x S^n for any n greater than or equal to the dimension of X. If the G-action on the universal cover of K is homologically trivial then so is the action on X x S^n.   Unlu and Yalcin recently showed that for every finite group G, there is a finite CW complex K with fundamental group G which acts homologicially trivially on the universal cover of K.   Thus every finite group acts smoothly, freely, and homologically trivially on a product of spheres. 
James F. Davis
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