A note on the Nielsen realization problem for connected sums of S² X S¹

Abstract

We consider finite group-actions on 3-manifolds Hg obtained as the connected sum of g copies of S² × S¹, with free funda- mental group Fg of rank g. We prove that, for g > 1, a finite group of diffeomorphisms of Hg inducing a trivial action on homology is cyclic and embeds into an S¹-action on Hg. As a consequence, no nontrivial element of the twist subgroup of the mapping class group of Hg (gen-erated by Dehn twists along embedded 2-spheres) can be realized by a periodic diffeomorphism of Hg (in the sense of the Nielsen realization problem). We also discuss when a finite subgroup of the outer automor- phism group Out(Fg) of the fundamental group of Hg can be realized by a group of diffeomorphisms of Hg.