Nielsen realization for sphere twists of 3-manifolds
with L. Chen
Israel J. Math., to appear
For a 3-manifold M, the twist group Twist(M) is the subgroup of the mapping class group Mod(M) generated by twists about embedded 2-spheres. We study the Nielsen realization problem for subgroups of Twist(M). We prove that a group G<Twist(M) is realized by diffeomorphisms if and only if G is cyclic and M is a connected sum of lens spaces.
Correction: The remark about a counterexample to equivariant geometrization of 3-manifolds is incorrect (i.e. there is no counterexample). Note that a geometric metric on S^2 ⨉ S^1 is not necessarily a product metric. This mistake does not affect anything else in the paper. We also note that a similar mistake appears in Zimmermann’s paper “A note on the Nielsen realization problem for connected sums of $S^2\times S^1$”, which is one paper that motivated our work.