Nielsen realization in dimension four and projective twists

Abstract

We demonstrate the existence of numerous non-spin 4-manifolds for which the smooth Nielsen realization problem fails; namely, there exist finite subgroups of their mapping class groups that cannot be realized by any group of diffeomorphisms. This extends and complements recent results for spin 4-manifolds. Our examples span virtually all possible intersection forms, both even and odd, indefinite and definite, and include many irreducible 4-manifolds. To derive these examples, we study multi-twists, projective twists, and multi-reflections, which are all mapping classes supported around collections of embedded spheres and projective planes. Our obstructions to Nielsen realization are based on the work of Konno. We investigate projective twists in further detail, and notably, employ them to show that, for many closed symplectic 4-manifolds, the symplectic Torelli group is not generated by squared Dehn twists.