Topological mapping class groups of non-simply connected 4-manifolds

Abstract

We construct topological isotopy invariants of homeomorphisms of non-simply connected 4-manifolds using the topological Dax invariant of disks and the Freedman-Quinn invariant of spheres. Our invariants are effective for various non-smoothable 4-manifolds without assuming that the fundamental groups are good. They detect infinite-rank free abelian direct factors of the pseudoisotopy-trivial and homotopy-trivial mapping class groups, and give lower bounds on the Prüfer rank of the Torelli groups modulo homotopy-trivial mapping classes, including examples of infinite Prüfer rank. Our method is also useful for smooth 4-manifolds. We construct stably smoothable homeomorphisms of the Enriques surface and many Dolgachev surfaces that act trivially on homology but are not homotopic to the identity.