Isotopy of the Dehn twist on K3#K3 after a single stabilization

Abstract

Kronheimer-Mrowka recently proved that the Dehn twist along a 3-sphere in the neck of $K3\#K3$ is not smoothly isotopic to the identity. This provides a new example of self-diffeomorphisms on 4-manifolds that are isotopic to the identity in the topological category but not smoothly so. (The first such examples were given by Ruberman.) In this paper, we use the Pin(2)-equivariant Bauer-Furuta invariant to show that this Dehn twist is not smoothly isotopic to the identity even after a single stabilization (connected summing with the identity map on $S^{2}\times S^{2}$). This gives the first example of exotic phenomena on simply connected smooth 4-manifolds that do not disappear after a single stabilization.Comment: 19 pages. Version 2:Added the reference to Gompf and Kreck's theorem and the reference to Szymik's work. Version 3: corrected several typos. Version 4: Added a few references. Version 5: corrected a few typo