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

TL;DR

This paper demonstrates that a specific Dehn twist on a connected sum of K3 surfaces remains non-isotopic to the identity even after stabilization, revealing new exotic phenomena in smooth 4-manifold topology.

Contribution

It shows the Dehn twist on K3#K3 is not smoothly isotopic to the identity after stabilization, using the Pin(2)-equivariant Bauer-Furuta invariant, a novel application in 4-manifold theory.

Findings

Dehn twist on K3#K3 is not smoothly isotopic to identity after stabilization.

First example of exotic phenomena persisting after a single stabilization.

Utilizes Pin(2)-equivariant Bauer-Furuta invariant for proof.

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.