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

Abstract

Isotopy of the Dehn twist on K 3 # K 3 after a single stabilization JIANFENG LINKronheimer and 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.) 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 S 2 ). This gives the first example of exotic phenomena on simply connected smooth 4-manifolds that do not disappear after a single stabilization.

Symplectic mapping class groups of blowups of tori

“…We refer to [23,24] for a comprehensive introduction to ordinary Seiberg-Witten invariants and to [19] for a detailed account of their family version. Some recent applications of family SW-invariants can be found in [4,16,21].…”