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Isotopy of the Dehn twist on $K3\mathbin{\#} K3$ after a single stabilization

Jianfeng Lin

Geometry & Topology 27 (2023) 1987–2012
Abstract

Kronheimer 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 S2 × S2). This gives the first example of exotic phenomena on simply connected smooth 4–manifolds that do not disappear after a single stabilization.

Keywords
4–manifolds, Bauer–Furuta invariant, exotic phenomena, stabilization
Mathematical Subject Classification
Primary: 57R50, 57R52, 57R57
Secondary: 55P91
References
Publication
Received: 13 December 2020
Revised: 27 October 2021
Accepted: 3 January 2022
Published: 27 July 2023
Proposed: András I Stipsicz
Seconded: Tomasz S Mrowka, Gang Tian
Authors
Jianfeng Lin
Yau Mathematical Sciences Center
Tsinghua University
Beijing
China

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