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

Lin, Jianfeng

doi:10.2140/gt.2023.27.1987

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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

DOI: 10.2140/gt.2023.27.1987

Abstract

Kronheimer and Mrowka recently proved that the Dehn twist along a $3$ –sphere in the neck of $K 3 # K 3$ 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 $S^{2} \times S^{2}$ ). This gives the first example of exotic phenomena on simply connected smooth $4$ –manifolds that do not disappear after a single stabilization.

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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Volume 27, issue 5 (2023)