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

Lin, Jianfeng¹

¹ Yau Mathematical Sciences Center, Tsinghua University, Beijing, China

Geometry & topology, Tome 27 (2023) no. 5, pp. 1987-2012.

Résumé

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 $\mathrm{Pin}(2)$–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.

Détail

DOI : 10.2140/gt.2023.27.1987

Keywords: 4–manifolds, Bauer–Furuta invariant, exotic phenomena, stabilization

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