##### Department of Mathematics,

University of California San Diego

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### Math 292 - Topology Seminar

## Jianfeng Lin

#### UC San Diego

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

##### 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 talk, we study the Bauer-Furuta invariant as an element in the Pin(2)-equivariant stable homotopy group of spheres. We use it 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 cross S2). This gives the first example of exotic phenomena on simply-connected smooth 4-manifolds that do not disappear after a single stabilization. In particular, it implies that one stabilization is not enough in the diffeomorphism isotopy problem for 4-manifolds. It gives an interesting comparison with Auckly-Kim-Melvin-Ruberman-Schwartz's theorem that one stabilization is enough in the surface isotopy problem.

Host: Zhouli Xu

### January 12, 2021

### 10:30 AM

### Zoom information: Meeting ID: 933 6734 4286 Password: topology

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