The Dehn twist on a connected sum of two homology tori

TL;DR

This paper extends the understanding of Dehn twists in connected sums of homology tori by developing a refined Pin(2)-equivariant invariant, demonstrating non-triviality of certain Dehn twists in broader contexts.

Contribution

It generalizes the Pin(2)-equivariant family Bauer-Furuta invariant to non-simply connected manifolds and applies it to show non-trivial Dehn twists in connected sums of homology tori.

Findings

Dehn twist along a 3-sphere in the neck of connected sums of homology tori is not smoothly isotopic to the identity.

Refined Pin(2)-equivariant invariant constructed for non-simply connected manifolds.

Non-triviality result holds when determinants of the homology tori are odd.

Abstract

Kronheimer-Mrowka shows that the Dehn twist along a $3$-sphere in the neck of two $K3$ surfaces is not smoothly isotopic to the identity. Their result requires that the manifolds are simply connected and the signature of one of them is $16 \mod 32$. We generalize the Pin$(2)$-equivariant family Bauer-Furuta invariant to nonsimply connected manifolds, and construct a refinement of this invariant. We use it to show that, if $X_1,X_2$ are two homology tori such that the determinants $r_1,r_2$ of them are odd, then the Dehn twist along a $3$-sphere in the neck of $X_1\# X_2$ is not smoothly isotopic to the identity.