Boundary Dehn twists on Milnor fibers and Family Bauer--Furuta invariants

TL;DR

This paper investigates the properties of boundary Dehn twists on Milnor fibers, demonstrating they are exotic diffeomorphisms under specific conditions by analyzing family Bauer--Furuta invariants.

Contribution

It establishes new conditions under which boundary Dehn twists on Milnor fibers are exotic diffeomorphisms using Bauer--Furuta invariants.

Findings

Boundary Dehn twist is an exotic diffeomorphism for certain Milnor fibers.

Conditions involve odd, coprime integers q, r greater than 3, with (q-1)(r-1)/4 odd.

Comparison of family Bauer--Furuta invariants proves the main result.

Abstract

We proved that the boundary Dehn twist on the Milnor fiber $M_c(2, q, r)$ is an exotic diffeomorphism relative to the boundary if $q, r$ are odd, coprime integers bigger than $3$ and $(q-1)(r-1)/4$ is an odd number. The proof is given by comparing the family relative Bauer--Furuta invariants of the mapping torus.