The Boundary Dehn Twist on a Punctured Connected Sum of Two K3 Surfaces is Nontrivial in the Smooth Mapping Class Group

TL;DR

This paper demonstrates that the boundary Dehn twist on a punctured connected sum of two K3 surfaces is a nontrivial element in the smooth mapping class group, revealing an exotic diffeomorphism in a simply-connected spin four-manifold.

Contribution

It introduces an algebraic criterion using equivariant invariants to detect nontrivial boundary Dehn twists on certain 4-manifolds, expanding understanding of exotic smooth structures.

Findings

Boundary Dehn twist is nontrivial in the smooth mapping class group.

Any smooth bundle with fiber K3#K3 over S^2 is spin.

Provides algebraic tools to distinguish exotic diffeomorphisms.

Abstract

We prove that the boundary Dehn twist on $K3\#K3\setminus B^4$ is nontrivial in the smooth mapping class group, providing another example of an exotic diffeomorphism on a simply-connected spin four-manifold. We do so by finding an algebraic criterion that must be satisfied if the two maps are smoothly isotopic. The main tools involved are the $\Pintwo$-equivariant families Bauer-Furuta invariant, equivariant topological $K$-theory, and the Atiyah-Hirzebruch spectral sequence to show this algebraic criterion cannot be satisfied, and this establishes the result. As a corollary, we find any smooth bundle $K3\#K3\into E\downarrow S^2$ has $w_2(T^vE)=0$, so $E$ is spin.