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

We prove that the boundary Dehn twist on $K3\mathrm{\#}K3\setminus B^4$ 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 $\text{Pintwo}$-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 $K3\mathrm{\#}K3\text{into}E\downarrow S^2$ has $w_2(T^{v}E)=0$, so $E$ is spin.