The Boundary Dehn Twist on a Punctured Connected Sum of Two K3 Surfaces is Nontrivial in the Smooth Mapping Class Group
Scotty Tilton
TL;DR
This work shows that the boundary Dehn twist on the punctured simply-connected spin 4-manifold $K3\#K3\setminus B^4$ is exotic, i.e., not smoothly isotopic to the identity. The authors develop a contradiction argument using the ${\mathsf{Pin}(2)}$-equivariant family Bauer-Furuta invariant, together with equivariant K-theory and the Atiyah-Hirzebruch spectral sequence, to derive an algebraic obstruction that cannot be satisfied by any hypothetical smooth isotopy. Central to the argument are detailed constructions of BF maps on mapping tori, a desuspension reduction, and explicit K-theory calculations that force a nonzero obstruction, thereby ruling out smooth isotopy to the identity. As a corollary, any smooth fiber bundle with fiber $K3\#K3$ over $S^2$ must be spin, i.e., $w_2(T^vE)=0$, revealing a rigidity phenomenon for families of such spin 4-manifolds. The results illuminate the subtle differences between smooth and topological mapping class groups in dimension four and demonstrate the effectiveness of equivariant stable homotopy-theoretic invariants in detecting 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.