Exotic Dehn twists on sums of 4-manifolds, nuclei, and exotic structures

Abstract

For closed 4-manifolds $X_1, X_2$, we give a sufficient condition, in terms of the Seiberg-Witten invariants of $X_i$, for the Dehn twist on $X_1\#X_2$ along the neck to be smoothly nontrivial. This provides a broad generalization of the result of Kronheimer-Mrowka for $K3\#K3$. We also show that the boundary Dehn twists on the punctured connected sums of many pairs of elliptic surfaces are smoothly nontrivial. We also detect new examples of exotic Dehn twists along Seifert fibered 3-manifolds in elliptic surfaces. As a consequence, we exhibit a new phenomenon in which "exotic diffeomorphisms detect exotic smooth structures on 4-manifolds": the triviality of the boundary Dehn twist characterizes the standard nucleus $N(n)$ among the nuclei $N(n)_{p,q}$ homeomorphic to it, provided that $n\equiv0 \bmod 4$. These results are proved by computing the families Bauer-Furuta invariant for a new framing of the families Dirac index.