Constraints on Lefschetz fibrations with four-dimensional fibers from Seiberg-Witten theory

Abstract

We establish constraints on the topology of smooth Lefschetz fibrations with $4$-dimensional fibers, by studying the family Bauer-Furuta invariant. To compute this invariant, we analyze the framed bordism class of 1-dimensional Seiberg-Witten moduli spaces using the local index theorem by Bismut-Freed. Using this, we deduce new obstructions to the smooth isotopy to the identity for compositions of Dehn twists on $(-2)$-spheres in closed $4$-manifolds. We obtain several applications: (1) We exhibit the first examples of closed simply-connected symplectic $4$-manifolds admitting Torelli symplectomorphisms which are smoothly non-trivial. In particular, their symplectic Torelli mapping class group is not generated by squared Dehn-Seidel twists on Lagrangian spheres -- providing a negative answer to a question of Donaldson. (2) We provide the first examples of irreducible closed $4$-manifolds (both symplectic and non-symplectic) that admit exotic diffeomorphisms given by Seifert-fibered Dehn twist.