Configurations of Lagrangian spheres in $K3$ surfaces

TL;DR

This paper investigates the algebraic properties of Dehn--Seidel twists on Lagrangian spheres in symplectic K3 surfaces, revealing faithfulness of certain representations and independence of twists, with implications for symplectic topology.

Contribution

It proves the faithfulness of a Braid group representation and algebraic independence of squared Dehn--Seidel twists in symplectic K3 surfaces, extending to fundamental groups of symplectic form spaces.

Findings

Representation of the Braid group is faithful after abelianisation.

Squared Dehn--Seidel twists are algebraically independent.

Results extend to fundamental groups of symplectic form spaces.

Abstract

We study Dehn--Seidel twists on configurations of Lagrangian spheres in symplectic $K3$ surfaces, using tools from Seiberg--Witten theory. In the case of $ADE$ configurations of Lagrangian spheres, we prove that a naturally associated representation of the generalised Braid group in the symplectic mapping class group is always faithful after abelianising, in a suitable sense. More generally, we prove that squared Dehn--Seidel twists on homologically-distinct Lagrangian spheres are algebraically independent in the abelianisation of the smoothly-trivial symplectic mapping class group, and deduce from this new infinite-generation results. Beyond symplectic $K3$ surfaces, we also establish analogues of these results at the level of the fundamental group of the space of symplectic forms.