Georgia Institute of Technology College of Sciences

On a closed, simply-connected, symplectic 4-manifold, the Dehn–Seidel twists on Lagrangian spheres and their products provide all known examples of non-trivial elements in the symplectic mapping class group. However, little is known in general about the relations that may hold among Dehn–Seidel twists. 

I will discuss the following result: on a symplectic K3 surface, the squared Dehn--Seidel twists on Lagrangian spheres with distinct fundamental classes are algebraically independent in the abelianization of the (smoothly-trivial) symplectic mapping class group. In a particular case, this establishes an abelianized form of a Conjecture by Seidel and Thomas on the faithfulness of certain Braid group representations in the symplectic mapping class group of K3 surfaces. The proof makes use of Seiberg--Witten gauge theory for families of symplectic 4-manifolds.