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.

It is a classical problem to study whether the h-principle holds for certain classes of maximally non-integrable distributions. The most studied case is that of contact structures, where there is a rich interplay between flexibility and rigidity, exemplified by the overtwisted vs tight dichotomy. For other types of maximally non-integrable distributions, no examples of rigidity are currently known.

In this talk I will discuss rigidity phenomena for fat distributions, which can be viewed as higher corank generalizations of contact structures. These admit natural symplectizations and contactizations. I will introduce a natural class of submanifolds in fat manifolds, called prelegendrians, which admit canonical Legendrian lifts to the contactization. The main result of the talk is that these submanifolds exhibit rigidity: in the “standard corank-2 fat manifold” there exists an infinite family of prelegendrian tori, all of them formally equivalent but pairwise not prelegendrian isotopic. In other words, the h-principle fails for prelegendrians. The talk is based on joint work with Álvaro del Pino and Wei Zhou.
 

A knot is slice if it bounds a smoothly embedded disk in the four-ball and a knot is ribbon if it bounds such a disk with no local maxima. The slice-ribbon conjecture posits that every slice knot is ribbon. We prove that a linear combination of iterated cables of tight fibered knots is ribbon if and only if it is of the form K # -K, generalizing work of Miyazaki and Baker. Consequently, either iterated cables of tight fibered knots are linearly independent in the smooth concordance group, or the slice–ribbon conjecture fails.