Symplectic foliated fillings of sphere cotangent bundles

TL;DR

This paper classifies symplectic foliated fillings of certain foliated manifolds, showing uniqueness in the case of the sphere cotangent bundle of the Reeb foliation on the three-sphere, and describing the structure of fillings in a related example.

Contribution

It provides a classification of symplectic foliated fillings for specific foliated manifolds, introducing invariants that determine the fillings up to deformation.

Findings

Unique strong symplectic foliated filling for the sphere cotangent bundle of the Reeb foliation.

Foliated fillings of the product of a circle and an annulus are not unique and are characterized by Lefschetz fibrations.

Foliated Lefschetz fibrations are determined by combinatorial invariants from the singular locus.

Abstract

We classify symplectically foliated fillings of certain foliated manifolds with a contact structure on the leaves. We show that for the foliated sphere cotangent bundle of the Reeb foliation on the three-sphere, the corresponding foliated disk cotangent bundle is the unique strong symplectic foliated filling up to blowups and symplectic deformation equivalence. En route to the proof, we study another foliated manifold, namely the product of a circle and an annulus with an almost horizontal foliation. In this case, the foliated filling of the foliated sphere cotangent bundle is not unique. We show that any such filling is a foliated Lefschetz fibration, and is determined up to symplectic deformation equivalence, by combinatorial invariants arising from the singular locus of the Lefschetz fibration.