On weakly exact Lagrangians in Liouville bi-fillings

TL;DR

This paper investigates the properties of weakly exact Lagrangians in Liouville bi-fillings, establishing non-trivial symplectic homology, criteria for Floer cohomology, and homotopy restrictions in specific Liouville domains.

Contribution

It introduces the notion of Liouville-Hamiltonian structures and provides new results on Floer cohomology and homotopy restrictions for Lagrangians in Liouville bi-fillings.

Findings

Symplectic homology of bi-fillings is non-trivial.

Connected Lagrangians with boundary in different components have non-vanishing wrapped Floer cohomology.

Homotopy restrictions apply to weakly exact Lagrangians in certain Liouville domains.

Abstract

Here we study several questions concerning Liouville domains that are diffeomorphic to cylinders, so called trivial bi-fillings, for which the Liouville skeleton moreover is smooth and of codimension one; we also propose the notion of a Liouville-Hamiltonian structure, which encodes the symplectic structure of a hypersurface tangent to the Liouville flow, e.g. the skeleta of certain bi-fillings. We show that the symplectic homology of a bi-filling is non-trivial, and that a connected Lagrangian inside a bi-filling whose boundary lives in different components of the contact boundary automatically has non-vanishing wrapped Floer cohomology. We also prove geometric vanishing and non-vanishing criteria for the wrapped Floer cohomology of an exact Lagrangian with disconnected cylindrical ends. Finally, we give homotopy-theoretic restrictions on the closed weakly exact Lagrangians in the McDuff and torus bundle Liouville domains.