Spherical Lagrangians via ball packings and symplectic cutting

TL;DR

This paper establishes the connectedness of symplectic ball packings around spherical Lagrangians in certain symplectic manifolds, extending previous results and enabling applications in knot theory, group actions, and Floer theory.

Contribution

It introduces a symplectic cutting approach to prove connectedness of ball packings near spherical Lagrangians, extending McDuff's results and facilitating new applications.

Findings

Connectedness of symplectic ball packings around S^2 and RP^2 Lagrangians.

Applications to knotting and unknottedness of Lagrangians.

Implications for symplectic Torelli group actions and Floer theory.

Abstract

In this paper we prove the connectedness of symplectic ball packings in the complement of a spherical Lagrangian, S^2 or RP^2, in symplectic manifolds that are rational or ruled. Via a symplectic cutting construction this is a natural extension of McDuff's connectedness of ball packings in other settings and this result has applications to several different questions: smooth knotting and unknottedness results for spherical Lagrangians, the transitivity of the action of the symplectic Torelli group, classifying Lagrangian isotopy classes in the presence of knotting, and detecting Floer-theoretically essential Lagrangian tori in the del Pezzo surfaces.