Lagrangian Non-Intersections

TL;DR

This paper employs Floer theory to explore topological constraints on Lagrangian embeddings, revealing homological rigidity and new insights into Hamiltonian displacement and intersection phenomena in symplectic manifolds.

Contribution

It introduces novel methods for analyzing Lagrangian embeddings, demonstrating conditions under which their topology is rigid and establishing new results on Lagrangian intersections.

Findings

Homological rigidity of Lagrangian submanifolds in certain symplectic manifolds.

Conditions on low-dimensional invariants determine entire Lagrangian homology.

New results on Lagrangian intersection properties.

Abstract

In this paper we use Floer theory to study topological restrictions on Lagrangian embeddings in closed symplectic manifolds. One of the phenomena arising from our results is ``homological rigidity'' of Lagrangian submanifolds. Namely, in certain symplectic manifolds, conditions on low dimensional topological invariants of a Lagrangian (such as its first homology) completely determine its entire homology. We also develop methods for studying Hamiltonian displacement of Lagrangian submanifolds and its relations to the topology of Lagrangians. Finally we present some new results on Lagrangian intersections.