Linking of Lagrangian Tori and Embedding Obstructions in Symplectic 4-Manifolds

TL;DR

This paper classifies certain Lagrangian tori in symplectic 4-manifolds, strengthens existing results on their isotopy classes, and explores linking and embedding obstructions based on topological and enumerative invariants.

Contribution

It provides a classification of weakly exact rational Lagrangian tori in cotangent bundles of tori and links their properties to topological data, advancing symplectic topology.

Findings

Classification of weakly exact rational Lagrangian tori up to Hamiltonian isotopy.

Strengthening of results on Lagrangian tori homologous to the zero section.

Linking properties of Lagrangian tori are determined by topological data.

Abstract

We classify weakly exact, rational Lagrangian tori in $T^* \mathbb{T}^2- 0_{\mathbb{T}^2}$ up to Hamiltonian isotopy. This result is related to the classification theory of closed $1$-forms on $\mathbb{T}^n$ and also has applications to symplectic topology. As a first corollary, we strengthen a result due independently to Eliashberg-Polterovich and to Giroux describing Lagrangian tori in $T^* \mathbb{T}^2-0_{\mathbb{T}^2}$ which are homologous to the zero section. As a second corollary, we exhibit pairs of disjoint totally real tori $K_1, K_2 \subset T^*\mathbb{T}^2$, each of which is isotopic through totally real tori to the zero section, but such that the union $K_1 \cup K_2$ is not even smoothly isotopic to a Lagrangian. In the second part of the paper, we study linking of Lagrangian tori in $(\mathbb{R}^4, \omega)$ and in rational symplectic $4$-manifolds. We prove that the linking properties of such tori are determined by purely algebro-topological data, which can often be deduced from enumerative disk counts in the monotone case. We also use this result to describe certain Lagrangian embedding obstructions.