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

Abstract

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 1st 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 2nd 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 2nd 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.