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

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