Lagrangian Spheres and Cyclic Quotient T-singularities

Abstract

We study the Lagrangian isotopy classification of Lagrangian spheres in the Milnor fibre, $B_{d,p,q}$, of the cyclic quotient surface T-singularity $\frac{1}{dp^2} (1,dpq-1)$. We prove that there is a finitely generated group of symplectomorphisms such that the orbit of a fixed Lagrangian sphere exhausts the set of Lagrangian isotopy classes. Previous classifications of Lagrangian spheres have been established in simpler symplectic $4$-manifolds, being either monotone, or simply connected, whilst the family studied here satisfies neither of these properties. We construct Lefschetz fibrations for which the Lagrangian spheres are isotopic to matching cycles, which reduces the problem to a computation involving the mapping class group of a surface. These fibrations are constructed using the techniques of $J$-holomorphic curves and Symplectic Field Theory, culminating in the construction of a $J$-holomorphic foliation by cylinders of $T^*S^2$. Our calculations provide evidence towards the symplectic mapping class group of $B_{d,p,q}$ being generated by Lagrangian sphere Dehn twists and another type of symplectomorphism arising as the monodromy of the $\frac{1}{p^2}(1,pq-1)$ singularity.