Lagrangian spheres and cyclic quotient T-singularities

Buck, Matthew and Evans, Jonny (2025) Lagrangian spheres and cyclic quotient T-singularities. PhD thesis, Lancaster University.

Text (2025buckphd)
2025buckphd.pdf - Published Version
Available under License Creative Commons Attribution.
Download (876kB)

Abstract

We study the Lagrangian isotopy classification of Lagrangian spheres in the Milnor fibre, Bd,p,q, of the cyclic quotient surface T-singularity 1/dp2(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 that admit global genus 0 Lefschetz fibrations, which Bd,p,q does not. 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*S2. Our calculations provide evidence towards the symplectic mapping class group of Bd,p,q being generated by Lagrangian sphere Dehn twists and another type of symplectomorphism arising as the monodromy of the 1/p2(1,pq-1) singularity.