2025/09/23 by Matthew R. Buck, Buck, Matthew R.
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Advanced Topics in Algebra #FOS: Mathematics #Nonlinear Waves and Solitons #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2509.18976
openalex publication_date 2025/09/23 · openalex created_date 2025/10/16 · openalex updated_date 2026/07/28
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, 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^*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.