Surgery and Invariants of Lagrangian Surfaces

TL;DR

This paper introduces a new Lagrangian surgery technique and a symplectic invariant called y-index, demonstrating their applications in distinguishing Lagrangian surfaces up to Hamiltonian isotopy.

Contribution

It presents a novel Lagrangian attaching disk surgery method and the y-index invariant, expanding tools for studying Lagrangian surface invariants and isotopy classes.

Findings

Lagrangian attaching disk surgery preserves smooth isotopy.

The y-index distinguishes Lagrangian surfaces not Hamiltonian isotopic.

New examples of nullhomologous Lagrangian tori are provided.

Abstract

We considered a surgery, called Lagrangian attaching disk surgery, that can be applied to a Lagrangian surface L at the presence of a Lagrangian attaching disk D, to obtain a new Lagrangian surface L' which is always smoothly isotopic to L. We showed that this type of surgery includes all even generalized Dehn twists as constructed by Paul Seidel. We also constructed a new symplectic invariant, called y-index, for orientable closed Lagrangian surfaces immersed in a parallelizable symplectic 4-manifold W. With y-index we proved that L and L' are not Hamiltonian isotopic. We also obtained new examples of nullhomologous Lagrangian tori which are smooth isotopic but not Hamiltonian isotopic.