Hamiltonian handleslides for Heegaard Floer homology

TL;DR

This paper proves that handleslides of certain circles in a Riemann surface produce Hamiltonian-isotopic Lagrangian tori in the symmetric product, offering an alternative proof of Heegaard Floer homology invariance.

Contribution

It establishes Hamiltonian-isotopy of tori under handleslides in symmetric products, providing a new geometric perspective on Heegaard Floer homology invariance.

Findings

Handleslides produce Hamiltonian-isotopic Lagrangian tori.

Provides an alternative proof of Heegaard Floer homology invariance.

Connects symplectic geometry with topological invariance in Floer theory.

Abstract

A $g$-tuple of disjoint, linearly independent circles in a Riemann surface of genus $g$ determines a `Heegaard torus' in its $g$-fold symmetric product. Changing the circles by a handleslide produces a new torus. It is proved that, for symplectic forms with certain properties, these two tori are Hamiltonian-isotopic Lagrangian submanifolds. This provides an alternative route to the handleslide-invariance of Ozsvath-Szabo's Heegaard Floer homology.