Higher-dimensional Heegaard Floer homology and Hecke algebras

TL;DR

This paper constructs an algebra isomorphism between higher-dimensional Heegaard Floer homology of cotangent fibers of a surface and the associated Hecke algebra, extending to punctured surfaces.

Contribution

It introduces a new algebraic map linking Floer homology and Hecke algebras, proving it is an isomorphism for closed and punctured surfaces.

Findings

The map $$ is an algebra isomorphism.

Established analogous results for punctured surfaces.

Provides a new algebraic perspective on Floer homology.

Abstract

Given a closed oriented surface $\Sigma$ of genus greater than 0, we construct a map $\mathcal{F}$ from the higher-dimensional Heegaard Floer homology of the cotangent fibers of $T^*\Sigma$ to the Hecke algebra associated to $\Sigma$ and show that $\mathcal{F}$ is an isomorphism of algebras. We also establish analogous results for punctured surfaces.