An example of higher-dimensional Heegaard Floer homology

TL;DR

This paper explores higher-dimensional Heegaard Floer homology by counting pseudoholomorphic curves, revealing an algebraic structure isomorphic to the Hecke algebra of the symmetric group, thus connecting geometric and algebraic frameworks.

Contribution

It introduces a novel approach to higher-dimensional Heegaard Floer homology by establishing an isomorphism with the Hecke algebra, linking geometric counts to algebraic structures.

Findings

The algebra from counting pseudoholomorphic curves is isomorphic to the Hecke algebra.

Establishes a connection between higher-dimensional Floer homology and symmetric group algebra.

Provides a new perspective on the algebraic structures underlying Floer homology.

Abstract

We count pseudoholomorphic curves in the higher-dimensional Heegaard Floer homology of disjoint cotangent fibers of a two dimensional disk. We show that the resulting algebra is isomorphic to the Hecke algebra associated to the symmetric group.