Folded Morse flow trees

TL;DR

This paper develops a novel Morse theory framework on symmetric products of surfaces using folded ribbon trees, leading to an $A_$-category with connections to Hecke algebras and wrapped cotangent fibers.

Contribution

It introduces a new approach to Morse theory on symmetric products via folded ribbon trees and constructs an $A_$-category with objects as $$-tuples of Morse functions, linking to Hecke algebras.

Findings

The $A_$-category's endomorphism algebra is the Hecke algebra of the symmetric group.

When the differential graph corresponds to wrapped cotangent fibers, the endomorphism matches the Hecke algebra.

The approach provides a new perspective on Morse theory and symplectic topology of symmetric products.

Abstract

We present an approach to Morse theory on symmetric products of surfaces using the notion of folded ribbon trees. We introduce an $A_\infty$-category with objects defined as $\kappa$-tuples of Morse functions, where the differential of the tuple has no self-intersection. We show that when the graph of the differential of the $\kappa$-tuple of Morse functions on $T^*\mathbb{R}^2$ is the wrapped $\kappa$ disjoint cotangent fibers, its endormorphism is the Hecke algebra associated to the symmetric group $\mathfrak{S}_\kappa$.