Non-kissing and non-crossing complexes for locally gentle algebras

TL;DR

This paper introduces non-kissing and non-crossing complexes for locally gentle algebras, establishing their geometric and combinatorial properties and connecting them to surface dissections and mutation theories.

Contribution

It constructs and proves the isomorphism of three complexes associated with locally gentle algebras and relates them to surface dissections and mutation frameworks.

Findings

All three complexes are isomorphic, pure, and thin.

Established inverse bijections between locally gentle algebras and surface dissections.

Defined mutation operations analogous to τ-tilting mutation.

Abstract

Starting from a locally gentle bound quiver, we define on the one hand a simplicial complex, called the non-kissing complex. On the other hand, we construct a punctured, marked, oriented surface with boundary, endowed with a pair of dual dissections. From those geometric data, we define two simplicial complexes: the accordion complex, and the slalom complex, generalizing work of A. Garver and T. McConville in the case of a disk. We show that all three simplicial complexes are isomorphic, and that they are pure and thin. In particular, there is a notion of mutation on their facets, akin to $\tau$-tilting mutation. Along the way, we also construct inverse bijections between the set of isomorphism classes of locally gentle bound quivers and the set of homeomorphism classes of punctured, marked, oriented surfaces with boundary, endowed with a pair of dual dissections.