On the Cluster Category of a Marked Surface

TL;DR

This paper explores the cluster category of a marked surface, describing its objects geometrically and linking algebraic properties to topological features, thus advancing understanding of cluster categories in geometric terms.

Contribution

It provides an explicit geometric description of objects in the cluster category of a marked surface and relates algebraic structures to topological features.

Findings

Objects correspond to homotopy classes of curves and closed curves

Objects without self-extensions are curves without self-intersections

Every rigid indecomposable object is reachable from an initial triangulation

Abstract

We study in this paper the cluster category C(S,M) of a marked surface (S,M). We explicitly describe the objects in C(S,M) as direct sums of homotopy classes of curves in (S,M) and one-parameter families related to closed curves in (S,M). Moreover, we describe the Auslander-Reiten structure of the category C(S,M) in geometric terms and show that the objects without self-extensions in C(S,M) correspond to curves in (S,M) without self-intersections. As a consequence, we establish that every rigid indecomposable object is reachable from an initial triangulation.