On the cluster category of a marked surface without punctures

Abstract

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