On the cluster category of a marked surface without punctures

Abstract

We study the cluster category l(S,M) of a marked surface (S,M) without punctures. We explicitly describe the objects in l(S,M) as direct sums of homotopy classes of curves in.(S,M) and one-parameter families related to noncontractible closed curves in.(S,M) Moreover, we describe the Auslander-Reiten structure of the category l(S,M) in geometric terms and show that the objects without self-extensions in l(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. © 2011 by Mathematical Sciences Publishers.