Cotorsion pairs in the cluster category of a marked surface

TL;DR

This paper explores the relationship between curves and indecomposable objects in the cluster category of a marked surface, establishing a link between extension spaces and intersection numbers, and classifying cotorsion pairs.

Contribution

It proves the dimension of extension spaces equals intersection numbers and classifies cotorsion pairs in the cluster category of a marked surface without punctures.

Findings

Extension space dimension equals intersection number of curves

No non-trivial t-structures in connected surface categories

Classification of cotorsion pairs in these categories

Abstract

We study extension spaces, cotorsion pairs and their mutations in the cluster category of a marked surface without punctures. Under the one-to-one correspondence between the curves, valued closed curves in the marked surface and the indecomposable objects in the associated cluster category, we prove that the dimension of extension space of two indecomposable objects in the cluster categories equals to the intersection number of the corresponding curves. By using this result, we prove that there are no non-trivial $t-$structures in the cluster categories when the surface is connected. Based on this result, we give a classification of cotorsion pairs in these categories. Moreover we define the notion of paintings of a marked surface without punctures and their rotations. They are a geometric model of cotorsion pairs and of their mutations respectively.