Cotorsion pairs in the cluster category of a marked surface

Dedicated to Professor Idun Reiten on the occasion of her seventieth birthday. 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 indecom-posable objects in the cluster categories equals to the intersection number of the correspond-ing 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 classifi-cation 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