%0 Journal Article %T HMS for punctured tori and categorical mapping class group actions %A Nicolo` Sibilla %J Mathematics %D 2011 %I arXiv %X Let X_n be a cycle of n projective lines, and T_n a symplectic torus with n punctures. In this paper we review results appeared in arXiv:1103.2462 and in arXiv:1109.6615, which establish a version of homological mirror symmetry relating X_n and T_n, and define on D^b(Coh(X_n)) an action of the pure mapping class group of T_n. %U http://arxiv.org/abs/1111.5284v1