  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.
