A bijection between m-cluster-tilting objects and (m + 2)-angulations in m-cluster categories

In this article, we study the geometric realizations of m-cluster categories Dynkin types A, D, A˜ and D˜. We show, in those four cases, that there is a bijection between (m+2)-angulations isoclasses basic tilting objects. Under these bijections, flips correspond to mutations Our strategy consists showing certain Iyama-Yoshino reductions under consideration can be described terms cutting along an arc corresponding realizations. This allows infer results from small cases general ones.