Our strategy consists in showing that certain Iyama-Yoshino reductions of the m-cluster categories under consideration can be described in terms of cutting along an arc the corresponding geometric realizations. Underthese bijections, flips of (m + 2)-angulations correspond to mutations of m-cluster tilting objects. We show, in those four cases, that there is a bijection between (m + 2)-angulations and isoclasses of basic m-cluster tilting objects. In this thesis, we study the geometric realizations of m-cluster categories of Dynkin types A, D, A ̃ and D ̃. We show that the mutation of colored quivers and m-cluster-tilting objects is compatible with the flip of an (m + 2)-angulation. We show that a subcategory of the m-cluster category of type D ̃n is isomorphic to a category consisting of arcs in an (n - 2)m-gon with two central (m - 1)-gons inside of it.
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