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The Discrete Fundamental Group of the Associahedron

Abstract : The associahedron is an object that has been well studied and has numerous applications, particularly in the theory of operads, the study of non-crossing partitions, lattice theory and more recently in the study of cluster algebras. We approach the associahedron from the point of view of discrete homotopy theory, that is we consider 5-cycles in the 1-skeleton of the associahedron to be combinatorial holes, but 4-cycles to be contractible. We give a simple description of the equivalence classes of 5-cycles in the 1-skeleton and then identify a set of 5-cycles from which we may produce all other cycles. This set of 5-cycle equivalence classes turns out to be the generating set for the abelianization of the discrete fundamental group of the associahedron. In this paper we provide presentations for the discrete fundamental group and the abelianization of the discrete fundamental group. We also discuss applications to cluster algebras as well as generalizations to type B and D associahedra. \par
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Christopher Severs, Jacob White. The Discrete Fundamental Group of the Associahedron. 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), 2009, Hagenberg, Austria. pp.781-792. ⟨hal-01185404⟩



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