  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. We study the abelianization of the discrete fundamental group, and show
that it is free abelian of rank $\binom{n+2}{4}$. We also find a combinatorial
description for a basis of this rank. We also introduce the exchange module of
the type $A_n$ cluster algebra, used to model the relations in the cluster
algebra. We use the discrete fundamental group to the study of exchange module,
and show that it is also free abelian of rank $\binom{n+2}{3}$.
