  An important problem in the theory of cluster algebras is to compute the
fundamental group of the exchange graph. A non-trivial closed loop in the
exchange graph, for example, generates a non-trivial identity for the classical
and quantum dilogarithm functions. An interesting conjecture, partly motivated
by dilogarithm functions, is that this fundamental group is generated by closed
loops of mutations involving only two of the cluster variables. We present
examples and counterexamples for this naive conjecture, and then formulate a
better version of the conjecture for acyclic seeds.
