  Inspired by the ideas and techniques used in the study of cluster algebras we
construct a new class of algebras, called bistellar cluster algebras, from
closed oriented triangulated even-dimensional manifolds by performing
middle-dimensional bistellar moves. This class of algebras exhibit the
algebraic behaviour of middle-dimensional bistellar moves but do not satisfy
the classical cluster algebra axiom: "every cluster variable in every cluster
is exchangeable". Thus the construction of bistellar cluster algebras is quite
different from that of a classical cluster algebra. Secondly, using bistellar
cluster algebras and the techniques of combinatorial topology, we construct a
direct system associated with a set of PL homeomorphic PL manifolds of
dimension 2 or 4, and show that the limit of this direct system is a PL
invariant.
