Bistellar Cluster Algebras and Piecewise Linear Invariants

TL;DR

This paper introduces bistellar cluster algebras derived from triangulated manifolds, which exhibit unique algebraic properties and are used to construct piecewise linear invariants through combinatorial topology techniques.

Contribution

It constructs a new class of algebras called bistellar cluster algebras from triangulated manifolds and demonstrates their application in defining PL invariants.

Findings

Bistellar cluster algebras do not satisfy classical cluster algebra axioms.

A direct system of PL manifolds is constructed using these algebras.

The limit of this system yields a PL invariant.

Abstract

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.