Pachner moves on geometric triangulation

Abstract

As every smooth manifold can be smoothly triangulated so triangulations are a useful tool to combinatorially study manifolds. It is known that any two smooth triangulations of a manifold are related by a finite sequence of smooth local transformations called Pachner moves. So quantities defined in terms of triangulations which are invariant under Pachner moves become invariants of the manifold. We have shown that geometric triangulations of constant curvature manifolds are related by Pachner moves through geometric triangulations (up to derived subdivisions). This gives rise to the possibility of defining geometric invariants using geometric triangulations.

A fundamental question in combinatorial topology is to combinatorially determine when two simplicial complexes realise the same manifold. We give an algorithm for this in the case of geometrically triangulated constant curvature manifolds. This is done by obtaining an explicit bound on the number of Pachner moves needed to relate any two geometric triangulations of the constant curvature manifold, with the bound expressed in terms of the number of top dimensional simplexes and bounds on the lengths of edges of the two triangulation