Traversing three-manifold triangulations and spines

Abstract

A celebrated result concerning triangulations of a given closed three-manifold is that any two triangulations with the same number of vertices are connected by a sequence of so-called 2–3 and 3–2 moves. A similar result is known for ideal triangulations of topologically finite non-compact three-manifolds. These results build on classical work that goes back to Alexander, Newman, Moise, and Pachner. The key special case of one-vertex triangulations of closed three-manifolds was independently proven by Matveev and Piergallini. The general result for closed three-manifolds can be found in work of Benedetti and Petronio, and Amendola gives a proof for topologically finite non-compact three-manif..