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-manifolds. These results (and their proofs) are phrased in the dual language of spines.

The purpose of this note is threefold. We wish to popularise Amendola’s result; we give a combined proof for both closed and non-compact manifolds that emphasises the dual viewpoints of triangulations and spines; and we give a proof replacing a key general position argument due to Matveev with a more combinatorial argument inspired by the theory of subdivisions.