# Traversing three-manifold triangulations and spines

## TL;DR

This paper discusses the connectivity of 3-manifold triangulations and spines, providing a unified, combinatorial proof that emphasizes dual viewpoints and aims to popularize existing results.

## Contribution

It offers a combined proof for closed and non-compact 3-manifolds, replacing a key argument with a more combinatorial approach and clarifying the duality between triangulations and spines.

## Key findings

- Unified proof for closed and non-compact 3-manifolds.
- Replaces a complex argument with a combinatorial one.
- Highlights the duality between triangulations and spines.

## Abstract

A celebrated result concerning triangulations of a given closed 3-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 3-manifolds. These results build on classical work that goes back to Alexander, Newman, Moise, and Pachner. The key special case of 1-vertex triangulations of closed 3-manifolds was independently proven by Matveev and Piergallini. The general result for closed 3-manifolds can be found in work of Benedetti and Petronio, and Amendola gives a proof for topologically finite non-compact 3-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.