2018/12/06 by J. Hyam Rubinstein, Rubinstein, J. Hyam, Henry Segerman +3 · 3 citations
Mathematics · #52B70 #57M27 #57M50 #57N10 #57Q15 #57Q25 #FOS: Mathematics #Geometric Topology (math.GT) #math.GT #msc:52B70 #msc:57M27 #msc:57M50 #msc:57N10 #msc:57Q15 #msc:57Q25
paper · pdf · doi:10.48550/arxiv.1812.02806
Minor corrections. To appear in L'Enseignement Mathématique. 41 pages, 42 figures
arxiv created 2019/06/27 · arxiv updated 2019/06/28
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.