  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.
