Does there exist a closed Riemann manifold $M$, two distinct irreducible triangulations $S_1$ and $S_2$ of $M$, and a triangulation $T$ of $M$ such that there exists a sequence of edge contractions on $T$ such that $T$ reduces into $S_1$ and another seuquence of edge contractions on $T$ such that $T$ reduces into $S_2$?
Perhaps being a closed manifold is an unnecesary condition, but I am partial to closed Riemann manifolds. My assumption is that the answer to my question is "Yes", but if the answer is "No", then that would mean that the irreducible triangulations of a manifold form an equivalence relation between triangulations of that manifold.
Here is a triangulation of the real projective plane which reduces into both irreducible triangulations of the real projective plane.