Connecting essential triangulations

Tejas Kalelkar (IISER Pune)

Abstract: Amendola, Matveev and Piergallini have shown that any two ideal triangulations of a compact 3-dimensional manifolds (with boundary) are connected by a sequence of ideal triangulations, where each triangulation in the sequence is obtained from the previous one by a local combinatorial change, namely a 2-3 or 3-2 move. For some applications however, we need our triangulations to have special properties, such as the condition that all edges are essential. An edge is called essential if it is not a homotopically trivial loop. Certain essential ideal triangulations are isolated, in the sense that any 2-3 or 3-2 move makes the triangulation inessential. We have shown that when the universal cover of M has infinitely many boundary components and the two given essential ideal triangulations are not isolated, then they are connected by a sequence of 2-3 and 3-2 moves through essential ideal triangulations. If in addition, we allow V-moves and their inverses, then any two essential ideal triangulations (including the isolated ones) are connected via essential ideal triangulations. Our results have applications such as proving that the 1-loop invariant is independent of the choice of triangulation. This is joint work with Henry Segerman and Saul Schleimer.