Geometry and Topology Seminar

There are infinitely many triangulations of a given 3-manifold
(here we restrict to one-vertex triangulations for closed manifolds, and
ideal triangulations for manifolds with boundary). For each manifold M we
form a graph T(M) whose vertices are the triangulations of M, and
for which two vertices are connected if the triangulations are related by
a Pachner 2-3 move. Matveev and Piergallini independently show that
for each manifold, T(M) is connected (other than for triangulations
consisting of a single tetrahedron). However, very little else is
known about the structure of T(M).

There are many useful properties a triangulation can have, for example
geometric triangulations, triangulations with angle structures, 0- and
1-efficient triangulations and triangulations with essential edges. Almost
nothing is known about the subgraphs of T(M) corresponding to
these kinds of triangulation. I will survey these properties and the
relations between them, and say something about how we can start to
investigate their connectivity in T(M).