One measure of complexity for a three-manifold is the minimum number of tetrahedra required triangulate it. Computer studies have determined the minimum number of tetrahedra required to triangulate hundreds of thousands of closed irreducible 3-manifolds. In 2009 Jaco, Rubinstein and Tillmann developed the first infinite family of examples, all lens spaces, for which they were able to demonstrate the minimum number of tetrahedra required to triangulate in the context of singular triangulations. We will show how to adapt their method to simplicial posets, abstract simplicial complexes where a set of vertices can support more than one face. Along the way we will discover a surprising property of the 'average discrete normal surface' for any simplicial decomposition of a closed three-manifold. We will also find a close connection to the Charney-Davis conjecture for flag spheres.

We will assume no specialized knowledge of three-manifolds. This is not entirely unrelated to the fact that the speaker has no such specialized knowledge.