Mogami manifolds, nuclei, and 3D simplicial gravity

TL;DR

This paper investigates Mogami pseudomanifolds, showing not all 3-balls are Mogami, and characterizes the Mogami property using nuclei, revealing only the tetrahedron qualifies as a Mogami nucleus.

Contribution

It demonstrates that not all 3-balls are Mogami and characterizes the Mogami property via nuclei, establishing the tetrahedron as the only Mogami nucleus.

Findings

Not all 3-balls are Mogami.

The Mogami property is characterized by nuclei.

Only the tetrahedron is a Mogami nucleus.

Abstract

Mogami introduced in 1995 a large class of triangulated 3-dimensional pseudomanifolds, henceforth called "Mogami pseudomanifolds". He proved an exponential bound for the size of this class in terms of the number of tetrahedra. The question of whether all 3-balls are Mogami has remained open since, a positive answer would imply a much-desired exponential upper bound for the total number of 3-balls (and 3-spheres) with N tetrahedra. Here we provide a negative answer: many 3-balls are not Mogami. On the way to this result, we characterize the Mogami property in terms of nuclei, in the sense of Collet-Eckmann-Younan: "The only three-dimensional Mogami nucleus is the tetrahedron".