Coverings and Minimal Triangulations of 3-Manifolds

TL;DR

This paper investigates minimal triangulations of 3-manifolds, specifically lens spaces and quaternionic spaces, establishing their complexity and uniqueness using covering space techniques.

Contribution

It extends classification results of minimal triangulations to new families of 3-manifolds, demonstrating their complexity and uniqueness.

Findings

Lens spaces $L(4k, 2k-1)$ have complexity $k$

Generalized quaternionic spaces $S^3/Q_{4k}$ have complexity $k$

Minimal triangulations of these spaces are unique

Abstract

This paper uses results on the classification of minimal triangulations of 3-manifolds to produce additional results, using covering spaces. Using previous work on minimal triangulations of lens spaces, it is shown that the lens space $L(4k, 2k-1)$ and the generalised quaternionic space $S^3/Q_{4k}$ have complexity $k,$ where $k\ge 2.$ Moreover, it is shown that their minimal triangulations are unique.