A characterization of tightly triangulated 3-manifolds

TL;DR

This paper characterizes $$-tight triangulations of closed 3-manifolds, proving they are exactly the $$-orientable, neighbourly, and stacked triangulations, thus confirming the K"uhnel-Lutz conjecture in dimension three.

Contribution

It provides a complete characterization of $$-tight triangulations of closed 3-manifolds, confirming the conjecture in dimension three.

Findings

$$-tight triangulations of 3-manifolds are exactly those that are $$-orientable, neighbourly, and stacked.

The K"uhnel-Lutz conjecture holds true for all closed 3-manifolds.

Boundary of a triangle is the only $$-tight triangulation of a closed 1-manifold.

Abstract

For a field $\mathbb{F}$, the notion of $\mathbb{F}$-tightness of simplicial complexes was introduced by K\"uhnel. K\"uhnel and Lutz conjectured that any $\mathbb{F}$-tight triangulation of a closed manifold is the most economic of all possible triangulations of the manifold. The boundary of a triangle is the only $\mathbb{F}$-tight triangulation of a closed 1-manifold. A triangulation of a closed 2-manifold is $\mathbb{F}$-tight if and only if it is $\mathbb{F}$-orientable and neighbourly. In this paper we prove that a triangulation of a closed 3-manifold is $\mathbb{F}$-tight if and only if it is $\mathbb{F}$-orientable, neighbourly and stacked. In consequence, the K\"uhnel-Lutz conjecture is valid in dimension $\leq 3$.