A characterization of tightly triangulated 3-manifolds

Abstract

For a field $\mathbb{F}$, the notion of $\mathbb{F}$-tightness of simplicial complexes was introduced by Kühnel. Kühnel 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 Kühnel-Lutz conjecture is valid in dimension $\le 3$.

Corollary 1.5 is new