A characterization of tightly triangulated 3-manifolds

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