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 $\leq 3$.
Categories: math.GT math.CO