Tight triangulations of closed 3-manifolds

A triangulation of a closed 2-manifold is tight with respect to a field of characteristic two if and only if it is neighbourly; and it is tight with respect to a field of odd characteristic if and only if it is neighbourly and orientable. No such characterization of tightness was previously known for higher dimensional manifolds. In this paper, we prove that a triangulation of a closed 3-manifold is tight with respect to a field of odd characteristic if and only if it is neighbourly, orientable and stacked. In consequence, the Kühnel–Lutz conjecture is valid in dimension three for fields of odd characteristic.Next let FF be a field of characteristic two. It is known that, in this case, any neighbourly and stacked triangulation of a closed 3-manifold is FF-tight. For closed, triangulated 3-manifolds with at most 71 vertices or with first Betti number at most 188, we show that the converse is true. But the possibility of the existence of an FF-tight, non-stacked triangulation on a larger number of vertices remains open. We prove the following upper bound theorem on such triangulations. If an FF-tight triangulation of a closed 3-manifold has nn vertices and first Betti number β1β1, then (n−4)(617n−3861)≤15444β1(n−4)(617n−3861)≤15444β1. Equality holds here if and only if all the vertex links of the triangulation are connected sums of boundary complexes of icosahedra.