Stacked polytopes and tight triangulations of manifolds

Tightness of a triangulated manifold is a topological condition, roughly meaning that any simplex-wise linear embedding of the triangulation into Euclidean space is “as convex as possible”. It can thus be understood as a generalization of the concept of convexity. In even dimensions, super-neighborliness is known to be a purely combinatorial condition which implies the tightness of a triangulation. Here, we present other sufficient and purely combinatorial conditions which can be applied to the odd-dimensional case as well. One of the conditions is that all vertex links are stacked spheres, which implies that the triangulation is in Walkupʼs class K ( d ) . We show that in any dimension d ⩾ 4 , tight-neighborly triangulations as defined by Lutz, Sulanke and Swartz are tight. Furthermore, triangulations with k-stacked vertex links and the centrally symmetric case are discussed.