Stacked polytopes and tight triangulations of manifolds

Abstract

Tightness of a triangulated manifold is a topological condition, roughly meaning that any simplexwise 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 Walkup's class $\mathcal{K}(d)$. We show that in any dimension $d\ge 4$ \emph{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.

arXiv comment28 pages, 2 figures

References in corpus (1)

• On Walkup's class $\mathcal{K}(d)$ and a minimal triangulation of $(S^3 \times \rotatebox{90}{\ltimes} S^1)^{\#3}$

Cited by in corpus (1)

• On Walkup's class $\mathcal{K}(d)$ and a minimal triangulation of $(S^3 \times \rotatebox{90}{\ltimes} S^1)^{\#3}$