A construction principle for tight and minimal triangulations of manifolds

TL;DR

This paper introduces a new combinatorial method to construct tight triangulations of manifolds, providing numerous new examples in dimensions three to five and a family of such manifolds with exponentially many distinct members.

Contribution

It presents a computer-friendly scheme for constructing tight triangulations and demonstrates the existence of many such manifolds in higher dimensions.

Findings

New examples of tight triangulations in dimensions three, four, and five.

A family of tight triangulated manifolds with exponentially many distinct members.

Evidence that tight triangulations are more abundant than previously known.

Abstract

Tight triangulations are exotic, but highly regular objects in combinatorial topology. A triangulation is tight if all its piecewise linear embeddings into a Euclidean space are as convex as allowed by the topology of the underlying manifold. Tight triangulations are conjectured to be strongly minimal, and proven to be so for dimensions $\leq 3$. However, in spite of substantial theoretical results about such triangulations, there are precious few examples. In fact, apart from dimension two, we do not know if there are infinitely many of them in any given dimension. In this paper, we present a computer-friendly combinatorial scheme to obtain tight triangulations, and present new examples in dimensions three, four and five. Furthermore, we describe a family of tight triangulated $d$-manifolds, with $2^{d-1} \lfloor d / 2 \rfloor ! \lfloor (d-1) / 2 \rfloor !$ isomorphically distinct members for each dimension $d \geq 2$. While we still do not know if there are infinitely many tight triangulations in a fixed dimension $d > 2$, this result shows that there are abundantly many.