Tight and stacked triangulations of manifolds

TL;DR

This paper surveys the current state of research on tight triangulated manifolds, discussing known results, examples, and open problems in the field of combinatorial topology.

Contribution

It provides a comprehensive overview of known tight triangulated manifolds and recent developments, highlighting gaps and directions for future research.

Findings

Most known tight triangulated manifolds are stacked.

Locally stacked tight manifolds are strongly minimal.

Few infinite series of tight triangulated manifolds are known.

Abstract

Tight triangulated manifolds are generalisations of neighborly triangulations of closed surfaces and are interesting objects in Combinatorial Topology. Tight triangulated manifolds are conjectured to be minimal. Except few, all the known tight triangulated manifolds are stacked. It is known that locally stacked tight triangulated manifolds are strongly minimal. Except for three infinite series and neighborly surfaces, very few tight triangulated manifolds are known. From some recent works, we know more on tight triangulation. In this article, we present a survey on the works done on tight triangulation. In Section 2, we state some known results on tight triangulations. In Section 3, we present all the known tight triangulated manifolds. Details are available in the references mentioned there. In Section 1, we present some essential definitions.