Geometric bistellar flips. The setting, the context and a construction

TL;DR

This paper introduces the theory of secondary polytopes and geometric bistellar flips, reviews known results, and presents a new construction of a point set with a disconnected triangulation space, revealing new topological properties.

Contribution

It provides a self-contained overview of the theory and introduces the first example of a point set with a disconnected space of triangulations.

Findings

Connectedness of the triangulation space is not guaranteed.

Construction of a point set with a disconnected triangulation space.

Links to algebraic geometry and topological combinatorics.

Abstract

We give a self-contained introduction to the theory of secondary polytopes and geometric bistellar flips in triangulations of polytopes and point sets, as well as a review of some of the known results and connections to algebraic geometry, topological combinatorics, and other areas. As a new result, we announce the construction of a point set in general position with a disconnected space of triangulations. This shows, for the first time, that the poset of strict polyhedral subdivisions of a point set is not always connected.