The graph of triangulations of a point configuration with d + 4 vertices is 3-connected

Abstract

We study the graph of bistellar flips between triangulations of a vector configuration A with d + 4 elements in rank d + 1 (i.e. with corank 3), as a step in the Baues problem. We prove that the graph is connected in general and 3-connected for acyclic vector configurations, which include all point configurations of dimension d with d + 4 elements. Hence, every pair of triangulations can be joined by a finite sequence of bistellar flips and, in the acyclic case, every triangulation has at least three geometric bistellar neighbours. In corank 4, connectivity is not known and having at least four flips is false. In corank 2, the results are trivial since the graph is a cycle. Our methods are based on a dualization of the concept of triangulation of a point or vector configuration A to that of a virtual chamber of its Gale transform B, introduced by de Loera et al. in 1996. As an additional result we prove a topological representation theorem for virtual chambers, stating that every virtual chamber of a rank 3 vector configuration B can be realized as a cell in some pseudo-chamber complex of B in the same way that regular triangulations appear as cells in the usual chamber complex. All the results in this paper generalize to triangulations of corank 3 oriented matroids and virtual chambers of rank 3 oriented matroids, realizable or not. The details for this generalization are given in the Appendix.