Circuit Admissible Triangulations of Oriented Matroids

All triangulations of euclidean oriented matroids are of the same PLhomeomorphism type by a result of Anderson. That means all triangulations of euclidean acyclic oriented matroids are PL-homeomorphic to PL-balls and that all triangulations of totally cyclic oriented matroids are PL-homeomorphic to PLspheres. For non-euclidean oriented matroids this question is wide open. One key point in the proof of Anderson is the following fact: for every triangulation of a euclidean oriented matroid the adjacency graph of the set of all simplices “intersecting” a segment [p-p+] is a path. We call this graph the [p-p+]adjacency graph of the triangulation. While we cannot solve the problem of the topological type of triangulations of general oriented matroids we show in this note that for every circuit admissible triangulation of an arbitrary oriented matroid the [p-p+]-adjacency graph is path. Triangulations of oriented matroids appeared in the literature as natural combinatorial models for triangulations of point configurations [2]. However, since not all oriented matroids model point configurations the notion of a triangulation of an oriented matroid gives rise to additional questions that do not come up in the theory of triangulations of point configurations. One of these questions is the following: is the abstract simplicial complex defined by a triangulation of an oriented matroid homeomorphic ot a sphere in the totally acyclic case or a ball in the acyclic case? The answer to this question in the realizable case is of course affirmative because in the case of point configurations the triangulation is naturally embedded as a convex set in a euclidean space. Why care about the general case? An application of triangulations of oriented matroids in their full generality is their appearance in the theory of combinatorial differential manifolds. “Good” topological properties in this context lead to the existence of differentiable strucures on these objects, making the combinatorial model more suitable [1, 4]. But also as an investigation of what weird things might happen in the theory of non-realizable oriented matroids this question has become a challenging open problem in its own right. (An in-depth study of triangulations of oriented matroids is presented in [5], background on oriented matroids can be found in [3].) For a euclidean oriented matroid Anderson has proved that the topological types of all its triangulations are the same. Since for all oriented matroids there are triangulations known that are homeomorphic to a sphere resp. to a ball—the socalled lifting triangulations—the answer to the above question is affirmative. One important building block in the construction of Anderson is the fact that the adjacency graph of the set of simplices in a triangulation “intersecting” an arbitrary segment is always a path. (For exact definitions see below.) In this note we show that this graph is also a path for general oriented matroids provided the triangulation respects an additional property: it does not contain a so-called intersection circuit, a circuit that has positive in one simplex and negative part in another simplex in the triangulation. We start by defining our main object of study. For simplicity, we call r-subsets of full rank r simplices; subsets of rank r − 1 are called facets.