Triangulations of oriented matroids (Q2783395)

This paper extends to oriented matroids the classic concepts of triangulations of polytopes and of point configurations -- both object of various contributions in recent literature. The gain, even for these concepts, is already clear in the paper, where, in a way that depends deeply on the concept of oriented matroid, two conjectures on the homotopy type of posets of subdivisions of different types of polytopes are shown to be equivalent. In fact, the paper reveals the oriented matroid as the natural setting for both concepts and the result is an elegant and robust theory. NEWLINENEWLINENEWLINEFollowing the definition of \textit{L. J. Billera} and \textit{B. S. Munson} [SIAM J. Algebraic Discrete Methods 5, 515-525 (1984; Zbl 0557.05026)] (``weak'' version), the author calls a set \(T\) of bases of an oriented matroid \({\mathcal M}\) a triangulation if: 1) for every \(\sigma\in T\), each facet of \(\sigma\) is contained either in a facet of \({\mathcal M}\) or in a different base \(\sigma'\in T\); 2) given two bases, \(\sigma_1\) and \(\sigma_2\) of \(T\), and a one-element extension \({\mathcal M}'={\mathcal M}\cup p\) of \({\mathcal M}\), if \(p\) belongs to the convex hull of both bases (in \({\mathcal M}'\)), then \(p\) belongs to the convex hull of their intersection.NEWLINENEWLINENEWLINEAfter a brief but clear introductory chapter in oriented matroids, in the second chapter the author shows that the above definition is essentially equivalent to other previous ones, namely the ``strong'' definition given in the cited paper and the one given in [Topology 38, No. 1, 197-221 (1999; Zbl 0924.57028)], by \textit{L. Anderson}, and to a number of apparently weaker definitions. He then considers various possibilities of ``editing'' triangulations, such as extending to an oriented matroid \({\mathcal M}\) a triangulation \(T\) of \({\mathcal M}\setminus a\), where \(a\) is an element of the ground set \(E\) of \({\mathcal M}\) (either in the convex hull of \(E\setminus a\) or not), or constructing from \(T\) a triangulation of \({\mathcal M}/a\) in the case where \(a\) does not belong to the convex hull of \(E\setminus a\). The chapter ends with the characterization in terms of pseudo-manifolds of \({\mathcal P}(T)\), the simplicial complex of which a given triangulation \(T\) is the set of maximal simplices, and with some results related with the problem of characterizing \({\mathcal P}(T)\) as a topological space; this part follows essentially the work of Anderson in the paper mentioned above.NEWLINENEWLINENEWLINEThe third and fourth chapters are the core of the article. They are devoted to the exhibition and study of the duality between lifting triangulations of an oriented matroid and interior extensions in general position of its dual. This duality is the oriented matroid version of the duality exhibited by \textit{L. J. Billera, P. Filliman} and \textit{B. Sturmfels} in [Adv. Math. 83, No. 2, 155-179 (1990; Zbl 0714.52004)] between regular triangulations of a point configuration \({\mathcal A}\) and chambers of its Gale transform \({\mathcal A}^*\). It is shown, in particular, that two extensions of an oriented matroid that differ by a flipping (or mutation, in the uniform case) will have corresponding lifting triangulations either equal or differing by a bistellar flip. The full extension of the duality appears when one considers generic extensions in general position, on the one hand, and lifting subdivisions -- instead of (simpler) triangulations, on the other hand. Then, the duals of extensions ordered by weak maps are subdivisions ordered by refinement. The poset homomorphism defined in this way is an isomorphism in case of Lawrence polytopes, for which all subdivisions are shown to be lifting. From here follows the equivalence between the generalized Baues conjecture, posed by Billera, Filliman and Sturmfels in the cited paper, for realized Lawrence polytopes and the extension space conjecture of oriented matroid theory. Finally, by using Bohne-Dress Theorem, it follows that the two conjectures mentioned above on the homotopy type of certain posets (again two forms of the generalized Baues conjecture) are equivalent. Chapter four ends by introducing a reoriented version of Lawrence's construction, with which one builds, from an acyclic non-polytopal oriented matroid \({\mathcal M}\), a matroid polytope \(\Sigma({\mathcal M})\) with the same collection of triangulations; hence, triangulations of polytopes are essentially the same as triangulations of point configurations.NEWLINENEWLINENEWLINEIn chapter five lifting triangulations are studied in more detail. They are compared with regular triangulations of point configurations, and interesting and rich examples of non-lifting triangulations are given. Finally triangulations (even more generally, subdivisions) that are lifting are characterized, both partially and in a complete way, the latter in two different, intrinsic characterizations. NEWLINENEWLINENEWLINEThe paper is an excellent introduction to the topic of triangulations of oriented matroids (probably the most important recent development in oriented matroid theory), not only for the thorough way the topics are studied, but also for being clear, concise and self-contained. It was already followed by several interesting results by different authors; yet, it still leaves various challenging open problems.