The Positive Bergman Complex of an Oriented Matroid

Abstract

We study the positive Bergman complex B+(M) of an oriented matroid M, which is a certain subcomplex of the Bergman complex B(M) of the underlying unoriented matroid. The positive Bergman complex is defined so that given a linear ideal I with associated oriented matroid M_I, the positive tropical variety associated to I is equal to the fan over B+(M_I). Our main result is that a certain "fine" subdivision of B+(M) is a geometric realization of the order complex of the proper part of the Las Vergnas face lattice of M. It follows that B+(M) is homeomorphic to a sphere. For the oriented matroid of the complete graph K_n, we show that the face poset of the "coarse" subdivision of B+(K_n) is dual to the face poset of the associahedron A_{n-2}, and we give a formula for the number of fine cells within a coarse cell.