The positive Bergman complex of an oriented matroid

We study the positive Bergman complex B+(M)B+(M) of an oriented matroid MM, which is a certain subcomplex of the Bergman complex B(M¯) of the underlying unoriented matroid M¯. The positive Bergman complex is defined so that given a linear ideal II with associated oriented matroid MIMI, the positive tropical variety associated with II is equal to the fan over B+(MI)B+(MI). Our main result is that a certain “fine” subdivision of B+(M)B+(M) is a geometric realization of the order complex of the proper part of the Las Vergnas face lattice of MM. It follows that B+(M)B+(M) is homeomorphic to a sphere. For the oriented matroid of the complete graph KnKn, we show that the face poset of the “coarse” subdivision of B+(Kn)B+(Kn) is dual to the face poset of the associahedron An−2An−2, and we give a formula for the number of fine cells within a coarse cell.