The positive Bergman complex of an oriented matroid

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 M ¯ . The positive Bergman complex is defined so that given a linear ideal I with associated oriented matroid M I , the positive tropical variety associated with 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.