The Bergman complex of a matroid and phylogenetic trees

We study the Bergman complex 13(M) of a matroid M: a polyhedral complex which arises in algebraic geometry, but which we describe purely combinatorially. We prove that a natural subdivision of the Bergman complex of M is a geometric realization of the order complex of the proper part of its lattice of flats. In addition, we show that the Bergman fan (B) over tilde (K-n) of the graphical matroid of the complete graph K-n is homeomorphic to the space of phylogenetic trees T-n x R. This leads to a proof that the link of the origin in T-n is homeomorphic to the order complex of the proper part of the partition lattice Pi(n). (c) 2005 Elsevier Inc. All rights reserved.