The Bergman complex of a matroid and phylogenetic trees

Abstract

We study the Bergman complex $\mathcal{B}(M)$ 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 $\tilde{\mathcal{B}}(K_n)$ of the graphical matroid of the complete graph $K_n$ is homeomorphic to the space of phylogenetic trees ${\mathcal{T}}_n\times R$. This leads to a proof that the link of the origin in ${\mathcal{T}}_n$ is homeomorphic to the order complex of the proper part of the partition lattice ${\mathrm{\Pi }}_n$.