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Summary: arXiv:math.CO/0311370v24May2005
The Bergman Complex of a Matroid
and Phylogenetic Trees
Federico Ardila Caroline J. Klivans
Abstract
We study the Bergman complex B(M) of a matroid M: a poly-
hedral complex which arises in algebraic geometry, but which we de-
scribe purely combinatorially. We prove that a natural subdivision
of the Bergman complex of M is a geometric realization of the order
complex of its lattice of flats. In addition, we show that the Bergman
fan B(Kn) of the graphical matroid of the complete graph Kn is home-
omorphic to the space of phylogenetic trees Tn.
1 Introduction
In [1], Bergman defined the logarithmic limit-set of an algebraic variety in or-
der to study its exponential behavior at infinity. We follow [15] in calling this
set the Bergman complex of the variety. Bergman conjectured that this set is
a finite, pure polyhedral complex. He also posed the question of studying the
geometric structure of this set; e.g., its connectedness, homotopy, homology
and cohomology. Bieri and Groves first proved the conjecture in [2] using
valuation theory.
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