On the Topology of the Combinatorial Flag Varieties

For any positive integer n, let [n] denote the set {1, . . . , n}, and letMn be the set of all matroids on [n]. Throughout this paper all matroids will have ground set [n], and we shall frequently omit the symbol n from our notation. Define a partial ordering on Mn by M ′ ≤ M if M ′ is a quotient of M . Let Ωn be the simplicial complex of chains in Mn ; every simplex s ∈ Ωn can be written as s = 〈M1, . . . ,Mr〉, with all Mi ∈ Mn and M1 . . . Mr.