Topological Representation of Dual Pairs of Oriented Matroids

Among the many ways to view oriented matroids as geometrical objects, we consider two that have special properties: • Bland’s analysis of complementary subspaces in IRn [2] has the special feature that it simultaneously and symmetrically represents a realizable oriented matroid and its dual; • Lawrence’s topological representation of oriented matroids by arrangements of pseudospheres [4] has the advantage of yielding a faithful picture also in the general case of non-realizable oriented matroids. In this note we prove a “Topological Representation Theorem for Dual Pairs”, which combines these two points of view. We refer to [1, Chap. 1] for an exposition of the theory of oriented matroids. Here we only review some notation and fix terminology. Bland’s [2, Sect. 3] [1, Sect. 1.2(d)] set-up is as follows. Let ξB be a subspace of IRn of dimension r. The intersections of the coordinate hyperplanes Hi = {x ∈ IRn : xi = 0} with ξ determine an arrangement of hyperplanes {ξ ∩ Hi : 1 ≤ i ≤ n} in ξ, and with it a (realizable) oriented matroid M of rank r on {1, . . . , n}. In the same way, the orthogonal complement ξ⊥ of dimension n− r determines an arrangement in ξ⊥ that represents M∗. Now write ξ and ξ⊥ as intersections ξ⊥ = ⋂n+r j=n+1 H ′ j and ξ = ⋂2n j=n+r+1 H ′ j of hyperplanes H ′ j ⊆ IRn. This construction encodes the realizable oriented matroid M and its dual M∗ into an arrangement of 2n hyperplanes Hi, H ′ j in IRn, for 1 ≤ i ≤ n and n+1 ≤ j ≤ 2n. In view of this, the Topological Representation Theorem of Lawrence suggests a generalization that encodes a general pair of dual oriented matroids into an arrangement of 2n pseudospheres in Sn−1, stated below as Theorem 1.