Higher Bruhat Orders and Cyclic Hyperplane Arrangements

We study the higher Bruhat orders B(n, k) of Manin & Schechtman [MaS] and-characterize them in terms of inversion sets,-identify them with the posets ZY(C n+1 ' r ,n+l) of uniform extensions of the alternating oriented matroids C n ' r for r := n—k (that is, with the extensions of a cyclic hyperplane arrangement by a new oriented pseudoplane),-show that B(n, k) is a lattice for k = 1 and for r < 3, but not in general,-show that B(n, k) is ordered by inclusion of inversion sets for k — 1 and for r < 4. However, 2?(8,3) is not ordered by inclusion. This implies that the partial order B c (n, k) defined by inclusion of inversion sets differs from B(n, k) in general. We show that the proper part of B c (n, k) is homotopy equivalent to S r ~ 2. Consequently,-£(n, k) ~ S r ~ 2 for Jf e = 1 and for r < 4. In contrast to this, we find that the uniform extension poset of an affine hyperplane arrangement is in general not graded and not a lattice even for r = 3, and that the proper part is not always homotopy equivalent to S r(-M '~ 2 .