  We define a partial ordering on the set Q = Q(M) of pairs of topes of an
oriented matroid M, and show the geometric realization |Q| of the order complex
of Q has the same homotopy type as the Salvetti complex of M. For any element e
of the ground set, the complex |Qe| associated to the rank-one oriented matroid
on {e} has the homotopy type of the circle. There is a natural free simplicial
action of Z4 on |Q|, with orbit space isomorphic to the order complex of the
poset Q(M,e) associated to the pointed (or affine) oriented matroid (M,e). If M
is the oriented matroid of an arrangement A of linear hyperplanes in real
space, the Z_4 action corresponds to the diagonal action of C* on the
complement M of the complexification of A: |Q| is equivariantly
homotopy-equivalent to M under the identification of Z_4 with {+1, i, -1, -i},
and |Q(M, e)| is homotopy-equivalent to the complement of the decone of A
relative to the hyperplane corresponding to e. All constructions and arguments
are carried out at the level of the underlying posets. If a group G acts on the
set of topes of M preserving adjacency, then G acts simplicially on |Q|. We
also show that the class of fundamental groups of such complexes is strictly
larger than the class of fundamental groups of complements of complex
hyperplane arrangements. Specifically, the group of the non-Pappus arrange-
ment is not isomorphic to any realizable arrangement group.
