Homotopy Sphere Representations for Matroids

For any rank r oriented matroid M, a construction is given of a ??topological representation?? of M by an arrangement of homotopy spheres in a simplicial complex which is homotopy equivalent to S ^r?C1 . The construction is completely explicit and depends only on a choice of maximal flag in M. If M is orientable, then all Folkman-Lawrence representations of all orientations of M embed in this representation in a homotopically nice way.