Oriented matroids from smooth manifolds

Results of Folkman and Lawrence and Mandel on representations of oriented matroids by topological spheres are used to prove a method of constructing oriented matroids from intersections of smooth topological hyperplanes. A class of such constructions is given corresponding to non real-representable matroids of rank ϱ on 2ϱ + 1 elements, ϱ ≥ 4.