On the Generation of Oriented Matroids

Abstract. We provide a multiple purpose algorithm for generating oriented matroids. An application disproves a conjecture of Grünbaum that every closed triangulated orientable 2-manifold can be embedded geometrically in R3 , i.e., with flat triangles and without self-intersections. We can show in particular that there exists an infinite class of orientable triangulated closed 2-manifolds for each genus g \geq 6 that cannot be embedded geometrically in Euclidean 3-space. Our algorithm is interesting in its own right as a tool for many investigations in which oriented matroids play a key role.

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