On the Topology and Geometric Construction of Oriented Matroids and Convex Polytopes

This paper develops new combinatorial and geometric techniques for studying the topology of the real semialgebraic variety 31 (M) of all realizations of an oriented matroid M . We focus our attention on point configurations in general position, and as the main result we prove that the realization space of every uniform rank 3 oriented matroid with up to eight points is contractible. For these special classes our theorem implies the isotopy property which states the spaces 32(M) are path-connected. We further apply our methods to several related problems on convex polytopes and line arrangements. A geometric construction and the isotopy property are obtained for a large class of neighborly polytopes. We improve a result of M. Las Vergnas by constructing a smallest counterexample to a conjecture of G Ringel, and, finally, we discuss the solution to a problem of R. Cordovil and P. Duchet on the realizability of cyclic matroid polytopes.