On the topology and geometric construction of oriented matroids and convex polytopes

Abstract

This paper develops new combinatorial and geometric techniques for studying the topology of the real semialgebraic variety R ( M ) \mathcal {R}(M) of all realizations of an oriented matroid M 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 3 oriented matroid with up to eight points is contractible. For these special classes our theorem implies the isotopy property which states the spaces R ( M ) \mathcal {R}(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.