Grassmanians and Pseudosphere Arrangements

We extend vector configurations to more general objects that have nicer combinatorial and topological properties, called weighted pseudosphere arrangements. These are defined as a weighted variant of arrangements of pseudospheres, as in the Topological Representation Theorem for oriented matroids. We show that in rank 3, the real Stiefel manifold, Grassmanian, and oriented Grassmanian are homotopy equivalent to the analagously defined spaces of weighted pseudosphere arrangements. We also show for all rank 3 oriented matroids, that the space of realizations by weighted pseudosphere arrangements is contractible. This is a sharp contrast with vector configurations, where the space of realizations can have the homotopy type of any primary semialgebraic set.