Title: Grassmanians and Pseudosphere Arrangements,
Abstract: 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.