Convex, acyclic, and free sets of an oriented matroid

We study the global and local topology of three objects associated to a simple oriented matroid: the lattice of convex sets, the simplicial complex of acyclic sets, and the simplicial complex of free sets. Special cases of these objects and their homotopy types have appeared in several places in the literature. The global homotopy types of all three are shown to coincide, and are either spherical or contractible depending on whether the oriented matroid is totally cyclic. Analysis of the homotopy type of links of vertices in the complex of free sets yields a generalization and more conceptual proof of a recent result counting the interior points of a point configuration.