Convex, acyclic, and free sets of an oriented matroid

We study the global and local topology of three objects associated to a sim ple oriented matroid: the lattice of convex sets, the simplicial complex of acyclic sets, and the simplicial complex of free sets. Special cases of th ese 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 eithe r spherical or contractible depending on whether the oriented matroid is to tally cyclic. Analysis of the homotopy type of links of vertices in the complex of free s ets yields a generalization and more conceptual proof of a recent result co unting the interior points of a point configuration.