Oriented Matroids and Combinatorial Manifolds

An oriented matroid lattice is a lattice arising from the span of cocircuits of an oriented matroid ordered by conformal relation. One important subclass of the o.m. lattices is the polars of face lattices of zonotopes. In this paper we show that every o.m. lattice is a (combinatorial) manifold. This brings out several interesting results on graphs associated with an o.m. lattice. For example, through Barnetteʹs theorem on connectivity of manifolds, we obtain the (r - 1)-connectivity of the graph of the Las Vergnas lattice and its polar, where r is the rank of the oriented matroid. Furthermore, we prove that the graph of an o.m. lattice is 2(r - 1)-connected, while the graph of its polar is only r-connected. These results are the best possible in the sense that each claimed connectivity is exact for some oriented matroid of rank r. Finally, we give an algorithmic proof of the Bjِrner-Edelman-Ziegler theorem: that an oriented matroid is determined by the cograph of the associated o.m. lattice.