Homotopy groups of the combinatorial Grassmannian

We prove that the homotopy groups of the oriented matroid Grassmannian MacP(k; n) are stable as n ! 1, that ß 1 (\Delta MacP(k; n)) ¸ = ß 1 (G(k; R n )), and that there is a surjection ß 2 (G(k; R n )) ! ß 2 (\Delta MacP(k; n)). The theory of oriented matroids gives rise to a combinatorial analog to the Grassmannian G(k; R n ). By thinking of an oriented matroid as a "combinatorial vector space", one is led to define the combinatorial Grassmannian (or MacPhersonian MacP(k; n)) as a partially ordered set, whose order complex is hoped to have topology similar that of G(k; R n ). While there are some immediate and natural correspondences between these two topological spaces, the topology of the MacPhersonian is on the whole a mystery. The topology of MacP(k; n) and the direct limit MacP(k; 1) is of interest from several perspectives. From the topological perspective, the MacPhersonian MacP(k; 1) is the classifying space for the combinatorial vector bundles (or matroid bundles..