Mod 2 Cohomology of Combinatorial Grassmannians

Oriented matroids have long been of use in various areas of combinatorics [BLS93]. Gelfand and MacPherson [GM92] initiated the use of oriented matroids in manifold and bundle theory, using them to formulate a combinatorial formula for the rational Pontrjagin classes of a differentiable manifold. MacPherson [Mac93] abstracted this into a manifold theory (combinatorial differential (CD) manifolds) and a bundle theory (which we call combinatorial vector bundles or matroid bundles). In this paper we explore the relationship between combinatorial vector bundles and real vector bundles. As a consequence of our results we get theorems relating the topology of the combinatorial Grassmannians to that of their real analogs. The theory of oriented matroids gives a combinatorial abstraction of linear algebra; a k-dimensional subspace of Rn determines a rank k oriented matroid with elements {1, 2, . . . , n}. Such oriented matroids can be given a partial order by using the notion of weak maps, which geometrically corresponds to moving the k-plane into more special position with respect to the standard basis of Rn . The poset MacP(k, n) of rank k oriented matroids with n elements was defined by MacPherson in [Mac93] and is often called the MacPhersonian. The limit of the finite MacPhersonians gives an infinite poset MacP(k,∞), and its geometric realization ‖MacP(k,∞)‖ is the classifying space for rank k matroid bundles. Our main results are the Combinatorialization Theorem, which associates a matroid bundle to a vector bundle, the Spherical Quasifibration Theorem, which associates a spherical quasifibration to a matroid bundle, and the Comparison Theorem, which shows that the composition of these two associations is essentially the forgetful functor. More precisely, if B is a regular cell complex, and if Vk(B), Mk(B), and Qk(B) denote the isomorphism classes of rank k vector bundles, matroid bundles, and spherical quasifibrations respectively, we construct maps Vk(B) → Mk(B) → Qk(B)