Title: Mod 2 Cohomology of Combinatorial Grassmannians
Authors: Laura Anderson and James F. Davis
Abstract: Matroid bundles, introduced by MacPherson, are combinatorial
analogues of real vector bundles. This paper sets up the foundations of
matroid bundles, and defines a natural transformation from isomorphism
classes of real vector bundles to isomorphism classes of matroid bundles,
as well as a transformation from matroid bundles to spherical
quasifibrations. The poset of oriented matroids of a fixed rank classifies
matroid bundles, and the above transformations give a splitting from
topology to combinatorics back to topology. This shows the mod 2
cohomology of the poset of rank k oriented matroids (this poset classifies
matroid bundles) contains the free polynomial ring on the first k
Stiefel-Whitney classes. The homotopy groups of this poset are related to
the image of the J-homomorphism from stable homotopy theory.