  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.
