Abstract.
Matroid bundles, introduced by MacPherson, are combinatorial analogues of real vector bundles. This paper sets up the foundations of matroid bundles. It defines a natural transformation from isomorphism classes of real vector bundles to isomorphism classes of matroid bundles. It then gives a transformation from matroid bundles to spherical quasifibrations, by showing that the geometric realization of a matroid bundle is a spherical quasifibration. 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. A consequence is that 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.
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Anderson, L., Davis, J. Mod 2 cohomology of combinatorial Grassmannians. Sel. math., New ser. 8, 161–200 (2002). https://doi.org/10.1007/s00029-002-8104-4
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DOI: https://doi.org/10.1007/s00029-002-8104-4