Mod 2 cohomology of combinatorial Grassmannians
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 StiefelWhitney classes. The homotopy groups of this poset are related to the image of the Jhomomorphism from stable homotopy theory.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 November 1999
 arXiv:
 arXiv:math/9911158
 Bibcode:
 1999math.....11158A
 Keywords:

 Mathematics  Geometric Topology;
 Mathematics  Algebraic Topology;
 Mathematics  Combinatorics;
 55R25 (primary);
 05B35;
 52B40;
 57R22 (secondary)
 EPrint:
 Selecta Mathematica 8 (2002), 161200