Mod 2 cohomology of combinatorial Grassmannians

• L. Anderson ^[1] ; J.F. Davis ^[2]
  1. [1] Binghamton University

     Binghamton University

     City of Binghamton, Estados Unidos

     
  2. [2] Indiana University, USA
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 8, Nº. 2, 2002, págs. 161-200
• Idioma: inglés
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• Resumen

  • 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.