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 Stiefel-Whitney classes. The homotopy groups of this poset are related to the image of the J-homomorphism from stable homotopy theory.
Cite
@article{arxiv.math/9911158,
title = {Mod 2 cohomology of combinatorial Grassmannians},
author = {Laura Anderson and James F. Davis},
journal= {arXiv preprint arXiv:math/9911158},
year = {2015}
}