The homotopy type of the matroid grassmannian

Abstract

Stiefel [26] and Whitney [28] and with integral coefficients by Pontrjagin [24], make up the primary source of first-order invariants of smooth manifolds. When their utility was first recognized, it became an obvious goal to study the ways in which they admitted extensions to other categories, such as the categories of topological or PL manifolds; perhaps a clean description of characteristic classes for simplicial complexes could even give useful computational techniques. Modulo 2, this hope was realized rather quickly: it is not hard to see that the Stiefel-Whitney classes are PL invariants. Moreover, Whitney was able to produce a simple explicit formula for the class in codimension i in terms of the i-skeleton of the barycentric subdivision of a triangulated manifold (for a proof of this result, see [13]). One would like to find an analogue of these results for the Pontrjagin classes. However, such a naive goal is entirely out of reach; indeed, Milnor's use of the Pontrjagin classes to construct an invariant which distinguishes between nondiffeomorphic manifolds which are homeomorphic and PL isomorphic