Jeffrey Giansiracusa: An E-infinity structure on MacPherson's matroid grassmannians

Abstract

In the early 90s, MacPherson introduced a combinatorial model of manifolds called CD-manifolds. The CD world sits somewhere between PL and smooth. The role of vector bundles in this world is played by certain data in terms of oriented matroids, and so vector bundles are classified by the space of oriented matroids, which is called the 'matroid grassmannian'. MacPherson conjectured that the matroid grassmannian is homotopy equivalent to the ordinary real grassmannian. The conjecture remains open. If true, then matroidal bundles would define a version of K-theory that is identical to KO-theory. In this talk I will describe how the matroid grassmannians carry an E-infinity structure so we can at least say that matroidal K-theory is a generalized cohomology theory.