Seminar

The oriented matroid Grassmannians, later also called MacPhersonians, were introduced in 1993 by Robert MacPherson as a combinatorial analogue to real Grassmann manifolds. The MacPhersonian MacP(r, n) is the order complex of the partially ordered set of all rank r oriented matroids on a labelled set of n elements, ordered by weak maps. It was a crucial ingredient for giving a combinatorial formula for Pontrjagin classes by Gelfand and MacPherson. Moreover, MacPherson constructed a canonical map µ : G(r, n) → MacP(r, n) from the Grassmannian to the oriented matroid Grassmannian. Since then, the main question is whether the map µ is a homotopy equivalence. Some progress towards understanding this map has been made in the literature, but the topology of the MacPhersonian is still unknown.

In this talk some properties of the MacPhersonian will be discussed and main results will be mentioned. This is an ongoing research project.