Homotopy groups of the combinatorial Grassmannian (Q1275678)

The combinatorial Grassmannian (or MacPhersonian, \(\text{MacP}(k,n)\)), is the order complex of the poset of all oriented matroids of rank \(k\) on \(n\) elements, ordered by weak maps. The topology of these combinatorially-defined, finite simplicial complexes is of utmost importance for MacPherson's theory of combinatorial differential manifolds [\textit{R. D. MacPherson}, Proc. ``Topological Methods in Modern Mathematics: a Symposium in Honor of John Milnor's Sixtieth Birthday'' (Stony Brook NY, 1991), Publish Perish Inc., 203-221 (1993; Zbl 0812.57019)]; in particular, their cohomology provides the characteristic classes for this theory. Results of Anderson, Babson, Davis and others indicate that \(\text{MacP}(k,n)\) has ``at least as much'' cohomology as the corresponding real Grassmannian \(G(k,{\mathbb R}^n)\). In this paper, L. Anderson provides more topological data that indicate that \(\text{MacP}(k,n)\) is ``similar'' to \(G(k,{\mathbb R}^n)\): she shows that the canonical homotopy class \(G(k,{\mathbb R}^n)\rightarrow \text{MacP}(k,n)\) induces maps of homotopy groups \(\pi_i(G(k,{\mathbb R}^n))\rightarrow \pi_i(\text{MacP}(k,n))\) that are isomorphisms for \(i\leq 1\), and surjections for \(i=2\). Furthermore, the homotopy groups \(\pi_i(\text{MacP}(k,n))\) stabilize for \(n\rightarrow\infty\).