Triangulations of Grassmannians and flag manifolds

Abstract

MacPherson conjectured that the Grassmannian $\mathrm{Gr}(2,{\mathbb{R}}^n)$ has the same homeomorphism type as the combinatorial Grassmannian $\|\mbox{MacP}(2,n)\|$, while Babson proved that the spaces $\mathrm{Gr}(2,{\mathbb{R}}^n)$ and $\mathrm{Gr}(1,2,{\mathbb{R}}^n)$ are homotopy equivalent to their combinatorial analogs $\Vert \mathrm{MacP}(2,n)\Vert$ and $\|\mbox{MacP}(1,2,n)\|$ respectively. We will prove that $\mathrm{Gr}(2,{\mathbb{R}}^n)$ and $\mathrm{Gr}(1,2,{\mathbb{R}}^n)$ are homeomorphic to $\|\mbox{MacP}(2,n)\|$ and $\|\mbox{MacP}(1,2,n)\|$ respectively.

arXiv comment29 pages