Triangulations of Grassmannians and flag manifolds

Abstract

Abstract MacPherson in\cite{macp:rob} 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\cite{bab:eric} proved that the spaces \(\mbox{Gr}(2,\mathbb{R}^n)\) and \(\mbox{Gr}(1,2,\mathbb{R}^n)\) are homotopy equivalent to their combinatorial analogs, the simplicial complexes \(\|\mbox{MacP}(2,n)\|\) and \(\|\mbox{MacP}(1,2,n)\|\) respectively. We will prove that \(\mbox{Gr}(2, \mathbb{R}^n)\) and \(\mbox{Gr}(1,2, \mathbb{R}^n)\) are homeomorphic to $\|\mbox{MacP}(2,n)\|$ and $\|\mbox{MacP}(1,2,n)\|$ respectively