Triangulations of Grassmannians and flag manifolds

TL;DR

This paper proves that certain real Grassmannians and flag manifolds are homeomorphic to their associated combinatorial complexes, confirming conjectures and extending known homotopy equivalences to homeomorphisms.

Contribution

The authors establish that $ ext{Gr}(2, extbf{R}^n)$ and $ ext{Gr}(1,2, extbf{R}^n)$ are homeomorphic to their combinatorial models, confirming MacPherson's conjecture for these spaces.

Findings

$ ext{Gr}(2, extbf{R}^n)$ is homeomorphic to $ ext{MacP}(2,n)$

$ ext{Gr}(1,2, extbf{R}^n)$ is homeomorphic to $ ext{MacP}(1,2,n)$

Confirmed conjecture relating Grassmannians and combinatorial complexes.

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 $\|\mathrm{MacP}(2,n)\|$ 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.