Toric topology of the complex Grassmann manifolds

TL;DR

This paper develops a toric topology framework to analyze the orbit spaces of complex Grassmann manifolds under torus actions, revealing their homotopy types and stratification structures.

Contribution

It introduces new methods for describing the topology of orbit spaces of Grassmannians using stratification, admissible polytopes, and parameter spaces, advancing the understanding of their toric topology.

Findings

Orbit space G_{4,2}/T^4 is homeomorphic to a join of a boundary and CP^1.

Orbit space G_{5,2}/T^5 is homotopy equivalent to a space formed by attaching a disc to a suspended RP^2.

The paper provides a method to describe the orbit space topology of G_{n,k} using stratification and parameter spaces.

Abstract

The family of the complex Grassmann manifolds $G_{n,k}$ with a canonical action of the torus $T^n=\mathbb{T}^{n}$ and the analogue of the moment map $\mu : G_{n,k}\to \Delta _{n,k}$ for the hypersimplex $\Delta _{n,k}$, is well known. In this paper we study the structure of the orbit space $G_{n,k}/T^n$ by developing the methods of toric geometry and toric topology. We use a subdivision of $G_{n,k}$ into the strata $W_{\sigma}$ and determine all regular and singular points of the moment map $\mu$, introduce the notion of the admissible polytopes $P_\sigma$ such that $\mu (W_{\sigma}) = \stackrel{\circ}{P_{\sigma}}$ and the notion of the spaces of parameters $F_{\sigma}$, which together describe $W_{\sigma}/T^{n}$ as the product $\stackrel{\circ}{P_{\sigma}} \times F_{\sigma}$. To find the appropriate topology for the set $\cup _{\sigma} \stackrel{\circ}{P_{\sigma}} \times F_{\sigma}$ we introduce the notions of the universal space of parameters $\tilde{\mathcal{F}}$ and the virtual spaces of parameters $\tilde{F}_{\sigma}\subset \tilde{\mathcal{F}}$ such that there exist the projections $\tilde{F}_{\sigma}\to F_{\sigma}$. Hence, we propose a method for the description of the orbit space $G_{n,k}/T^n$. Earlier we proved that the orbit space $G_{4,2}/T^4$, defined by the canonical $T^4$-action of complexity $1$, is homeomorphic to $\partial \Delta _{4,2}\ast \mathbb{C} P^1$. We prove here that the orbit space $G_{5,2}/T^5$, defined by the canonical $T^5$-action of complexity $2$, is homotopy equivalent to the space obtained by attaching the disc $D^8$ to the space $\Sigma ^{4}\mathbb{R} P^2$ by the generator of the group $\pi _{7}(\Sigma ^{4}\mathbb{R} P^2)=\mathbb{Z} _{4}$. In particular, $(G_{5,2}/G_{4,2})/T^5$ is homotopy equivalent to $\partial \Delta _{5,2}\ast \mathbb{C} P^2$. The methods and the results of this paper are fundaments for our theory of $(2l,q)$-manifolds.