Toric topology of the complex Grassmann manifolds

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.

Section 12 improved, corrected the description of the homotopy type for G_{5,2}/T^5 and subsection 3.2 extended; final version, to appear in Moscow Mathematical Journal