Moscow Mathematical Journal

Volume 19, Issue 3, July–September 2019 pp. 397–463.

Toric Topology of the Complex Grassmann Manifolds

Authors: V. M. Buchstaber (1) and S. Terzić (2)
Author institution:(1) Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow State University M.V.Lomonosov, Skolkovo Institute of Science and Technology, Moscow, Russia
(2) Faculty of Science and Mathematics, University of Montenegro, Podgorica, Montenegro

Summary:

The family of the complex Grassmann manifolds $G_{n,k}$ with the canonical action of the torus $T^n=\mathbb{T}^{n}$ and the analogue of the moment map $\mu \colon G_{n,k}\to \Delta _{n,k}$ 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 $W_{\sigma}$. Relying on this subdivision we determine all regular and singular points of the moment map $\mu$, introduce the notion of the admissible polytopes $P_\sigma$ such that $\mu (W_{\sigma}) = \mathring{P}_{\sigma}$ and the notion of the spaces of parameters $F_{\sigma}$, which together describe $W_{\sigma}/T^{n}$ as the product $\mathring{P}_{\sigma}\times F_{\sigma}$. To find the appropriate topology for the set $\bigcup_{\sigma} \mathring{P}_{\sigma} \times F_{\sigma}$ we introduce also 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 $\tilde{F}_{\sigma}\to F_{\sigma}$. Having this in mind, we propose a method for the description of the orbit space $G_{n,k}/T^n$. The existence of the action of the symmetric group $S_{n}$ on $G_{n,k}$ simplifies the application of this method. In our previous paper we proved that the orbit space $G_{4,2}/T^4$, which is defined by the canonical $T^4$-action of complexity $1$, is homeomorphic to $\partial \Delta _{4,2}\ast \mathbb{C}P^1$. We prove in this paper that the orbit space $G_{5,2}/T^5$, which is defined by the canonical $T^5$-action of complexity $2$, is homotopy equivalent to the space which is 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 very important for the construction of the theory of $(2l,q)$-manifolds we have been recently developing, and which is concerned with manifolds $M^{2l}$ with an effective action of the torus $T^{q}$, $q\leq l$, and an analogue of the moment map $\mu \colon M^{2l}\to P^{q}$, where $P^{q}$ is a $q$-dimensional convex polytope.

2010 Math. Subj. Class. 57S25, 57N65, 53D20, 14M25, 52B11, 14B05.

Keywords: Grassmann manifold, Thom spaces, torus action, orbit spaces, spaces of parameters.

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