The foundations of $(2n,k)$-manifolds

TL;DR

This paper develops a foundational axiomatic framework for $(2n,k)$-manifolds, generalizing toric geometry, and demonstrates its application through the example of complex Grassmann manifolds with torus actions.

Contribution

It introduces a system of axioms for $(2n,k)$-manifolds, constructs a model space with torus action, and extends the theory beyond toric cases to homogeneous spaces with non-zero complexity.

Findings

Established a model space $rak{E}$ with a $T^k$-action for $(2n,k)$-manifolds.

Proved the existence of a $T^k$-equivariant homeomorphism between $rak{E}$ and $M^{2n}$.

Applied the theory to complex Grassmann manifolds $G_{k+1,q}$ with torus actions.

Abstract

In the focus of our paper is a system of axioms that serves as a basis for introducing structural data for $(2n,k)$-manifolds $M^{2n}$, where $M^{2n}$ is a smooth, compact $2n$-dimensional manifold with a smooth effective action of the $k$-dimensional torus $T^k$. In terms of these data a construction of the model space $\mathfrak{E}$ with an action of the torus $T^k$ is given, such that there exists a $T^k$-equivariant homeomorphism $\mathfrak{E}\to M^{2n}$. This homeomorphism induces a homeomorphism $\mathfrak{E}/T^k\to M^{2n}/T^k$. The number $d=n-k$ is called the complexity of an $(2n,k)$-manifold. Our theory comprises toric geometry and toric topology, where $d=0$. It is shown that the class of homogeneous spaces $G/H$ of compact Lie groups, where rk$G=$rk$H$, contains $(2n,k)$-manifolds that have non zero complexity. The results are demonstrated on the complex Grassmann manifolds $G_{k+1,q}$ with an effective action of the torus $T^k$.

Equations

$$M\cong (T^n\times P^n)/\approx ,(t_1,p_1)\approx (t_2,p_2)\text{if and only if }{p}_1=p_2,t_1{t}_2^{-1}\in \Lambda (F(p_1)),$$

$$P(A(L))=(P^J(A(L)))=(\det A_J(L)),$$

$$P_f^J(L)=(\sum _{\sigma \in S_k}\mathrm{sign}(\sigma ){\alpha }_{\sigma (1)}^1\cdots {\alpha }_{\sigma (k)}^k)P_e^J(L)=\text{det}(\alpha ),$$

$$\mu (L)=\frac{{\sum }_J|P^J(L){|}^2{\delta }_J}{{\sum }_J|P^J(L){|}^2},$$

$$({\delta }_J{)}_i=1,i\in J,({\delta }_J{)}_i=0,i\notin J,$$

$$W_{\sigma }=M_{i_1}\cap \cdots \cap M_{i_l}\cap Y_{i_{l+1}}\cap \cdots \cap Y_{i_m},$$

$$s(\sigma )=P_{\sigma },\text{where}{P}_{\sigma }=\text{convhull}(v_{i_1},\dots ,v_{i_l}).$$

$$T^{\sigma }\to W_{\sigma }\to W_{\sigma }/T^{\sigma }.$$

$$\begin{array}{ccc}{W}_{\sigma }/T^{\sigma }& \to & \stackrel{\circ }{P_{\sigma }}\times F_{\sigma }\\ \downarrow \hat{\mu }& & \downarrow \\ \stackrel{\circ }{P_{\sigma }}& =& \stackrel{\circ }{P_{\sigma }}.\end{array}$$

$$W_{\sigma }[{\xi }_{\sigma },c_{\sigma }]=({\pi }_{\sigma }^{-1}\circ {\xi }_{\sigma }^{-1})(c_{\sigma }),$$

$${\eta }_{\sigma ,\bar{\sigma }}:F_{\sigma }\to F_{\bar{\sigma }}$$

$$W_{\overline{\stackrel{ˉ}{\sigma }}}[{\xi }_{\overline{\stackrel{ˉ}{\sigma }}},c_{\overline{\stackrel{ˉ}{\sigma }}}]\subset \bar{\partial }{W}_{\bar{\sigma }}[{\xi }_{\bar{\sigma }},c_{\bar{\sigma }}]\subset \overline{W_{\bar{\sigma }}[{\xi }_{\bar{\sigma }},c_{\bar{\sigma }}]},W_{\bar{\sigma }}[{\xi }_{\bar{\sigma }},c_{\bar{\sigma }}]\subset \bar{\partial }{W}_{\sigma }[{\xi }_{\sigma },c_{\sigma }]\subset \overline{W_{\sigma }[{\xi }_{\sigma },c_{\sigma }]}$$

$$W_{\overline{\stackrel{ˉ}{\sigma }}}[{\xi }_{\overline{\stackrel{ˉ}{\sigma }}},c_{\overline{\stackrel{ˉ}{\sigma }}}]\subset \overline{W_{\sigma }[{\xi }_{\sigma },c_{\sigma }]}\setminus W_{\sigma }[{\xi }_{\sigma },c_{\sigma }]=\bar{\partial }{W}_{\sigma }[{\xi }_{\sigma },c_{\sigma }].$$

$$\sigma (x)=\{P_{\sigma }\in P_{\mathfrak{S}}:x\in {\stackrel{\circ }{P}}_{\sigma }\}.$$

$${\hat{\mu }}^{-1}(x)={\cup }_{\sigma \in \mathfrak{S}(x)}\{y\in W_{\sigma }/T^k:\hat{\mu }(y)=x\}.$$

$$\overline{F_x}={\hat{\mu }}^{-1}(x)\cong {\cup }_{\sigma \in \mathfrak{S}(x)}{F}_{\sigma },$$

$${\cup }_{\sigma \in \mathfrak{S}({\sigma }_0)}{F}_{\sigma }\cong \overline{F}.$$

$$(x,f_1)\approx (y,f_2)\iff x=y\in P_{{\sigma }_0}\subset \partial P^k\text{and}{\eta }_{{\sigma }_1,{\sigma }_0}(f_1)={\eta }_{{\sigma }_2,{\sigma }_0}(f_2).$$

$$M^{2n}/T^k\cong S^{k-1}\ast \bar{F}.$$

$$M^{2n}/T^k\cong P^k\times \overline{F}/\approx ,(x,f_1)\approx (y,f_2)\iff x=y\in \partial P^k.$$

$$M^{2n}/T^k\cong C{S}^{k-1}\times \overline{F}/\approx ,(x,0,f_1)\approx (x,0,f_2),$$

$$C(M^{2n},P^k)={\cup }_{P_{\sigma }\in P_{\mathfrak{S}}}\stackrel{\circ }{P_{\sigma }}.$$

$$d_C(P_{\sigma }^{{}^{\prime }})=(d{P}_{\sigma }{)}_C,$$

$$\mathcal{C}=\left(\begin{array}[]{cccc}1&0&c&1\\ 0&1&1&1\end{array}\right),\;\;c\neq 0,1.$$

$$C_{0}=\left(\begin{array}[]{cccc}1&0&0&1\\ 0&1&1&1\end{array}\right).$$

$$C_{1}=\left(\begin{array}[]{cccc}1&0&1&1\\ 0&1&1&1\end{array}\right).$$

$$\left(\begin{array}[]{cccc}1&0&z_{1}&z_{2}\\ 0&1&z_{3}&z_{4}\end{array}\right)\;\;\text{and}\;\;\left(\begin{array}[]{cccc}w_{1}&1&0&w_{2}\\ w_{3}&0&1&w_{4}\end{array}\right).$$

$$w_1=-\frac{z_3}{z_1},w_2=z_4-\frac{z_2{z}_3}{z_1},w_3=\frac{1}{z_1},w_4=\frac{z_2}{z_1}.$$

$$\mathcal{C}=\left(\begin{array}[]{cccc}-\frac{1}{c}&1&0&1-\frac{1}{c}\\ \frac{1}{c}&0&1&\frac{1}{c}\end{array}\right),\;\;c\neq 0,1,$$

$$C_{\infty}=\left(\begin{array}[]{cccc}0&1&0&1\\ 0&0&1&0\end{array}\right).$$

Full text

1. 1 Introduction

2. 2 The key examples of manifolds with torus actions

3. 2.1 Quasitoric manifolds

4. 2.2 Complex Grassmann manifolds $G_{k+1,q}$

5. 3 Definition of $(2n,k)$-manifolds

6. 3.1 A smooth manifold structure.

7. 4 Almost moment map, strata and admissible polytopes

8. 5 Stabilizers for the torus action on the strata

9. 6 Orbit spaces of the strata

10. 7 On singular and regular values of the almost moment map

11. 7.1 The main stratum and its orbit space

12. 8 Complex of admissible polytopes

13. 8.1 CW-topology on $C(M^{2n},P^{k})$

14. 8.2 Quotient topology on $C(M^{2n},P^{k})$

15. 8.2.1 An example of a closed set in $CQ(G_{4,2},\Delta_{4,2})$ that is not closed in $CW(G_{4,2},\Delta_{4,2})$

16. 8.3 Induced partial ordering on $CQ(M^{2n},P^{k})$

17. 9 The space $\mathfrak{E}(M^{2n},P^{k})$

18. 10 The gluing of the orbit spaces of strata in $\mathfrak{E}(M^{2n},P^{k})/T^{k}$

19. 11 A universal space of parameters

20. 12 Quasitoric manifolds $M^{2n}$ as $(2n,n)$-manifolds

21. 13 $(2n,1)$-manifolds

22. 14 Complex Grassmann manifolds $G_{k+1,q}$ as $(2n,k)$-manifolds with $n=q(k+1-q)$

23. 14.1 Proof of Axiom 5

24. 14.2 Axiom 6 for the manifolds $G_{k+1,q}$

25. 15 The complex manifold of complete flags

26. 16 The orbit spaces of some key examples

27. 17 Examples for the construction of virtual spaces of parameters for $G_{k+1,q}$.

28. 18 Gel’fand-Serganova example

.

1 Introduction

The goal of this paper is to extend the issues from [5], specify the axioms and present the new results of our theory of $(2n,k)$ - manifolds. It is about a wide class of $2n$ - dimensional smooth, compact, oriented manifolds with an effective action of the compact torus $T^{k}$ having only isolated fixed points. We propose the tools for an effective description of the equivariant structure of such manifolds as well as the structure of their orbit spaces.

This class contains toric and quasitoric manifolds $M^{2n}$ , whose effective description in toric topology (see [4]) is given by the combinatorial data $(P^{n},\Lambda)$, where $P^{n}$ is a $n$-dimensional simple polytope and $\Lambda$ is a characteristic function from the set of facets of the polytope $P^{n}$ to the lattice $\mathbb{Z}^{n}$ that satisfies Davis-Januszkiewicz (*) condition [9]. In this case, a $(2n,n)$-manifold is obtained, while the orbit space $M^{2n}/T^{n}$ is homeomorphic to the polytope $P^{n}$. In this paper are described the key examples of the manifolds $M^{2n}$ with an effective action of the torus $T^{k}$ which form a basis for the theory of $(2n,k)$-manifolds.

Any $(2n,k)$-manifold is equipped with the so called almost moment map $\mu:M^{2n}\to\mathbb{R}^{k}$, whose image is a convex polytope $P^{k}$. The polytope $P^{k}$ does not need to be simple and the orbit space $M^{2n}/T^{k}$ for $k<n$ is not homeomorphic to $P^{k}$ in general. For the Grassmann manifolds $G_{k+1,q},$ the polytope $P^{k}$ is the hypersimplex $\Delta_{k+1,q}$ and for the complex flag manifolds $F_{k+1}$, the polytope $P^{k}$ is the permutahedron $P{e}^{k}$. Note that $\Delta_{k+1,q}$ is a simple polytope only for $q=1$ or $q=k$, while $P{e}^{k}$ is a simply polytope for any $k$.

One of the main tools, which we introduce, is a family of admissible polytopes $P_{\sigma}$ which are spanned by some subsets $\sigma$ of vertices of the polytope $P^{k}$. In the quasitoric case the family of admissible polytopes coincides with the family of the faces of the simple polytope $P^{k}$ including the polytope $P^{k}$. In the case of Grassmann manifolds $G_{k+1,q}$ our admissible polytopes coincide with the admissible polytopes from the paper [14] in which a number of results about these polytopes are obtained .

To each admissible polytope $P_{\sigma}$ corresponds a $T^{k}$-invariant subspace $W_{\sigma}\subset M^{2n},$ called the stratum, and a subtorus $T_{\sigma}\subseteq T^{k}$ which acts trivially on $W_{\sigma}$ such that the torus $T^{\sigma}=T^{k}/T_{\sigma}$, $\dim T^{\sigma}=\dim P_{\sigma}$ acts freely on $W_{\sigma}$. The polytope $P^{k}$ is considered to be an admissible polytope and the corresponding stratum $W$ is called the main stratum. One of our axioms requires that $\mu(W_{\sigma})=\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}$ and that the restriction of the induced almost moment map $\hat{\mu}$ to $W_{\sigma}/T^{\sigma}$ is a projection of a fiber bundle with the base $\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}$. This implies that $W_{\sigma}/T^{\sigma}\cong\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}\times F_{\sigma}$, for some topological space $F_{\sigma}$, which is called the space of parameters of the stratum $W_{\sigma}$. In the case of Grassmann manifolds, our strata $W_{\sigma}$ coincide with the strata of Gel’fand, MacPherson, Goresky and Serganova, which they introduced using the action of the algebraic torus $(\mathbb{C}^{*})^{k+1}$ on $G_{k+1,q}$. In this case, the subtorus $(\mathbb{C}^{*})^{\sigma}$ acts freely on $W_{\sigma}$ and $F_{\sigma}=W_{\sigma}/(\mathbb{C}^{*})^{\sigma}$. Note that in general case, an action of the compact torus $T^{k}$ on a $(2n,k)$-manifold $M^{2n}$ does not extend to an effective action of the algebraic torus $(\mathbb{C}^{*})^{k}$.

The union of all admissible polytopes $P_{\sigma}$ is the polytope $P^{k}$ and to each point $x\in P^{k}$ we assign the cortége $\sigma(x)=\{P_{\sigma}:x\in\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}\}$. We assume that $\mu$ is a smooth map and obtain that if $\dim P_{\sigma}=k$ for any $P_{\sigma}\in\sigma(x)$ then $x$ is a regular value for $\mu$.

Towards our goal to describe the equivariant topology of an $(2n,k)$-manifold $M^{2n}$ and its orbit space $M^{2n}/T^{k}$, we introduce structural data and obtain a model for the space $M^{2n}/T^{k}$ in terms of these structural data. Our structural data consist of the virtual spaces of parameters $\tilde{F_{\sigma}}$ together with the continuous projections $p_{\sigma}:\tilde{F}_{\sigma}\to F_{\sigma}$, and the universal space of parameters $\mathcal{F}$ together with the embeddings $I_{\sigma}:\tilde{F}_{\sigma}\to\mathcal{F}$. Note that for the main stratum, the virtual space of parameters $\tilde{F}$ coincides with the space of parameters $F$. We use the fact that the orbit space of the main stratum $W/T^{k}\cong\stackrel{{\scriptstyle\circ}}{{P^{k}}}\times F$ is a dense set in $M^{2n}/T^{k}$. It is required that the compactification of $\stackrel{{\scriptstyle\circ}}{{P^{k}}}\times F$, which corresponds to the compactification $\overline{W/T^{k}}=M^{2n}/T^{k}$, is realized by the topology of the universal space of parameters $\mathcal{F}$, which is a compactification of the space of parameters $F$ of the main stratum such that $\mathcal{F}=\cup_{\sigma}I_{\sigma}(\tilde{F}_{\sigma})$. We realized in detail this approach in [6] for the Grassmann manifold $G_{4,2}$ and in [7] for the Grassmann manifold $G_{5,2}$. In general case, we obtain the orbit spaces $M^{2n}/T^{k}$ as a quotient space of the union $\cup_{\sigma}(\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}\times\tilde{F}_{\sigma})$ by an equivalence relation which is defined in terms of the structural maps $I_{\sigma}:\tilde{F}_{\sigma}\to\mathcal{F}$ and $p_{\sigma}:\tilde{F}_{\sigma}\to F_{\sigma}$.

The complexity of an $(2n,k)$-manifold is defined to be the number $d=n-k$. Our definition of the complexity of an action generalizes the definition of the complexity of an algebraic torus action $(\mathbb{C}^{*})^{k}$ in algebraic geometry and symplectic geometry. The complexity $1$ torus actions, under some appropriate assumptions, are widely studied in algebraic and symplectic geometry (see  [22, 18]). In the recent paper [2], inspired by our work [6, 7], the approach is described for solving the classification problem of complexity $1$ torus actions in terms of equivariant topology. The problem of the torus actions of complexity $2$ or more is still not well understood and, in the literature, it is considered to be quite difficult. Our theory leads to the results in this direction. In the papers  [6, 7] the orbit spaces $\mathbb{C}P^{5}/T^{4}$ and $G_{5,2}/T^{5}$ of the complexity $2$ actions on an $(10,3)$ and $(12,4)$-manifolds, respectively are described in detail.

The axioms and methods of this paper rely on the well known results on the algebraic torus action on Grassmann manifolds [13, 14, 15, 16], as well as on our results on the orbit spaces of the compact torus action on Grassmann manifolds [6, 7]. We emphasize the Grassmann manifolds $G_{k+1,2}$ with an effective action of the torus $T^{k}$ of complexity $k-2$, $k\geq 2$, since many papers have been devoted to them in the recent time, due to their connection with the moduli spaces of curves (see, for example, [16, 17, 20]).

2 The key examples of manifolds with torus actions

In this section we provide the key examples and the basic facts, which served as a starting point for establishing axioms for the theory of $(2n,k)$-manifolds.

2.1 Quasitoric manifolds

A quasitoric manifold is a topological analog of a non-singular projective toric variety from algebraic geometry. We follow  [4] and recall that a quasitoric manifold is a smooth, closed manifold $M^{2n}$ equipped with a smooth action of the torus $T^{n}$ such that:

• •

the action of the torus $T^{n}$ on $M^{2n}$ is locally standard;

• •

the orbit space $M^{2n}/T^{n}$ is diffeomorphic to a simple polytope $P^{n}$ as a manifold with corners.

The second condition gives that there exists a smooth map $\mu:M^{2n}\to P^{n}$ that is constant on $T^{n}$-orbits and maps an $p$-dimensional orbit to an interior point of some $p$-dimensional face of $P^{n}$. In follows that $\mu^{-1}(\stackrel{{\scriptstyle\circ}}{{P^{n}}}),$ is a dense set in $M^{2n}$, the action of the torus $T^{n}$ is free on $\mu^{-1}(\stackrel{{\scriptstyle\circ}}{{P^{n}}})$ and the vertices of the polytope $P^{n}$ correspond to the fixed points of $T^{n}$-action. We recall the notion of the characteristic map and the characteristic matrix for a quasitoric manifold $M^{2n}$ which, together with the combinatorics of the polytope $P^{n}$, determine an equivariant topology and cohomology of a manifold $M^{2n}$.

Let $\{F_{1},\ldots,F_{m}\}$ be the set of all facets of $P^{n}$. The stationary subgroups $T(F_{i})$ for the faces $F_{i}$ are one-dimensional connected subgroups in $T^{n}$ and they can be written as $T(F_{i})=(e^{2\pi\sqrt{-1}\lambda_{1i}\varphi},\ldots,e^{2pi\sqrt{-1}\lambda_{ni}\varphi})$, where $\varphi\in\mathbb{R}$ and $\lambda_{i}=(\lambda_{1i},\ldots,\lambda_{ni})\in\mathbb{Z}^{n}$. Denote by $S(T^{n})$ the set of all connected subgroups of the torus $T^{n}$. The characteristic map $l:\{F_{i}\}\to S(T^{n})$, which is defined by $l:F_{i}\to T(F_{i})$, can be described using the characteristic matrix $\Lambda$ whose columns are the integer vectors $\lambda_{i}$, $1\leq i\leq m$ which satisfy the following condition: if the intersection $F=F_{i_{1}}\cap\ldots\cap F_{i_{n}}$ is a vertex of the polytope $P^{n}$, then the vectors $\lambda_{i_{1}},\ldots,\lambda_{i_{n}}$ form a basis for $\mathbb{Z}^{n}$. Due to Davis-Januszkiewicz theorem (see [9]), the matrix $\Lambda$ and the combinatorics of the polytope $P^{n}$ determine together the cohomology of $M^{2n}$.

Let $\bf{F}$ be the partially ordered set of all faces for $P^{n}$. The points from $\mu^{-1}(F)\subset M^{2n}$ have the same stabilizer for any $F\in\bf{F}$, so the characteristic map extends to the map ${\bf F}\to S(T^{n})$, which to each face $F$ assigns the stationary subgroup of the set $\mu^{-1}(F)$. More precisely, the face $F=F_{i_{1}}\cap\ldots\cap F_{i_{k}}$ maps to the image of the subgroup $T(F_{i_{1}})\times\cdots\times T(F_{i_{k}})\subset T^{n}$. The map ${\bf F}\to S(T^{n})$ is completely determined by the matrix $\Lambda$ and it is denoted by $\Lambda$ as well. A quasitoric manifold $M^{2n}$ can be recovered, up to diffeomorphism, using the characteristic pair $(P^{n},\Lambda)$. In other words, it can be constructed a model for $M^{2n}$ by:

[TABLE]

where $F(p_{1})$ is the smallest face of the polytope $P^{n}$ that contains $p_{1}$.

In this case, for any point $p\in P^{n}$, the cortége $\sigma(p)=\{P_{\sigma}:p\in\stackrel{{\scriptstyle\circ}}{{P}}_{\sigma}\}$ consists of one polytope, that is the face $F(p)$.

2.2 Complex Grassmann manifolds $G_{k+1,q}$

The complex Grassmann manifold $G_{k+1,q}$ consists of all $q$-dimensional complex subspaces in the complex vector space $\mathbb{C}^{k+1}$. The canonical action of the torus $T^{k+1}$ on $\mathbb{C}^{k+1}$, considered in the canonical basis, induces the action of the torus $T^{k+1}$ on the manifold $G_{k+1,q}$. This action is not effective, as the diagonal subgroup $\Delta=\{(t,\ldots,t),t\in S^{1}\}$ acts trivially on $G_{k+1,q}$. The torus $T^{k}=T^{k+1}/H$ acts effectively on $G_{k+1,q}$.

We recall some classical constructions on the complex Grassmann manifolds. After fixing a basis in an $q$-dimensional subspace $L\subset\mathbb{C}^{k+1}$, this subspace can be represented by the $q\times(k+1)$ matrix $A(L)$ such that $\text{rank}A(L)=q$. For any subset $J\subset\{1,\ldots,k+1\}$ consisting of $q$ elements, $|J|=q$, denote by $A_{J}(L)$ the matrix of dimension $q\times q$ given by the columns of the matrix $A_{L}$ that are indexed by $J$. We will assume that the set of all subsets $J=\{j_{1}<j_{2}\ldots<j_{q}\}\subset\{1,\ldots,k+1\}$ is ordered lexicographically. Using this ordering define the vector

[TABLE]

whose coordinates are called the Plücker coordinates of a point $L\in G_{k+1,q}$.

The Plücker coordinates depend on a fixed basis for $L$ and they are, up to constant, uniquely defined. More precisely, two bases $f_{1},\ldots,f_{q}$ and $e_{1},\ldots,e_{q}$ for a subspace $L$ are related by $f_{j}=\sum_{i=1}^{q}\alpha_{j}^{i}e_{i}$, $1\leq j\leq q$. It implies that the Plücker coordinates for $L$ in these two bases are related by

[TABLE]

where $\alpha=(\alpha_{j}^{i})$ is a transition matrix between these two bases. In this way the Plücker coordinates produce an embedding of the Grassmann manifold $G_{k+1,q}$ into $\mathbb{C}P^{N-1}$, where $N={k+1\choose q}$.

The Plücker coordinates define the smooth atlas $\{(M_{I},u_{I})\}$ on $G_{k+1,q}$, where $I$ runs through all $q$-element subsets of the set $\{1,\ldots,k+1\}$, as follows. Here $M_{I}=\{L\in G_{k+1,q}:P^{I}(L)\neq 0\}$ and the coordinate map $u_{I}:M_{I}\to\mathbb{C}^{q(k+1-q)}$ is defined by the Plücker coordinates $P^{J}(L)$, $J=(I\setminus\{i_{p}\})\cup\{j_{s}\}$, $i_{p}\in I,j_{s}\in\{1,\ldots,k+1\}\setminus I$, in such a basis of a subspace $L$ that the $(q\times q)$-dimensional sub-matrix of the matrix $A(L)$ whose columns are indexed by $I$ is an identity matrix.

Let us consider the action of $T^{k+1}$ on $\mathbb{C}P^{N-1}$, which is given by the composition of the $q$-th exterior power representation $T^{k+1}\to T^{N}$ and the standard action of $T^{N}$ on $\mathbb{C}P^{N-1}$. The standard moment map $\mathbb{C}P^{N-1}\to\mathbb{R}^{k+1}$ for such an action of the torus $T^{k+1}$ induces the moment map $\mu:G_{k+1,q}\to\mathbb{R}^{k+1}$ ( see [21]), which is defined by

[TABLE]

where $\delta_{J}\in\mathbb{R}^{k+1}$ are the vectors whose coordinates are given by

[TABLE]

and $J$ runs through the $q$-element subsets of $\{1,\ldots,k+1\}$.

The map $\mu$ is $T^{k+1}$-invariant and the image of $\mu$ is, by its definition, a convex hull over the points $\delta_{J}$. The convex polytope obtained in this way is known as the hypersimplex $\Delta_{k+1,q}$, see [23]. In particular, $\Delta_{4,2}$ is the octahedron.

Recall that the Grassmann manifold $G_{k+1,q}$ admits as well an action of the algebraic torus $(\mathbb{C}^{*})^{k+1}$, which is induced by the coordinate wise action of $(\mathbb{C}^{*})^{k+1}$ on $\mathbb{C}^{k+1}$. The orbit $(\mathbb{C}^{*})^{k+1}\cdot L$ is a smooth submanifold in $G_{k+1,q}$ for any point $L\in G_{k+1,q}$. The stationary subgroup $(\mathbb{C}^{*})^{k+1}_{L}$ of a point $L$ is a toral subgroup in $(\mathbb{C}^{*})^{k+1}$ and the algebraic torus $(\mathbb{C}^{*})^{L}=(\mathbb{C}^{*})^{k+1}/(\mathbb{C}^{*})^{k+1}_{L}$ acts freely on the orbit $(\mathbb{C}^{*})^{k+1}\cdot L$. Moreover, $\mu((\mathbb{C}^{*})^{k+1}\cdot L)=\stackrel{{\scriptstyle\circ}}{{P_{L}}}$, where $P_{L}$ is a convex polytope spanned by the vertices $\delta_{J}$ of the hypersimplex $\Delta_{n,k}$ indexed by those $J$ such that $P^{J}(L)\neq 0$.

3 Definition of $(2n,k)$-manifolds

We assume the following to be given:

• •

a smooth, closed, oriented manifold $M^{2n}$;

• •

a smooth, effective action $\theta$ of the torus $T^{k}$ on $M^{2n}$, where $1\leq k\leq n$, such that the stabilizer of any point is a connected subgroup of $T^{k}$;

• •

a smooth, $\theta$-equivariant map $\mu:M^{2n}\to\mathbb{R}^{k}$, whose image is a $k$-dimensional convex polytope $P^{k}$, where $\mathbb{R}^{k}$ is considered with the trivial $T^{k}$ - action. We assume $\mu:M^{2n}\to\mathbb{R}^{k}$ to be an open map.

The map $\mu$ we call an almost moment map for the given $T^{k}$-action on $M^{2n}$.

We say that the triple $(M^{2n},\theta,\mu)$ is an $(2n,k)$-manifold if it satisfies the six axioms which we formulate below.

3.1 A smooth manifold structure.

Axiom 1**.**

There exists a smooth atlas $\mathfrak{M}=\{(M_{i},\varphi_{i})\}_{i\in I}$, where $M_{i}$ are an open subsets in $M^{2n}$ and $\varphi_{i}:M_{i}\to\mathbb{R}^{2n}$ are coordinate homeomorphisms. Any chart $M_{i}$ is $T^{k}$-invariant, contains exactly one fixed point $x_{i}$ with $\varphi_{i}(x_{i})=(0,\ldots,0)$, such that $x_{i}\neq x_{j}$ for $i\neq j$, The closure of any chart $M_{i}$ is the whole manifold $M^{2n}$.

Any atlas that satisfies Axiom 1 has finitely many charts, since $M^{2n}$ is a compact manifold. It implies:

Corollary 3.1**.**

The action of $T^{k}$ on $M^{2n}$ has finitely many isolated fixed points.

By $m$ denote the number of fixed points for $T^{k}$-action on $M^{2n}$. We enumerate as $(M_{1},\varphi_{1}),\ldots,(M_{m},\varphi_{m})$ the charts given by Axiom 1 The sets $Y_{i}=M^{2n}\setminus M_{i}$ are closed in $M^{2n}$ and $T^{k}$-invariant, by their definition.

There is a standard concept of the boundary for a subset $Y$ of a topological space $X$ in general topology. There is also a concept of the boundary of a manifold, manifold with corners and, in that context the boundary of a convex polytope, used in algebraic topology and differential geometry. In all these cases $\partial$ is a standard notation for the boundary. It is not difficult to realize that all these concepts are not always appropriate for some purposes in the theory of $(2n,k)$-manifolds. Therefore, we introduce the new notion of the boundary and the corresponding symbol. The boundary $\bar{\partial}$ of a subset $Y$ in a topological space $X$ is the set $\bar{\partial}Y=\bar{Y}\setminus Y$, where $\bar{Y}$ is the closure of a set $Y$ in a space $X$. Note that if $Y$ is an open set in $X$ , then $\bar{\partial}Y=\partial Y$, where $\partial Y=\bar{Y}\cap\overline{X\setminus Y}$ is the standard boundary as defined in general topology. In the sequel, set $\bar{\partial}Y\subset X$ is called the $\bar{\partial}-$boundary of a set $Y$ as well.

Therefore, the fact that the above defined sets $M_{i}$ are dense, open sets in $M^{2n}$, implies that $Y_{i}=\partial M_{i}=\bar{\partial}M_{i}$.

For any $\sigma=\{i_{1},\ldots,i_{l}\}\subseteq[1,m]$ let us consider the set :

[TABLE]

where $\{i_{l+1},\ldots,i_{m}\}=[1,m]\setminus\{i_{1},\ldots,i_{l}\}$.

Definition 3.2**.**

The non-empty set $W_{\sigma}$ is said to be a stratum. The index set $\sigma\subset[1,m]$ of a stratum $W_{\sigma 7}$ is said to be an admissible set.

Lemma 3.3**.**

The strata $W_{\sigma}$ are $T^{k}$-invariant, pairwise disjoint and their union is the whole manifold $M^{2n}$.

Proof.

Any stratum $W_{\sigma}$ is $T^{k}$-invariant since the sets $M_{i}$ and $Y_{i}$ are $T^{k}$- invariant. Further, if $W_{\sigma_{1}}\neq W_{\sigma_{2}}$ then $\sigma_{1}\neq\sigma_{2}$ and, thus, we can assume that there exists $i\in\sigma_{1}$ such that $i\notin\sigma_{2}$, $1\leq i\leq m$. Therefore, for $x\in W_{\sigma_{1}}$ it follows that $x\in M_{i}$ and, thus, $x\notin Y_{i}$, which gives that $x\notin W_{\sigma_{2}}$. Similarly, if $x\in W_{\sigma_{2}}$ then $x\in Y_{i}$ and thus $x\notin M_{i}$, which implies $x\notin W_{\sigma_{1}}$. The union of all strata is the manifold $M^{2n}$, since the charts cover $M^{2n}$. Note that it is uniquely defined an admissible set $\sigma=\sigma(x)=\{i\in[1,m]|x\in M_{i}\}$ for any point $x\in M^{2n}$. ∎

This Lemma together with the fact that the $T^{k}$-action on $M^{2n}$ is continuous implies that the closure $\overline{W_{\sigma}}$ is a $T^{k}$-invariant set for any stratum $W_{\sigma}$.

Example 3.4**.**

The set $W_{[1,m]}=M_{1}\cap\cdots\cap M_{m}$ is non-empty and, thus, it is a stratum and the set $\sigma=[1,m]$ is an admissible set. The set $W_{[1,m]}$ is an open dense set in $M^{2n}$ since, by Axiom 1 any chart $M_{i}$, $1\leq i\leq m$ is an open, dense set in $M^{2n}$. The stratum $W_{[1,m]}$ is called the main stratum and it is further denoted by $W=W_{[1,m]}$.

Example 3.5**.**

The set $W_{\{i\}}=M_{i}\cap\left(\bigcap_{j\neq i}Y_{j}\right)$ is a stratum and the set $\sigma=\{i\}$ is an admissible set for any $1\leq i\leq m$. It follows from the observation that the set $W_{i}$ is non-empty since, by Axiom 1 the fixed point $x_{i}$ which belongs to $M_{i}$ also belongs to all $Y_{j}$, $j\neq i$, $1\leq j\leq m$.

Remark 3.6*.*

Since $M^{2n}=M_{1}\cup\cdots\cup M_{m}$, we see that $W_{\emptyset}=Y_{1}\cap\cdots\cap Y_{m}=\emptyset$, which implies that $W_{\emptyset}$ is not a stratum and $\emptyset$ is not an admissible set.

Remark 3.7*.*

A stratum $W_{\sigma}$ different from the main stratum $W$ is not an open set in $M^{2n}$. It follows from the observations that in this case there exists a chart $M_{i}$ such that $M_{i}\cap W_{\sigma}=\emptyset$ and that $M_{i}$ is a dense set in $M^{2n}$.

Lemma 3.8**.**

The boundary $\bar{\partial}W_{\sigma}$ of a stratum $W_{\sigma}$ is contained in the union of the strata $W_{\tilde{\sigma}}$, where $\tilde{\sigma}$ are the admissible sets such that $\tilde{\sigma}\subset\sigma$.

Proof.

Let $W_{\sigma}=M_{i_{1}}\cap\cdots\cap M_{i_{l}}\cap Y_{i_{l+1}}\cap\ldots\cap Y_{i_{m}}$. Then the boundary $\partial W_{\sigma}$ is contained in the union of the sets $\left(\bigcap\limits_{q=1}^{p}\bar{\partial}M_{i_{j_{q}}}\right)\cap\left(\bigcap\limits_{s\neq j_{1},\ldots,j_{p}}M_{i_{s}}\right)\cap\left(\bigcap\limits_{j=l+1}^{m}Y_{i_{j}}\right)=\left(\bigcap\limits_{q=1}^{p}Y_{i_{j_{q}}}\right)\cap\left(\bigcap\limits_{s\neq j_{1},\ldots,j_{p}}M_{i_{s}}\right)\cap\left(\bigcap\limits_{j=l+1}^{m}Y_{i_{j}}\right)$, where $j_{1}<\ldots<j_{p}$ and $1\leq p\leq l$. Such nonempty sets give $W_{\tilde{\sigma}}$, where $\tilde{\sigma}=\sigma\setminus\{i_{j_{1}},\ldots i_{j_{p}}\}$. Hence, $\bar{\partial}W_{\sigma}\subseteq\cup W_{\tilde{\sigma}}$, where ${\tilde{\sigma}}$ goes through all proper admissible subsets of $\sigma$. ∎

4 Almost moment map, strata and admissible polytopes

Axiom 2**.**

The map $\mu$ is a bijection between the set of fixed points and the set of vertices of the polytope $P^{k}$.

Since an $k$-dimensional polytope has at least $k+1$ vertices, it follows:

Corollary 4.1**.**

The number of fixed points for $T^{k}$-action on $M^{2n}$ is not less then $k+1$.

Let $S(P^{k})$ be a family of all convex polytopes that are spanned by the vertices of the polytope $P^{k}$. By $\mathfrak{S}$ denote the set of all admissible sets. Using the almost moment map $\mu$, we define the map $s:\mathfrak{S}\to S(P^{k})$ as follows. Put $v_{i}=\mu(x_{i})$, $1\leq i\leq m$ and let $\sigma=\{i_{1},\ldots,i_{l}\}\subseteq[1,m]$. By Axiom 1, for any $i_{j}\in\sigma$ there exists a unique fixed point $x_{i_{j}}\in M_{i_{j}}$ . We put

[TABLE]

Definition 4.2**.**

The polytope $P_{\sigma}\in S(P^{k})$ is said to be an admissible polytope if it is in the image of the map $s:\mathfrak{S}\to S(P^{k})$.

Remark 4.3*.*

Since $W_{\emptyset}=\emptyset$, it follows that $\emptyset$ is not an admissible polytope.

Example 4.4**.**

The polytope $P^{k}$ is an admissible polytope. The set $\sigma=\{1,\ldots,m\}$ is an admissible set as it is shown in Example 3.4. In addition, by Axiom 2, $s(\sigma)$ is a convex hull of all vertices of $P^{k}$, which implies that $s(\sigma)=P^{k}$.

Example 4.5**.**

Any vertex $v_{i}$ of $P^{k}$ is an admissible polytope. To see that, by Axiom 2, take the fixed point $x_{i}$ such that $\mu(x_{i})=v_{i}$. Then the set $\sigma=\{i\}$ is an admissible set as it is shown in Example 3.5 and $s(\sigma)=v_{i}$.

Definition 4.6**.**

The set of all admissible polytopes is said to be pure if any admissible polytope of the dimension $\leq k-1$ is a face of some admissible polytope of the dimension $k$.

Example 4.7**.**

The set of admissible polytopes of a quasitoric manifold is a pure set, Moreover it follows from [6] and [7] (Proposition 9, Propositions 11-15 and Corollary 18) that the set of admissible polytopes for the Grassmann manifolds $G_{4,2}$ and $G_{5,2}$ are pure sets as well.

Remark 4.8*.*

For a general $(2n,k)$-manifold, two admissible polytopes $P_{\sigma_{1}}$ and $P_{\sigma_{2}}$ may have nonempty intersection $\stackrel{{\scriptstyle\circ}}{{P}}_{\sigma_{1}}\cap\stackrel{{\scriptstyle\circ}}{{P}}_{\sigma_{2}}$. For example, one can verify this in the case of complex Grassmann manifold $G_{4,2}$ which is an $(8,3)$-manifold, see [6].

Definition 4.9**.**

The point $p\in P^{k}$ is said to be an exceptional point if $p\in\stackrel{{\scriptstyle\circ}}{{P}}_{\sigma_{1}}\cap\stackrel{{\scriptstyle\circ}}{{P}}_{\sigma_{2}}$ for some different admissible polytopes $P_{\sigma_{1}},P_{\sigma_{2}}$. Otherwise, it is said to be simple.

In this way, the set of exceptional points $S\subseteq P^{k}$ is defined .

5 Stabilizers for the torus action on the strata

By $S(T^{k})$ denote, as above, the set of all connected subgroups of the torus $T^{k}$. Note that a connected subgroup of the torus $T^{k}$ is a torus. Let us consider a function $\chi:M^{2n}\to S(T^{k})$ which to any point $x$ assigns its stabilizer $\chi(x)$ regarded to the given $T^{k}$-action on $M^{2n}$. It follows from the set-up assumptions that $\chi(x)$ is a connected subgroup of the torus $T^{k}$. We assume the following to be satisfied:

Axiom 3**.**

The characteristic function $\chi$ is constant on any stratum $W_{\sigma}$.

Using Axiom 3, the torus $T^{\sigma}=T^{k}/\chi(W_{\sigma})$ can be defined for any stratum $W_{\sigma}$.

Corollary 5.1**.**

The torus $T^{\sigma}$ acts freely on $W_{\sigma}$, which gives the principal bundle

[TABLE]

It is shown in [7] (Remark 3) that, in the case of Grassmann manifolds, the notion of the strata as defined in [14], coincides with our notion of the strata. In addition, one verifies [7] (Proposition 4) that the characteristic function is constant on any stratum of the Grassmann manifolds.

6 Orbit spaces of the strata

By its definition the almost moment map $\mu:M^{2n}\to P^{k}$ is $T^{k}$ - invariant . Therefore, it induces the map $\widehat{\mu}:M^{2n}/T^{k}\to P^{k}$.

Axiom 4**.**

The almost moment map $\mu$:

• a)

maps a stratum $W_{\sigma}$ onto $\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}$,

• b)

induces the fiber bundle $\widehat{\mu}_{\sigma}:W_{\sigma}/T^{\sigma}\to\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}$,

• c)

$\dim P_{\sigma}=\dim T^{\sigma}$.

An immediate consequence of this Axiom is:

Corollary 6.1**.**

• It holds
• •

$\widehat{\mu}(W/T^{k})=\stackrel{{\scriptstyle\circ}}{{P^{k}}}$* for the main stratum $W=M_{1}\cap\cdots\cap M_{m}$,*

• •

$\mu(W_{\{i\}})=\{v_{i}\}$, where $v_{i}$ is a vertex and $W_{\{i\}}=M_{i}\cap\left(\bigcap_{j\neq i}\right)Y_{j}$.

Remark 6.2*.*

Note that Axiom 4 does not require a fiber bundle $\widehat{\mu}_{\sigma}:W_{\sigma}/T^{\sigma}\to\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}$ to be smooth, since there is no argument to claim that, in general, a stratum $W_{\sigma}$ is a smooth submanifold in $M^{2n}$. The main stratum being open is of course a smooth submanifold, but for the other strata it does not have to be the case. Even for the Grassmann manifolds, the differential geometry of the strata can be very complicated, see [19].

By $[F_{\sigma}]$ denote the homeomorphic type of a fiber for the fiber bundle $\widehat{\mu}_{\sigma}:W_{\sigma}/T^{\sigma}\to\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}$.

Definition 6.3**.**

The space $F_{\sigma}$ of a homeomorphic type $[F_{\sigma}]$ is called the space of parameters of a stratum $W_{\sigma}$.

Since $\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}$ is contractible, for the fiber bundle $W_{\sigma}/T^{\sigma}\to\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}$ we conclude that the following holds:

Corollary 6.4**.**

The fiber bundle $\widehat{\mu}_{\sigma}:W_{\sigma}/T^{\sigma}\to\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}$ is isomorphic to the trivial bundle. That is $W_{\sigma}/T^{\sigma}$ is homeomorphic to $\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}\times F_{\sigma}$ by the fiber wise homeomorphism

[TABLE]

Definition 6.5**.**

For any $\sigma\in\mathfrak{S}$, we fix the space $F_{\sigma}$ and the trivialization $h_{\sigma}:W_{\sigma}/T^{\sigma}\to\stackrel{{\scriptstyle\circ}}{{P}}_{\sigma}\times F_{\sigma}$ as structural data of $(2n,k)$-manifolds.

Let $\overline{W_{\sigma}/T^{\sigma}}$ denote the closure of $W_{\sigma}/T^{\sigma}$. It is a compact subset in $M^{2n}/T^{k}$ since we assume $M^{2n}$ to be a compact manifold. We obtain:

Corollary 6.6**.**

$\widehat{\mu}(\overline{W_{\sigma}/T^{\sigma}})=P_{\sigma}$.

Proof.

It holds $\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}\subset\widehat{\mu}(\overline{W_{\sigma}/T^{\sigma}})\subseteq P_{\sigma}$ since $\widehat{\mu}:M^{2n}/T^{k}\to P^{k}$ is continuous and $\widehat{\mu}(W_{\sigma}/T^{\sigma})=\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}$. Furthermore, since $\widehat{\mu}(\overline{W_{\sigma}/T^{k}})$ is a compact set, it follows $\widehat{\mu}(\overline{W_{\sigma}/T^{\sigma}})=P_{\sigma}$. ∎

The trivialization $h_{\sigma}:W_{\sigma}/T^{\sigma}\to\stackrel{{\scriptstyle\circ}}{{P}}_{\sigma}\times F_{\sigma}$ induces the projection $\xi_{\sigma}:W_{\sigma}/T^{\sigma}\to F_{\sigma}$. For any point $c_{\sigma}\in F_{\sigma}$, define a subspace $W_{\sigma}[\xi_{\sigma},c_{\sigma}]$ of $W_{\sigma}$ by

[TABLE]

where $\pi_{\sigma}:W_{\sigma}\to W_{\sigma}/T^{\sigma}$ is a projection.

Definition 6.7**.**

The space $W_{\sigma}[\xi_{\sigma},c_{\sigma}]$ is said to be the leaf of a stratum $W_{\sigma}$ given by the trivialization $h_{\sigma}$.

Note that, by its definition, a leaf $W_{\sigma}[\xi_{\sigma},c_{\sigma}]$ is invariant under the action of the torus $T^{k}$ and $W_{\sigma}=\cup_{c_{\sigma}\in F_{\sigma}}W_{\sigma}[\xi_{\sigma},c_{\sigma}]$.

The definition of a leaf also implies:

Lemma 6.8**.**

Let $\widehat{\mu}_{\xi_{\sigma},c_{\sigma}}$ denote the restriction of the map $\widehat{\mu}:M^{2n}/T^{k}\to P^{k}$ to $W_{\sigma}[\xi_{\sigma},c_{\sigma}]/T^{\sigma}$. Then the map $\widehat{\mu}_{\xi_{\sigma},c_{\sigma}}:W_{\sigma}[\xi_{\sigma},c_{\sigma}]/T^{\sigma}\to\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}$ is a homeomorphism for any $c_{\sigma}\in F_{\sigma}$.

Moreover, we obtain:

Lemma 6.9**.**

$\widehat{\mu}(\overline{W_{\sigma}[\xi_{\sigma},c_{\sigma}]/T^{\sigma}})=P_{\sigma}$.

Proof.

It follows from Lemma 6.8 that $\widehat{\mu}(\overline{W_{\sigma}[\xi_{\sigma},c_{\sigma}]/T^{\sigma}})\subseteq P_{\sigma}$, since $\widehat{\mu}$ is a continuous map. On the other hand, $\overline{W_{\sigma}[\xi_{\sigma},c_{\sigma}]/T^{\sigma}}$ is a closed subset in the compact space $M^{2n}/T^{k}$ and, thus, it is a compact set as well. It implies that $\widehat{\mu}(\overline{W_{\sigma}[\xi_{\sigma},c_{\sigma}]/T^{\sigma}})$ is a compact set in $P_{\sigma}$ which contains the interior of $P_{\sigma}$, which further implies that $\widehat{\mu}(\overline{W_{\sigma}[\xi_{\sigma},c_{\sigma}]/T^{\sigma}})=P_{\sigma}$. ∎

Suppose it is given an $2n$-dimensional manifold $M^{2n}$ with an effective action of the algebraic torus $(\mathbb{C}^{*})^{k}$. Assume that the induced action of the compact torus $T^{k}\subset(\mathbb{C}^{*})^{k}$ on $M^{2n}$ satisfies Axioms 1-4 and the following folds:

1. (1)

Any stratum $W_{\sigma}$ is $(\mathbb{C}^{*})^{k}$- invariant; 2. (2)

The free action of the torus $T^{\sigma}$ on $W_{\sigma}$ extends to a free action of the algebraic torus $(\mathbb{C}^{*})^{\sigma}$ on $W_{\sigma}$; 3. (3)

The projections $\hat{\mu}:W_{\sigma}/T^{\sigma}\to\stackrel{{\scriptstyle\circ}}{{P}}_{\sigma}$ and $\hat{\pi}:W_{\sigma}/T^{\sigma}\to W_{\sigma}/(\mathbb{C}^{*})^{\sigma}$ define the homeomorphism $h_{\sigma}=(\hat{\mu},\hat{\pi}):W_{\sigma}/T^{\sigma}\to\stackrel{{\scriptstyle\circ}}{{P}}_{\sigma}\times W_{\sigma}/(\mathbb{C}^{*})^{\sigma}$.

Then, the space of parameters $F_{\sigma}$ of a stratum $W_{\sigma}$ can be identified with $F_{\sigma}\cong W_{\sigma}/(\mathbb{C}^{*})^{\sigma}$.

Definition 6.10**.**

We say than an action of the compact torus $T^{k}$ on an $(2n,k)$-manifold $M^{2n}$ extends to the compatible action of the algebraic torus $(\mathbb{C}^{*})^{k}$ if it is defined an action of the algebraic torus $(\mathbb{C}^{*})^{k}$ on $M^{2n}$, which satisfies the conditions 1-3 given above.

We use the description of the strata as spaces consisting of leafs to formulate the properties which allow to describe the gluing of the strata. Recall that we have fixed the projection $\xi_{\sigma}:W_{\sigma}/T^{\sigma}\to F_{\sigma}$ as a part of our structural data.

Axiom 5**.**

For any leaf $W_{\sigma}[\xi_{\sigma},c_{\sigma}]$ of $W_{\sigma}$ it holds:

• a)

it is a smooth submanifold in $M^{2n}$ and the induced map $\mu_{\xi_{\sigma},c_{\sigma}}:W_{\sigma}[\xi_{\sigma},c_{\sigma}]\to\stackrel{{\scriptstyle\circ}}{{P}}_{\sigma}$ is a smooth fiber bundle,

• b)

its boundary $\bar{\partial}W_{\sigma}[\xi_{\sigma},c_{\sigma}]$ is the union of leafs $W_{\bar{\sigma}}[\xi_{\bar{\sigma}},c_{\bar{\sigma}}]$ for exactly one $c_{\bar{\sigma}}\in F_{\bar{\sigma}}$, where $P_{\bar{\sigma}}$ runs through some admissible faces for $P_{\sigma}$ and $\sigma\in\mathfrak{S}$,

• c)

the map $\eta_{\sigma,\bar{\sigma}}:F_{\sigma}\to F_{\bar{\sigma}}$, $\eta_{\sigma,\bar{\sigma}}(c_{\sigma})=c_{\bar{\sigma}}$ given by b) is a continuous map.

Remark 6.11*.*

Axiom 5, as it will be seen in Section 2.2, is motivated by the results of Atiyah, Guillemin-Sternberg and Gel’fand-MacPherson about $(\mathbb{C}^{*})^{n}$-action on the complex Grassmann manifolds $G_{n,k}$.

We deduce the following important consequence of the statement b) of Axiom 5.

Proposition 6.12**.**

A face of any admissible polytope is an admissible polytope.

Proof.

Let us fix some face $P_{\tilde{\sigma}}$ of an admissible polytopes $P_{\sigma}$ and let us consider a point $p\in\stackrel{{\scriptstyle\circ}}{{P_{\tilde{\sigma}}}}$. By Lemma 6.9, we see that $\widehat{\mu}^{-1}(p)\cap\bar{\partial}W_{\sigma}[\xi_{\sigma},c_{\sigma}]/T^{k}\neq\emptyset$. Therefore, there exists a point $x$ from $\bar{\partial}$-boundary of the leaf $W_{\sigma}[\xi_{\sigma},c_{\sigma}]$ such that $\pi(x)\in\widehat{\mu}^{-1}(p)$. It implies that $\mu(x)=p$. By Axiom 5, we have that $x$ belongs to some leaf $W_{\bar{\sigma}}[\xi_{\bar{\sigma}},c_{\bar{\sigma}}]$, where $P_{\bar{\sigma}}$ is a face of $P_{\sigma}$. This implies that $P_{\bar{\sigma}}$ is an admissible polytope and that $\mu(x)=p\in\stackrel{{\scriptstyle\circ}}{{P_{\bar{\sigma}}}}$. Therefore, $p\in\stackrel{{\scriptstyle\circ}}{{P_{\tilde{\sigma}}}}\cap\stackrel{{\scriptstyle\circ}}{{P_{\bar{\sigma}}}}$, and, since $P_{\bar{\sigma}}$ and $P_{\tilde{\sigma}}$ are faces of the same polytope $P_{\sigma}$, it follows that $P_{\tilde{\sigma}}=P_{\bar{\sigma}}$. Therefore, $P_{\tilde{\sigma}}$ is an admissible polytope. ∎

Remark 6.13*.*

In the case of the canonical action of the algebraic torus $(\mathbb{C}^{*})^{k+1}$on $G_{k+1,q}$, this result is obtained in [1, 14].

Remark 6.14*.*

Note that combining Axiom 5 and the proof of Proposition 6.12 we obtain that if $P_{\bar{\sigma}}$ is a face of $P_{\sigma}$ then there exists a leaf $W_{\bar{\sigma}}[\xi_{\bar{\sigma}},c_{\tilde{\sigma}}]$ which is contained in the $\bar{\partial}$-boundary of the leaf $W_{\sigma}[\xi_{\sigma},c_{\sigma}]$.

It follows that the condition b) of Axiom 5 can be strengthen:

Corollary 6.15**.**

For any $c_{\sigma}\in F_{\sigma}$, the boundary $\bar{\partial}W_{\sigma}[\xi_{\sigma},c_{\sigma}]$ of a leaf $W_{\sigma}[\xi_{\sigma},c_{\sigma}]$ of a stratum $W_{\sigma}$ is the union of leafs $W_{\bar{\sigma}}[\xi_{\bar{\sigma}},c_{\bar{\sigma}}]$ for exactly one $c_{\bar{\sigma}}\in F_{\bar{\sigma}}$, where $P_{\bar{\sigma}}$ runs through all faces for $P_{\sigma}$.

The statement a) of Axiom 5 combining with Corollary 6.15 directly implies that Lemma 6.9 can be strengthen:

Corollary 6.16**.**

The map $\widehat{\mu}:\overline{W_{\sigma}[\xi_{\sigma},c_{\sigma}]/T^{\sigma}}\to P_{\sigma}$ is a homeomorphism.

Let $P_{\bar{\sigma}}$ be a face of $P_{\sigma}$. Then Corollary 6.15 implies an existence of the map

[TABLE]

defined by: $\eta_{\sigma,\bar{\sigma}}(c_{\sigma})$ is a unique point $c_{\bar{\sigma}}$ from $F_{\bar{\sigma}}$ such that $W_{\bar{\sigma}}[\xi_{\bar{\sigma}},c_{\bar{\sigma}}]\subset\bar{\partial}W_{\sigma}[\xi_{\sigma},c_{\sigma}]$. The statement c) of Axiom 5 states that the map $\eta_{\sigma,\bar{\sigma}}$ is a continuous map.

Let now $P_{\bar{\bar{\sigma}}}$ be a face of $P_{\bar{\sigma}}$ and $P_{\bar{\sigma}}$ be a face of $P_{\sigma}$. Then Axiom 5 states an existence of the projections $\xi_{\bar{\bar{\sigma}}}:W_{\bar{\bar{\sigma}}}/T^{k}\to F_{\bar{\bar{\sigma}}}$ and $\xi_{\sigma}:W_{\sigma}/T^{k}\to F_{\sigma}$ such that

[TABLE]

Since the leafs are disjoint, it follows that

[TABLE]

Altogether this implies:

Corollary 6.17**.**

For any pair $P_{\bar{\sigma}}\subset P_{\sigma}$ there exists the map $\eta_{\sigma,\sigma^{{}^{\prime}}}:F_{\sigma}\to F_{\bar{\sigma}}$ such that if $P_{\bar{\bar{\sigma}}}\subset P_{\bar{\sigma}}\subset P_{\sigma}$ then $\eta_{\bar{\sigma},\bar{\bar{\sigma}}}\circ\eta_{\sigma,\bar{\sigma}}=\eta_{\sigma,\bar{\bar{\sigma}}}$.

7 On singular and regular values

of the almost moment map

We characterize the singular and regular values of the almost moment map $\mu:M^{2n}\to P^{k}$.

Definition 7.1**.**

The cortége $\sigma(x)$ of a point $x\in P^{k}$ is a set of admissible polytopes defined by:

[TABLE]

Obviously, $\sigma(x)\neq\emptyset$ for any point $x\in P^{k}$, since $\cup_{\sigma\in\mathfrak{S}}\stackrel{{\scriptstyle\circ}}{{P}}_{\sigma}=P^{k}$. Moreover, $P^{k}\in\sigma(x)$ for any point $x\in\stackrel{{\scriptstyle\circ}}{{P^{k}}}$ .

Definition 7.2**.**

The point $x\in P^{k}$ is said to be a regular point if $\dim P_{\sigma}=k$ for all $P_{\sigma}\in\sigma(x)$.

Note that if $x\in P^{k}$ is a regular point then $x\in\stackrel{{\scriptstyle\circ}}{{P^{k}}}$. By $P^{k}_{r}$ we denote the set of regular points in $P^{k}$.

Remark 7.3*.*

Note that the set of regular points in $\stackrel{{\scriptstyle\circ}}{{P^{k}}}$ is a non-empty set and moreover it is a dense set in $\stackrel{{\scriptstyle\circ}}{{P^{k}}}$. This follows from the fact that there are finitely many admissible polytopes, so the union of admissible polytopes of the dimension less then $k$ has the dimension less then $k$.

For a given $(2n,k)$-manifold $M^{2n}$, let $Z$ denote the union of all open admissible polytopes $\stackrel{{\scriptstyle\circ}}{{P}}_{\sigma}$ whose dimension is $<k$. The following immediately holds:

Lemma 7.4**.**

The set $\stackrel{{\scriptstyle\circ}}{{P^{k}}}\setminus(Z\cap\stackrel{{\scriptstyle\circ}}{{P^{k}}})$ coincides with $P^{k}_{r}$. In particular $P^{k}_{r}$ is an open set in $P^{k}$ which has finitely many connected components.

Example 7.5**.**

For the Grassmann manifold $G_{4,2}$, the set $P^{3}_{r}\subset\stackrel{{\scriptstyle\circ}}{{\Delta}}_{4,2}$ is the complement to the union of three open diagonal squares $\stackrel{{\scriptstyle\circ}}{{P}}_{12,34}$, $\stackrel{{\scriptstyle\circ}}{{P}}_{13,24}$, $\stackrel{{\scriptstyle\circ}}{{P}}_{14,23}$ in $\stackrel{{\scriptstyle\circ}}{{\Delta}}_{4,2}$ (see [6]).

For the Grassmann manifold $G_{5,2}$, the set $P^{4}_{r}\subset\stackrel{{\scriptstyle\circ}}{{\Delta}}_{5,2}$ is the complement to the union of of ten open prisms $\stackrel{{\scriptstyle\circ}}{{P}}_{i}$, $1\leq i\leq 10$ in $\stackrel{{\scriptstyle\circ}}{{\Delta}}_{5,2}$ (see [7],Proposition 9).

Recall that $x$ is a regular value of the almost moment map $\mu:M^{2n}\to P^{k}$ if and only if any $y\in\mu^{-1}(x)$ is a regular point, that is the differential of $\mu$ at $y$ has rank equal to $k$.

Theorem 7.6**.**

If $x\in P^{k}_{r}$ then $x$ is a regular value for the almost moment map $\mu:M^{2n}\to P^{k}$.

Proof.

Let $x\in P^{k}$ be a regular point. It holds that $\mu^{-1}(x)\subset\cup_{\sigma}W_{\sigma}$, where the union goes over all admissible sets $\sigma$ such that $P_{\sigma}\in\sigma(x)$. Now, if $y\in\mu^{-1}(x)\cap W_{\sigma}$, then $y$ belongs to a unique leaf $W_{\sigma}[\xi_{\sigma},c_{\sigma}]$. The differential of the map $\mu_{\xi_{\sigma},c_{\sigma}}:W_{\sigma}[\xi_{\sigma},c_{\sigma}]\to\stackrel{{\scriptstyle\circ}}{{P}}_{\sigma}$ is an epimorphism, according to Axiom 5. Since $P_{\sigma}\in\sigma(x)$, it follows that the rank of the differential of $\mu_{\xi_{\sigma},c_{\sigma}}$ at $y$ is equal to $\dim P_{\sigma}=k$, which proves the statement. ∎

Note that the above proof implies that any point of the main stratum $W$ is a regular point of the almost moment map.

Corollary 7.7**.**

For any point $x\in\stackrel{{\scriptstyle\circ}}{{P^{k}}}$ the preimage $M_{x}^{2n-k}=\mu^{-1}(x)$ is a closed submanifold in $M^{2n}$ of the dimension $2n-k$. The torus $T^{k}$ acts freely on the manifold $M_{x}^{2n-k}$ and the orbit space $M_{x}^{2n-k}/T^{k}$ is a smooth manifold of the dimension $2n-2k$, which can be identified with some compactification of the space of parameters $F$ of the main stratum.

From Remark 7.3, it follows the well-known fact:

Corollary 7.8**.**

The set of regular values of the almost moment map $\mu:M^{2n}\to P^{k}$ is a dense set in $P^{k}$.

Remark 7.9*.*

From our axioms, in general case, does not follow that the set of regular values of the almost moment map $\mu$ coincides with the set $P^{k}_{r}$. Nevertheless, in the case of Grassmann manifolds $G_{k+1,q}$, it is proved in [7] that these two sets coincide.

We can push up this further.

Lemma 7.10**.**

The set $M^{2n}_{r}=\mu^{-1}(P^{k}_{r})$ is a dense submanifold in $M^{2n}$ with a free action of the torus $T^{k}$. The orbit space $M^{2n}_{r}/T^{k}$ is a smooth manifold which is dense in $M^{2n}/T^{k}$. Moreover, the maps $\mu:M^{2n}_{r}\to P^{k}_{r}$ and $\hat{\mu}:M^{2n}_{r}\to P^{k}_{r}$ are projections of the smooth fiber bundles.

Proof.

The first statement follows from the observations that $P^{k}_{r}$ is an open set in $P^{k}$ and that, by Axiom 4, $T^{k}$ acts freely on $M^{2n}_{r}$.

The second statement follows from the observation that the almost moment map $\mu:M^{2n}_{r}\to P^{k}_{r}$ is, by Theorem 7.6, a submersion and moreover, being open, $\mu$ is a proper map. Since $\mu$ is $T^{k}$-invariant for the free $T^{k}$-action on $M_{r}^{2n}$, it follows that the same holds for the induced map $\hat{\mu}:M^{2n}_{r}/T^{k}\to P^{k}_{r}$. Then, classical Ehressmann’s fibration theorem implies that the maps $\mu$ and $\hat{\mu}$ produce the stated fiber bundles. ∎

Let $P^{k}_{r,1},\ldots,P^{k}_{r,s}$ be the connected components of the open set $P^{k}_{r}$ and $M^{2n}_{r,i}=\mu^{-1}(P^{k}_{r,i})$, $1\leq i\leq s$. Then, the maps $\mu:M^{2n}_{r,i}\to P^{k}_{r,i}$ define smooth fiber bundles with connected base. In this way, we obtain:

Corollary 7.11**.**

The manifolds $M_{x}^{2n-k}$ and $M_{y}^{2n-k}$ are diffeomorphic for any $x,y\in P^{k}_{r,i}$, $1\leq i\leq s$.

7.1 The main stratum and its orbit space

As noted in Example 3.4 the main stratum $W$ is a dense set in $M^{2n}$, which implies that $W/T^{k}$ is a dense set in the orbit space $M^{2n}/T^{k}$. Moreover, $\widehat{\mu}(W/T^{k})=\stackrel{{\scriptstyle\circ}}{{P^{k}}}$ and $W/T^{k}\cong\stackrel{{\scriptstyle\circ}}{{P^{k}}}\times F$, where $F$ denotes the space of parameters of the main stratum $W$, as introduced by Definition 6.3. It follows that there exists such a compactification of the product $\stackrel{{\scriptstyle\circ}}{{P^{k}}}\times F$ that is homeomorphic to $M^{2n}/T^{k}$.

The first result in this direction is the following.

Proposition 7.12**.**

If the space $F$ is a point then the orbit space $M^{2n}/T^{k}$ is homeomorphic to the polytope $P^{k}$.

Proof.

If $F$ is a point, then $W$ consists of one leaf. Thus, Corollary 6.16 implies that the closure $\overline{W/T^{k}}$ is homeomorphic to the polytope $P^{k}$, which further implies that $M^{2n}/T^{k}$ is homeomorphic to the polytope $P^{k}$. ∎

Example 7.13**.**

If $M^{2n}$ is a quasitoric manifold, then $F$ is a point.

Assume now that the space of parameters $F$ of the main stratum is not a point. Let $x\in\stackrel{{\scriptstyle\circ}}{{P^{k}}}$ and put $\mathfrak{S}(x)=\{\sigma\in\mathfrak{S}|x\in\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}\}$, that is $\sigma\in\mathfrak{S}(x)$ if and only if $P_{\sigma}\in\sigma(x)$. Then $\widehat{\mu}^{-1}(x)$ is a closed set in $M^{2n}/T^{k}$ and

[TABLE]

Since the subspace $\{y\in W_{\sigma}/T^{k}:\widehat{\mu}(y)=x\}$ is homeomorphic to $F_{\sigma}$, we introduce a topology on the union $\cup_{\sigma\in\mathfrak{S}(x)}F_{\sigma}$ such that the space obtained as this union becomes homeomorphic to $\widehat{\mu}^{-1}(x)$. Let $\overline{F_{x}}$ denote the closure of the space $F_{x}=\widehat{\mu}^{-1}(x)\cap W/T^{k}\cong F$. Since $\overline{W/T^{k}}=M^{2n}/T^{k}$, it follows that

[TABLE]

for any $x\in\stackrel{{\scriptstyle\circ}}{{P^{k}}}$. From Corollary 7.7 and Corollary 7.11 it follows:

Corollary 7.14**.**

If $x\in P^{k}_{r}$ then $\overline{F_{x}}=M_{x}^{2n-k}/T^{k}$ is a smooth manifold in $M^{2n}/T^{k}$. Moreover, the manifolds $\overline{F_{x}}$ and $\overline{F_{y}}$ are homeomorphic for any $x,y\in P^{k}_{r,i}$, $1\leq i\leq s$.

Remark 7.15*.*

We want to note that although a space $F_{x}$ is homeomorphic to the space of parameters $F$ for any $x\in\stackrel{{\scriptstyle\circ}}{{P^{k}}}$, there is no argument to claim that, in general, their compactifications $\overline{F_{x}}$ are homeomorphic. But, Corollary 7.14 implies that the set of manifolds $\overline{F_{x}}$, $x\in P^{k}_{r}$, is finite, up to diffeomorphism.

We first point the following:

Lemma 7.16**.**

The space of parameters $F$ of the main stratum is a compact space if and only if all points from $\stackrel{{\scriptstyle\circ}}{{P^{k}}}$ are simple.

Proof.

If all points from $\stackrel{{\scriptstyle\circ}}{{P^{k}}}$ are simple, then the condition $P_{\sigma}\cap\stackrel{{\scriptstyle\circ}}{{P^{k}}}\neq\emptyset$ implies that $P_{\sigma}=P^{k}$. This means that $\widehat{\mu}^{-1}(x)\cong F$ for any point $x\in\stackrel{{\scriptstyle\circ}}{{P^{k}}}$. Therefore, $F$ is a compact space. In opposite direction, if $F$ is a compact space, from the fact that $F\cong F_{x}\subseteq\widehat{\mu}^{-1}(x)$ for any point $x\in\stackrel{{\scriptstyle\circ}}{{P^{k}}}$, it follows that $F\cong\widehat{\mu}^{-1}(x)$, which implies that all points from $\stackrel{{\scriptstyle\circ}}{{P^{k}}}$ are simple. ∎

In the case when the spaces $\overline{F_{x}}$, $x\in\stackrel{{\scriptstyle\circ}}{{P^{k}}}$ are all homeomorphic and all points from the boundary of the polytope $P^{k}$ are simple, we provide an explicit topological description of the orbit space $M^{2n}/T^{k}$.

Let $P_{\sigma_{0}}$ be a face of the polytope $P^{k}$ and by $\mathfrak{S}(\sigma_{0})$ denote the set of those admissible sets $\sigma\in\mathfrak{S}$ for which $P_{\sigma_{0}}$ is a face of the polytope $P_{\sigma}$ and $P_{\sigma}$ is not a face of the polytope $P^{k}$.

Proposition 7.17**.**

Assume that all points from $\partial P^{k}$ are simple, that a space $\overline{F_{x}}$ is homeomorphic to the space $\bar{F}$ for all $x\in\stackrel{{\scriptstyle\circ}}{{P^{k}}}$. Then for any $\sigma_{0}\in\mathfrak{S}$, such that $P_{\sigma_{0}}$ is a face of the polytope $P^{k},$ there exists a point $x\in\stackrel{{\scriptstyle\circ}}{{P^{k}}}$ such that $\widehat{\mu}^{-1}(x)\cong\cup_{\sigma\in\mathfrak{S}(\sigma_{0})}F_{\sigma}$ and, thus

[TABLE]

Proof.

We first note that the assumption that all points from $\partial P^{k}$ are simple implies that if $P_{\sigma_{0}}$ is not a face of some admissible polytope $P_{\sigma}$ then $\partial P_{\sigma}\cap P_{\sigma_{0}}=\emptyset$. Otherwise we would have a point $z\in\partial P_{\sigma}\cap P_{\sigma_{0}}$ meaning that $z$ belongs to some face $P_{\bar{\sigma}}$ of $P_{\sigma}$, which is an admissible polytope. Since $z$ is a simple point, it follows that $P_{\bar{\sigma}}=P_{\sigma_{0}}$. Therefore, since there are only finitely many admissible polytopes, we see that there exists a neighborhood $U$ of $P_{\sigma_{0}}$ in $P^{k}$ such that $U\cap P_{\sigma}=\emptyset$ for any admissible polytope $P_{\sigma}$ for which $P_{\sigma_{0}}$ is not a face of $P_{\sigma}$. Further $U\cap(\cap_{\sigma\in\mathfrak{S}(\sigma_{0})}P_{\sigma})\neq\emptyset$ as $P_{\sigma_{0}}$ belongs to all polytopes from $\mathfrak{S}(\sigma_{0})$. For a point $x\in U\cap(\cap_{\sigma\in\mathfrak{S}(\sigma_{0})}P_{\sigma})$, we obtain $\overline{F}\cong\widehat{\mu}^{-1}(x)\cong\cup_{\sigma\in\mathfrak{S}(\sigma_{0})}F_{\sigma}$, since $x\in\stackrel{{\scriptstyle\circ}}{{P^{k}}}$. ∎

By $\stackrel{{\scriptstyle\circ}}{{P_{\mathfrak{S}}}}$ denote the set of interiors of all admissible polytopes that are not proper faces of the polytope $P^{k}$.

Theorem 7.18**.**

Assume that all points from $\partial P^{k}$ are simple and that a space $\widehat{\mu}^{-1}(x)\subset M^{2n}/T^{k}$ is homeomorphic to the space $\bar{F}$ for all $x\in\stackrel{{\scriptstyle\circ}}{{P^{k}}}$. Assume also that the set $\stackrel{{\scriptstyle\circ}}{{P_{\mathfrak{S}}}}$ can be divided into subsets $\stackrel{{\scriptstyle\circ}}{{P_{\mathfrak{S}_{1}}}},\ldots,\stackrel{{\scriptstyle\circ}}{{P_{\mathfrak{S}_{l}}}}$ such that the areas from $\stackrel{{\scriptstyle\circ}}{{P_{\mathfrak{S}_{i}}}}$ give the polytopal decomposition for the area $\stackrel{{\scriptstyle\circ}}{{P^{k}}}$ and $F_{\sigma}\cong F_{\mathfrak{S}_{i}}$ for any $\sigma\in\mathfrak{S}_{i}$, where $1\leq i\leq l$. Then the orbit space $M^{2n}/T^{k}$ is homeomorphic to the quotient space $P^{k}\times\bar{F}/\approx,$ where

[TABLE]

Here $\sigma_{1},\sigma_{2}\in\mathfrak{S}(\sigma_{0})$ are such that $f_{1}\in F_{\sigma_{1}}$, $f_{2}\in F_{\sigma_{2}}$ and the map $\eta_{\sigma_{i},\sigma_{0}}$ is defined by the formula (6).

Proof.

Put $W_{\mathfrak{S}_{i}}/T^{k}=\widehat{\mu}^{-1}(\stackrel{{\scriptstyle\circ}}{{P^{k}}})\cap(\cup_{\sigma\in\mathfrak{S}_{i}}W_{\sigma}/T^{k})$. Since $\widehat{\mu}_{\sigma}:W_{\sigma}/T^{k}\to\stackrel{{\scriptstyle\circ}}{{P}}_{\sigma}$ is a fiber bundle and, by the assumption, all $F_{\sigma}$ for $\sigma\in\mathfrak{S}_{i}$ are homeomorphic to the space $F_{\mathfrak{S}_{i}}$, it follows that $\widehat{\mu}:W_{\mathfrak{S}_{i}}/T^{k}\to\stackrel{{\scriptstyle\circ}}{{P^{k}}}$ is a fiber bundle with a fiber $F_{\mathfrak{S}_{i}}$. Since $\stackrel{{\scriptstyle\circ}}{{P^{k}}}$ is contractible, it follows that $W_{\mathfrak{S}_{i}}/T^{k}\cong\stackrel{{\scriptstyle\circ}}{{P^{k}}}\times F_{\mathfrak{S}_{i}}$. Since $\overline{F}\cong\widehat{\mu}^{-1}(x)$, for $x\in\stackrel{{\scriptstyle\circ}}{{P^{k}}}$, it follows that $\widehat{\mu}^{-1}(\stackrel{{\scriptstyle\circ}}{{P^{k}}})=\cup_{i=1}^{l}W_{\mathfrak{S}_{i}}\cong\stackrel{{\scriptstyle\circ}}{{P^{k}}}\times(\cup_{i=1}^{l}F_{\mathfrak{S}_{i}})\cong\stackrel{{\scriptstyle\circ}}{{P^{k}}}\times\bar{F}$, where the last homeomorphism holds according to (7).

Let $P_{\sigma_{0}}$ be a face of the polytope $P^{k}$. It is an admissible polytope. Since any point from $\partial P^{k}$ is simple, it follows that $W_{\sigma_{0}}/T^{k}=\widehat{\mu}^{-1}(\stackrel{{\scriptstyle\circ}}{{P}}_{\sigma_{0}})$ and, thus, the projection $\widehat{\mu}^{-1}:\widehat{\mu}^{-1}(\stackrel{{\scriptstyle\circ}}{{P}}_{\sigma_{0}})\to\stackrel{{\scriptstyle\circ}}{{P}}_{\sigma_{0}}$ is a fiber bundle with the fiber $F_{\sigma_{0}}$. It implies that the space $\widehat{\mu}^{-1}(\stackrel{{\scriptstyle\circ}}{{P}}_{\sigma_{0}})$ is homeomorphic to the direct product $\stackrel{{\scriptstyle\circ}}{{P}}_{\sigma_{0}}\times F_{\sigma_{0}}$. We also have that $F_{\sigma_{0}}\subseteq\overline{F}$ since $\widehat{\mu}^{-1}(\stackrel{{\scriptstyle\circ}}{{P^{k}}})$ is an everywhere dense set in $M^{2n}/T^{k}$ and $\overline{F}$ is a closed set. Combining this and Proposition 7.17, we obtain $F_{\sigma_{0}}\subseteq\cup_{\sigma\in\mathfrak{S}(\sigma_{0})}F_{\sigma}$. Therefore, the family of maps $\eta_{\sigma,\sigma_{0}}:F_{\sigma}\to F_{\sigma_{0}}$ produces the map $\eta:\overline{F}\to F_{\sigma_{0}}$ that is surjective. It implies that $\stackrel{{\scriptstyle\circ}}{{P}}_{\sigma_{0}}\times F_{\sigma_{0}}$ is homeomorphic to a quotient of the space $\stackrel{{\scriptstyle\circ}}{{P}}_{\sigma_{0}}\times\overline{F}$ by the equivalence relation $(x,c_{1})\approx(x,c_{2})$ if and only if $\eta(c_{1})=\eta(c_{2})$, that is $\eta_{\sigma_{1},\sigma_{0}}(c_{1})=\eta_{\sigma_{2},\sigma_{0}}(c_{2})$, where $c_{1}\in F_{\sigma_{1}},c_{2}\in F_{\sigma_{2}}$ and $\sigma_{2},\sigma_{2}\in\mathfrak{S}$. As the faces of $P^{k}$ are the only admissible polytopes from $\partial P^{k}$, this proves the statement. ∎

Theorem 7.19**.**

Assume that

all points from the boundary $\partial P^{k}$ are simple;

• 2)

a space $\widehat{\mu}^{-1}(x)\subset M^{2n}/T^{k}$ is homeomorphic to the space $\bar{F}$ for any $x\in\stackrel{{\scriptstyle\circ}}{{P^{k}}}$;

• 3)

the set $\stackrel{{\scriptstyle\circ}}{{P_{\mathfrak{S}}}}$ can be divided into subsets $\stackrel{{\scriptstyle\circ}}{{P_{\mathfrak{S}_{1}}}},\ldots,\stackrel{{\scriptstyle\circ}}{{P_{\mathfrak{S}_{l}}}}$ such that the areas from $\stackrel{{\scriptstyle\circ}}{{P_{\mathfrak{S}_{i}}}}$, $1\leq i\leq l$ give the polytopal decomposition for the area $\stackrel{{\scriptstyle\circ}}{{P^{k}}}$;

• 4)

$F_{\sigma}\cong F_{\mathfrak{S}_{i}}$* for any $\sigma\in\mathfrak{S}_{i}$, where $1\leq i\leq l$.*

If $F_{\sigma_{0}}$ is a point for any $\sigma_{0}\in\mathfrak{S}$ such that $P_{\sigma_{0}}$ is a face of the polytope $P^{k}$ then:

[TABLE]

Proof.

Recall that the joint $X\ast Y$ for topological spaces $X$ and $Y$ is defined to be a quotient space of the product $CX\times Y$ by the equivalence relation $(x_{1},0,y_{1})\approx(x_{2},0,y_{2})\Leftrightarrow x_{1}=x_{2}$, where $CX$ is a cone over $X$. In our case Theorem 7.18 gives that

[TABLE]

We identify the polytope $P^{k}$ with the closed disc $\overline{D^{k}}$ and further identify $\overline{D^{k}}$ with the cone $CS^{k-1}$ over its boundary $S^{k-1}$. It implies that

[TABLE]

which proves the statement. ∎

8 Complex of admissible polytopes

Let $P_{\mathfrak{S}}$ denote the family of all admissible polytopes. Combining Example 4.4, Example 4.5 and Proposition 6.12 we obtain:

• •

$P^{k}\in P_{\mathfrak{S}}$ and $v\in P_{\mathfrak{S}}$ for any vertex $v$.

• •

If $P_{\sigma}\in P_{\mathfrak{S}}$ and $P_{\bar{\sigma}}$ is a face of the polytope $P_{\sigma}$, then $P_{\bar{\sigma}}\in P_{\mathfrak{S}}$.

Let us consider the complex $C(M^{2n},P^{k})$, which is obtained a formal disjoint union of the interiors of all admissible polytopes:

[TABLE]

By the defintion (8), there is a bijection between the set of cells in $C(M^{2n},P^{k})$ and the set of admissible polytopes $P_{\mathfrak{S}}$.

Lemma 8.1**.**

There is the canonical map $\widehat{\pi}:C(M^{2n},P^{k})\to P^{k}$.

Proof.

For any polytope $P_{\sigma}^{{}^{\prime}}$ in $C(M^{2n},P^{k})$ there is a unique admissible polytope $P_{\sigma}\in P_{\mathfrak{S}}$ such that $\stackrel{{\scriptstyle\circ}}{{P}}_{\sigma}$ corresponds to the polytope $P_{\sigma}^{{}^{\prime}}$ and we define $\widehat{\pi}(P_{\sigma}^{{}^{\prime}})=\stackrel{{\scriptstyle\circ}}{{P}}_{\sigma}$. ∎

Corollary 8.2**.**

For any $P_{\sigma}\in P_{\mathfrak{S}}$ there exists a unique polytope $P_{\sigma}^{{}^{\prime}}$ in $C(M^{2n},P^{k})$ such that the map $\widehat{\pi}:P_{\sigma}^{{}^{\prime}}\to\stackrel{{\scriptstyle\circ}}{{P}}_{\sigma}$ is a homeomorphism.

Corollary 8.3**.**

The canonical map $\widehat{\pi}:C(M^{2n},P^{k})\to P^{k}$ is a quotient map.

Proof.

Define an equivalence relation $\approx$ on $C(M^{2n},P^{k})$ by $x\approx y$ if and only if $\widehat{\pi}(x)=\widehat{\pi}(y)$. It is obvious that $P^{k}=C(M^{2n},P^{k})/\approx$ and that $\widehat{\pi}:C(M^{2n},P^{k})\to P^{k}$ is a quotient map. ∎

Corollary 8.4**.**

The canonical map $\widehat{\pi}:C(M^{2n},P^{k})\to P^{k}$ is a bijection if and only if the only admissible polytopes are the whole $P^{k}$ and its faces.

Example 8.5**.**

For a quasitoric manifold $M^{2n}$ the canonical map $\widehat{\pi}:C(M^{2n},P^{n})\to P^{n}$ is a bijection.

For the almost moment map $\mu:M^{2n}\to P^{k}$ we prove the following result:

Lemma 8.6**.**

There is the canonical map $f:M^{2n}\to C(M^{2n},P^{k})$ such that $\mu=\widehat{\pi}\circ f$.

Proof.

For any point $x\in M^{2n}$ there exists a unique stratum $W_{\sigma}$ such that $x\in W_{\sigma}$. Then $\mu(x)\in\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}$ and by Corollary 8.2 there exists a unique polytope $P_{\sigma}^{{}^{\prime}}\subseteq C(M^{2n},P^{k})$ such that $\widehat{\pi}:P_{\sigma}^{{}^{\prime}}\to\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}$ is a homeomorphism. It implies that there exists a unique $y\in P_{\sigma}^{{}^{\prime}}$ such that $\widehat{\pi}(y)=\mu(x)$. In this way, the map $f:M^{2n}\to C(M^{2n},P^{k})$ is defined by $f(x)=y$. ∎

For the induced map $\widehat{\mu}:M^{2n}/T^{k}\to P^{k}$ we obtain:

Corollary 8.7**.**

There is the canonical map $\widehat{f}:M^{2n}/T^{k}\to C(M^{2n},P^{k})$ such that $\widehat{\mu}=\widehat{\pi}\circ\widehat{f}$.

We further assume that $P_{\mathfrak{S}}$ is partially ordered by the inclusion of admissible polytopes. It is defined the boundary operator $d$ which to each admissible polytope $P_{\sigma}$ assigns the disjoint union of of all its faces. The operator $d$ induces the operator $d_{C}$ on the cells of the complex $C(M^{2n},P^{k})$ by

[TABLE]

where $(dP_{\sigma})_{C}\subset C(M^{2n},P^{k})$ corresponds to $dP_{\sigma}$ by the bijection between $P_{\mathfrak{S}}$ and $C(M^{2n},P^{k})$.

It is important to emphasize the following. One can try to ask if it is possible to define similarly the boundary operator $\tilde{d}$ on the set of all strata $W_{\mathfrak{S}}$ by $\tilde{d}(W_{\sigma})=\overline{W_{\sigma}}\setminus W_{\sigma}$. The answer is negative in general. Namely, as it will be shown in Section 18, the Grassmann manifold $G_{7,3}$ is an example of a manifold that belongs to our class, but which contains a stratum $W_{\sigma}$ whose boundary is not the union of the strata. Therefore, the operator $\tilde{d}$ is not defined for $G_{7,3}$. This example is taken from [14].

We still want to note that there are important examples of $(2n,k)$-manifolds for which the operator $\tilde{d}$ is well defined. It is proved in the paper [14] that the operator $\tilde{d}$ is well defined for the manifolds $G_{k+1,2}$ and $G_{6,3}$. In the paper [6], the cases of Grassmann manifold $G_{4,2}$ and complex projective space $\mathbb{C}P^{5}$ with the canonical action of $T^{4}$ are studied in detail . In these cases the operator $\tilde{d}$ is defined on the complex of the strata $W_{\mathfrak{S}}$. Moreover, the canonical map $f:M^{2n}\to C(M^{2n},P^{k})$ gives a mapping from the complex of strata $W_{\mathfrak{S}}$ to the complex $C(M^{2n},P^{k})$. More precisely:

Lemma 8.8**.**

If the operator $\tilde{d}(W_{\sigma})=\overline{W_{\sigma}}-W_{\sigma}$ is defined on the set of strata $W_{\mathfrak{S}}$ then the map $f:M^{2n}\to C(M^{2n},P^{k})$ induces the map $f_{\mathfrak{S}}:W_{\mathfrak{S}}\to C(M^{2n},P^{k})$ which is a bijective map between these complexes and commutes with the boundary operators.

Proof.

Let $W_{\sigma}$ be a stratum. Then $\mu(W_{\sigma})=\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}$ and Lemma 8.6 implies that $\widehat{\pi}(f(W_{\sigma}))=\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}$. By Corollary 6.4 we conclude that $f(W_{\sigma})=P_{\sigma}^{{}^{\prime}}$, which is a cell of the complex $C(M^{2n},P^{k})$. Therefore, the map $f_{\mathfrak{S}}:W_{\mathfrak{S}}\to C(M^{2n},P^{k})$ is defined by $f_{\mathfrak{S}}(W_{\sigma})=f(W_{\sigma})$. The map $f_{\mathfrak{S}}$ is a bijection because of the above stated bijection between the set of strata $W_{\mathfrak{S}}$ and the set of admissible polytopes $P_{\mathfrak{S}}$. Moreover, it follows from Corollary 6.15 that $d\circ f_{\mathfrak{S}}=f_{\mathfrak{S}}\circ\tilde{d}$ which means that $f_{\mathfrak{S}}$ commutes with the boundary operators. ∎

8.1 CW-topology on $C(M^{2n},P^{k})$

The complex of all admissible polytopes $C(M^{2n},P^{k})$ can be naturally endowed with a topology such that $C(M^{2n},P^{k})$ becomes CW-complex, which we denote by $CW(M^{2n},P^{k})$.

• •

The cells of these complex are the open polytopes $\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}$ for $P_{\sigma}\in P_{\mathfrak{S}}$.

• •

The characteristic function on the boundary of the cells, which defines their attaching, is defined by the operator $d_{C}$. The skeletons are defined inductively by the dimension of the cells. The definition of the operator $d_{C}$ verifies that the cell axiom of CW-complex is satisfied.

• •

According to the axioms of CW-complex, it is defined on $CW(M^{2n},P^{k})$ the weak topology compatible with the cell decomposition.

Then the following is satisfied:

Lemma 8.9**.**

The canonical map $\widehat{\pi}:CW(M^{2n},P^{k})\to P^{k}$

• •

is a continuous map;

• •

is a cell map for the standard cell decomposition of the polytope $P^{k}$ if and only if $\hat{\pi}$ is a homeomorphism.

Proof.

To prove that the map $\widehat{\pi}$ is continuous, it is enough to notice that if $U\subseteq P^{k}$ is a closed set in $P^{k}$ then $U\cap P$ is a closed set in $P$, for any polytope $P$ over some subsets of vertices of the polytope $P^{k}$. This will be true as well for the admissible polytopes, which implies that $\widehat{\pi}^{-1}(U)$ is a closed set in $CW(M^{2n},P^{k})$. The second statement follows directly from Corollary 8.4. ∎

Example 8.10**.**

For a quasitoric manifold the canonical map $\widehat{\pi}:CW(M^{2n},P^{n})\to P^{n}$ is a homeomorphism.

Proposition 8.11**.**

If the set of admissible polytopes $P_{\mathfrak{S}}$ contains a $k$-dimensional polytope different from $P^{k}$, the canonical map $f:M^{2n}\to CW(M^{2n},P^{k})$ is not a continuous map.

Proof.

Let $P_{\sigma}\in P_{\mathfrak{S}}$, $\dim P_{\sigma}=k$ and $P_{\sigma}\neq P^{k}$. Then $P_{\sigma}^{{}^{\prime}}\cong\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}$ is an open set in $CW(M^{2n},P^{k})$. Let us consider the corresponding stratum $W_{\sigma}$, that is $\mu(W_{\sigma})=\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}$. Since $\mu=\widehat{\pi}\circ f$, it follows that $f^{-1}(P_{\sigma}^{{}^{\prime}})=W_{\sigma}$, which is, by Remark 3.7, not an open set in $M^{2n}$. ∎

Remark 8.12*.*

Note that the same assumption as in Proposition 8.11 leads that the canonical map $\widehat{f}:M^{2n}/T^{k}\to CW(M^{2n},P^{k})$ is not continuous as well.

Example 8.13**.**

We demonstrate Proposition 8.11 in the case of Grassmann manifold $G_{4,2}$ endowed with the canonical action of $T^{4}$. Following [6], any four sided pyramid $P$ in $\Delta_{4,2}$ is an admissible polytope, which implies that $P^{{}^{\prime}}\cong\stackrel{{\scriptstyle\circ}}{{P}}$ is an open set in $CW(G_{4,2},\Delta_{4,2})$. On the other hand $f^{-1}(P^{{}^{\prime}})$ is an $(\mathbb{C}^{*})^{3}$-orbit in the eight-dimensional manifold $G_{4,2}$, so it can not be an open set.

8.2 Quotient topology on $C(M^{2n},P^{k})$

As Proposition 8.11 points, the CW-topology on $C(M^{2n},P^{k})$ is not compatible with the topology of $M^{2n}$. Therefore, the CW-topology is not quite appropriate for the description of a topology of the orbit space $M^{2n}/T^{k}$.

We define another topology on the complex $C(M^{2n},P^{k})$ such that the space $CQ(M^{2n},P^{k})=C(M^{2n},P^{k})$ becomes a quotient space of $M^{2n}$ by the canonical map $f:M^{2n}\to C(M^{2n},P^{k})$. More precisely, we consider a subset $U\subseteq C(M^{2n},P^{k})$ to be open if and only if the subset $f^{-1}(U)\subseteq M^{2n}$ is open. This is equivalent to say that $CQ(M^{2n},P^{k})$ is a quotient space of $M^{2n}/T^{k}$ by the canonical map $\widehat{f}:M^{2n}/T^{k}\to C(M^{2n},P^{k})$.

Lemma 8.14**.**

The maps $\widehat{\pi}:CQ(M^{2n},P^{k})\to P^{k}$, $f:M^{2n}\to CQ(M^{2n},P^{k})$ and $\widehat{f}:M^{2n}/T^{k}\to CQ(M^{2n},P^{k})$ are continuous, canonical maps.

• •

Axiom 5 implies that a face of any polytope $P^{{}^{\prime}}\in CQ(M^{2n},P^{k})$ is contained in the boundary of $P^{{}^{\prime}}$ regarded to the quotient topology.

• •

Note also that $P^{k^{{}^{\prime}}}=\stackrel{{\scriptstyle\circ}}{{P^{k}}}$ is a dense set in $CQ(M^{2n},P^{k})$ since $f^{-1}(P^{k^{{}^{\prime}}})=W$, the main stratum which is a dense set in $M^{2n}$.

Lemma 8.15**.**

If the set of admissible polytopes consists of $P^{k}$ and its faces, that is if the canonical map map $\widehat{\pi}:C(M^{2n},P^{k})\to P^{k}$ is a bijection, then the space $CQ(M^{2n},P^{k})$ is a Hausdorff topological space.

Lemma 8.16**.**

If the set of admissible polytopes $P_{\mathfrak{S}}$ contains a polytope $P_{\sigma}$ such that $\stackrel{{\scriptstyle\circ}}{{P}}_{\sigma}\subset\stackrel{{\scriptstyle\circ}}{{P^{k}}}$, then the space $CQ(M^{2n},P^{k})$ is not a Hausdorff topological space.

Proof.

Let $P_{\sigma}$ be an admissible polytope as stated in the formulation. Then $\widehat{\pi}(P_{\sigma}^{{}^{\prime}})\subset\stackrel{{\scriptstyle\circ}}{{P^{k}}}$ and for any point $x\in P_{\sigma}^{{}^{\prime}}\subset CQ(M^{2n},P^{k})$ there exists a point $y\in P^{k^{{}^{\prime}}}\subset CQ(M^{2n},P^{k})$ such that $\widehat{\pi}(x)=\widehat{\pi}(y)$. Let further $V$ be an open set in $CQ(M^{2n},P^{k})$ containing the point $x$. Then $\widehat{f}^{-1}(V)$ is an open set in $M^{2n}/T^{k}$ and it contains all points from $W_{\sigma}/T^{k}$ which map to $\widehat{\pi}(x)$ by the map $\widehat{\mu}$. On the other hand, by (7) there exists a point $m\in\widehat{\mu}^{-1}(\widehat{\pi}(x))\cap W/T^{k}$ such that $m\in\widehat{f}^{-1}(V)$. Note that $\widehat{f}(\widehat{\mu}^{-1}(\widehat{\pi}(x))\cap W/T^{k})=y$ which implies that $y\in V$. Thus every neighborhood of the point $x$ in $CQ(M^{2n},P^{k})$ contains a point $y\in P^{k^{{}^{\prime}}}$. ∎

Corollary 8.17**.**

If the set of admissible polytopes $P_{\mathfrak{S}}$ is a pure set and contains a polytope different from $P^{k}$ and its faces then the space $CQ(M^{2n},P^{k})$ is not a Hausdorff topological space.

Let us discuss the relation between the CW-topology and the CQ-topology on $C(M^{2n},P^{k})$.

Lemma 8.18**.**

If the canonical map $\widehat{\pi}:C(M^{2n},P^{k})\to P^{k}$ is a bijection then the CW-topology and the CQ-topology on $C(M^{2n},P^{k})$ coincide.

Example 8.19**.**

For a quasitoric manifold $M^{2n}$ these two topologies on $C(M^{2n},P^{n})=P^{n}$ coincide.

As a direct consequence of Proposition 8.11, we also deduce the following:

Lemma 8.20**.**

If the set of admissible polytopes $P_{\mathfrak{S}}$ contains a $k$-dimensional polytope different from $P^{k}$, then there exists a set which is open in $CW(M^{2n},P^{k})$, but which is not open in $CQ(M^{2n},P^{k})$.

Example 8.13 demonstrates the situation described in this Lemma. The interior of any four-sided pyramid is an open set in $CW(G_{4,2},\Delta_{4,2})$, but it is not an open set in $CQ(G_{4,2},\Delta_{4,2})$.

Therefore, in general, the sets in $C(M^{2n},P^{k})$ which are open in the CW-topology are not necessarily open in the CQ-topology.

The inverse inclusion does not hold as well, in general the sets in $C(M^{2n},P^{k})$ which are open in the quotient topology are not necessarily open in the CW-topology. We demonstrate this in the case of Grassmann manifold $G_{4,2}$.

8.2.1 An example of a closed set in $CQ(G_{4,2},\Delta_{4,2})$ that is not closed in $CW(G_{4,2},\Delta_{4,2})$

We follow the notation and the methods from [6]. Let us consider the set $\mathcal{C}$ in $G_{4,2}$ given by the matrices

[TABLE]

This set belongs to the main stratum since all its points have all non-zero Plücker coordinates. We obtain the closure of $\mathcal{C}$ by attaching the limit points when $c\rightarrow 0,1,\infty.$

• •

When $c\rightarrow 0$, we obtain the point

[TABLE]

• •

When $c\rightarrow 1$, we obtain the point

[TABLE]

• •

When $c\rightarrow\infty$, in order to see which point is obtaining, we can proceed as follows. Since the set $\mathcal{C}$ belongs to all charts, it follows that it can be written down in the local coordinates of the chart $M_{23}$. The points in the charts $M_{12}$ and $M_{23}$ are uniquely expressed by the matrices:

[TABLE]

Therefore, the transition map, on the intersection of these charts, from the coordinates $(z_{1},z_{2},z_{3},z_{4})$ in the chart $M_{12}$ to the coordinates $(w_{1},w_{2},w_{3},w_{4})$ in the chart $M_{23}$ is given by the formulas

[TABLE]

This implies that the set $\mathcal{C}$ writes in the chart $M_{23}$ as

[TABLE]

and, when $c\rightarrow\infty$, we obtain the point

[TABLE]

Let us consider the closed set $\overline{\mathcal{C}}=\mathcal{C}\cup\{C_{0},C_{1},C_{\infty}\}.$ Its image by the canonical map $f:G_{4,2}\to C(G_{4,2},\Delta_{4,2})$ will be as follows:

• •

$f(\mathcal{C})\subset\stackrel{{\scriptstyle\circ}}{{\Delta}}_{4,2}$, since $\mathcal{C}$ belongs to the main stratum.

• •

$f(C_{0})\in\stackrel{{\scriptstyle\circ}}{{P_{23}}}$, where $P_{23}$ is a four sided pyramid that does not contain the vertex $\delta_{23}$.

• •

$f(C_{1})\in\stackrel{{\scriptstyle\circ}}{{P_{34}}}$, where $P_{34}$ is a four sided pyramid that does not contain the vertex $\delta_{34}$.

• •

$f(C_{\infty})\in(\delta_{23},\delta_{34})$, where $(\delta_{23},\delta_{34})$ is an edge with the vertices $\delta_{23}$ and $\delta_{34}$.

Therefore, the set $f(\overline{\mathcal{C}})$ is not a closed set in $CW(G_{4,2},\Delta_{4,2})$ since $f(\mathcal{C})\cap\Delta_{4,2}=\mu(\mathcal{C})\cup\mu(\{C_{\infty}\})$ which it is not a closed set in $\Delta_{4,2}$, the points $\mu(C_{0}),\mu(C_{1})$ from the closure of $\mu(\mathcal{C})$ are missing. These points belong to the open cells $\stackrel{{\scriptstyle\circ}}{{P}}_{23}$ and $\stackrel{{\scriptstyle\circ}}{{P}}_{34}$, respectively.

On the other hand, $f(\overline{\mathcal{C}})$ is obviously a closed set in the quotient topology $CQ(G_{4,2},\Delta_{4,2})$.

Therefore, the CW topology and the CQ topology on $C(G_{4,2},\Delta_{4,2})$ are essentially different: for each of these two topologies there is a set which is open in one topology, but which is not open in the other.

Note that, in a general case when the operator $\tilde{d}$ is defined on the set of all strata $W_{\mathfrak{S}}$, it can be also defined a weak topology on $W_{\mathfrak{S}}$ such that the boundary of any stratum is given by the operator $\tilde{d}$. In this case $A\subseteq W_{\mathfrak{S}}$ is closed if and only if its intersection with the closure of any stratum is closed. It is obvious that this topology coincide with the topology induced from $M^{2n}$, since the topology on strata is induced from the topology of a manifold $M^{2n}$.

8.3 Induced partial ordering on $CQ(M^{2n},P^{k})$

There is a canonical way [8] to introduce a preorder on $CQ(M^{2n},P^{k})$ using the quotient topology. More precisely for $x,y\in CQ(M^{2n},P^{k})$ one defines

[TABLE]

where $\overline{y}$ denotes the closure of $y$ in the quotient topology. Note that on a general Hausdorff topological space this preorder becomes trivial, which means that $x\leq y$ implies $x=y$. Since $\widehat{\pi}:CQ(M^{2n},P^{k})\to P^{k}$ is a continuous map, it follows:

[TABLE]

Obviously if the set of all admissible polytopes consists of $P^{k}$ and its faces this preorder will be trivial because in this case, by Lemma 8.15, $CQ(M^{2n},P^{k})$ is a Hausdorff space. If this is not the case from the definition of the preorder, it directly follows:

Lemma 8.21**.**

If $x\in P^{k^{{}^{\prime}}}$ then there is no $y\in CQ(M^{2n},P^{k})$, $y\neq x$ such that $x\leq y$. On the other hand, if $P_{\sigma}\neq P^{k}$ is an admissible polytope that is not a face of $P^{k}$ then for any $x\in P_{\sigma}^{{}^{\prime}}$ there exists a unique $y\in P^{k^{{}^{\prime}}}$ such that $x\leq y$.

Recall that, in a partially ordered set $(X,\leq)$, an upper set is defined to be a subset $U$ of $X$ such that if $x\in U$ and $x\leq y$ then $y\in U$ and, accordingly, it is defined a lower set. It is a standard fact that, regarded to the specialization preorder on a topological space $X$, every open set in $X$ is an upper set and every closed set in $X$ is a lower set. Recall also that a topological space $X$ is said to be an P. S. Alexandrov space if the intersection of any family of open sets is an open set. Alexandrov topologies on $X$ are in one-to-one correspondence with preorders on $X$ meaning that $X$ is an Alexandrov space if and only if every upper set regarded to the specialization preorder is an open set. In particular, it implies that a Hausdorff topological space is an Alexandrov space if and only it is a discrete space.

As for the space $CQ(M^{2n},P^{k})$, for simplicity, we consider the case when the set of all admissible polytopes $P_{\mathfrak{S}}$ is a pure set. If $P_{\mathfrak{S}}$ consists only of $P^{k}$ and its faces, it follows from Lemma 8.15 that $CQ(M^{2n},P^{k})$ is a Hausdorff topological space, which is obviously not discrete, so it is not an Alexandrov space. We prove that the same holds in general:

Lemma 8.22**.**

Assume that the set of admissible polytopes is a pure set and contains a polytope different from $P^{k}$ and its faces. Then the space $CQ(M^{2n},P^{k})$ is not a Hausdorff topological space and it is not an Alexandrov space as well.

Proof.

Since $P_{\mathfrak{S}}$ contains a polytope $P_{\sigma}$ different from $P^{k}$ and its faces, it follows that there exists a face $P_{\bar{\sigma}}$ of $P_{\sigma}$ such that $\stackrel{{\scriptstyle\circ}}{{P_{\bar{\sigma}}}}\subset\stackrel{{\scriptstyle\circ}}{{P^{k}}}$. Let $V\subset CQ(M^{2n},P^{k})$ be the smallest, under inclusion, upper set that contains $P^{k^{{}^{\prime}}}$ and $P_{\bar{\sigma}}^{{}^{\prime}}$. Then $P_{\sigma}^{{}^{\prime}}\not\subset V$ since there is no $x\in P_{\bar{\sigma}}^{{}^{\prime}}$ such that $x\leq y$ for some $y\in P_{\sigma}^{{}^{\prime}}$ as $\widehat{\pi}(x)\neq\widehat{\pi}(y)$. The set $V$ is not open in $CQ(M^{2n},P^{k})$. Namely $f^{-1}(V)$ contains the stratum over $\stackrel{{\scriptstyle\circ}}{{P_{\bar{\sigma}}}}$ but it does not contain the stratum over $\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}$, while from Axiom 5 it follows that there exist points in the stratum over $\stackrel{{\scriptstyle\circ}}{{P_{\bar{\sigma}}}}$ which are in the closure of the stratum over $\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}$. ∎

Example 8.23**.**

For the sake of clearness we provide an explicit example of a set which is an upper set in $CQ(G_{4,2},\Delta_{4,2})$, but which is not open. Let $V\subset CQ(G_{4,2},\Delta_{4,2})$ is given as the union of $\Delta_{4,2}^{{}^{\prime}}$ and $P_{12,34}^{{}^{\prime}}$, where $P_{12,34}$ is a square in $\Delta_{4,2}$ which does not contain the vertices $\delta_{12}$ and $\delta_{34}$, see [6]. Then $V$ is not an open set in $CQ(G_{4,2},\Delta_{4,2})$ since $f^{-1}(V)$ in not open in $G_{4,2}$. Namely, $f^{-1}(V)$ consists of the main stratum and of the four-dimensional $(\mathbb{C}^{*})^{3}$-orbit which maps to the square $\stackrel{{\scriptstyle\circ}}{{P}}_{12,34}$. On the one hand, this orbit is contained in the closures of the two six-dimensional $(\mathbb{C}^{*})^{3}$-orbits, which map to the pyramids $P_{12}$ and $P_{34}$ by the moment map. Therefore, the set $f^{-1}(V)$ is not open in $G_{4,2}$, which implies that the set $V$ is not open in $CQ(G_{4,2},\Delta_{4,2})$. On the other hand, it is obvious that $V$ in an upper set since for any $x\in V$ if $x\leq y$ it follows that $x=y$ or $x\in P_{12,34}^{{}^{\prime}}$, $y\in\Delta_{4,2}^{{}^{\prime}}$ and $\widehat{\pi}(x)=\widehat{\pi}(y)$.

9 The space $\mathfrak{E}(M^{2n},P^{k})$

Using results from previous sections, we can define the set $\mathfrak{E}(M^{2n},P^{k})$ over the complex $C(M^{2n},P^{k})$ by:

[TABLE]

A topology on $\mathfrak{E}(M^{2n},P^{k})$ is defined as the induced topology by the embedding $\mathfrak{E}(M^{2n},P^{k})\to CQ(M^{2n},P^{k})\times M^{2n}$. In this case the topology on $CQ(M^{2n},P^{k})$ can be obtained as a quotient topology defined by the map $p:\mathfrak{E}(M^{2n},P^{k})\to CQ(M^{2n},P^{k})$.

Lemma 9.1**.**

The space $\mathfrak{E}(M^{2n},P^{k})$ is a compact Hausdorff topological space.

Proof.

The canonical map $f:M^{2n}\to CQ(M^{2n},P^{k})$ is a continuous, surjective map. The space $\mathfrak{E}(M^{2n},P^{k})$ can be identified with the graph of the map $f$. Since the manifold $M^{2n}$ is a Hausdorff topological space, it follows that $\mathfrak{E}(M^{2n},P^{k})$ is a Hausdorff space as well. As for the compactness let us consider a covering $E_{i},i\in I$ of the space $\mathfrak{E}(M^{2n},P^{k})$ by an open sets. Without loss of generality, we may assume that $E_{i}=(U_{i}\times V_{i})\cap\mathfrak{E}(M^{2n},P^{k})$ for $i\in I$, where $U_{i}$ and $V_{i}$ are open sets in $CQ(M^{2n},P^{k})$ and $M^{2n}$ respectively. Note that for any point $y\in M^{2n}$ there exists a point $x\in CQ(M^{2n},P^{k})$ such that $(x,y)\in\mathfrak{E}(M^{2n},P^{k})$, which implies that $V_{i},i\in I$ is an open covering for $M^{2n}$. In addition, for any point $x\in CQ(M^{2n},P^{k})$, there exists a point $y\in M^{2n}$ such that $(x,y)\in\mathfrak{E}(M^{2n},P^{k})$, which implies that $U_{i},i\in I$ is an open covering for $CQ(M^{2n},P^{k})$. The manifold $M^{2n}$ is assumed to be a compact space, which implies that its quotient space $CQ(M^{2n},P^{k})$ is a compact space as well. Therefore, there are finite sub-coverings $U_{1},\ldots,U_{s}$ for $CQ(M^{2n},P^{k})$ and $V_{1},\ldots,V_{l}$ for $M^{2n}/T^{k}$, which implies that $X_{ij}=(U_{i}\times V_{j})\cap\mathfrak{E}(M^{2n},P^{k})$ is a finite sub-covering of $E_{i}$, $i\in I$ for $\mathfrak{E}(M^{2n},P^{k})$. ∎

There are two natural projections:

[TABLE]

• •

The maps $G_{1}$ and $G_{2}$ are obviously surjective. For the map $G_{1}$, this follows from the observation that for any point $x\in P_{\sigma}^{{}^{\prime}}\subset CQ(M^{2n},P^{k})$ there exists a point $y\in W_{\sigma}\subset M^{2n}$, such that $\widehat{\pi}(x)=\mu(y)$. This is because $\mu(W_{\sigma})=\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}=\widehat{\pi}(P_{\sigma}^{{}^{\prime}})$, which implies that $(x,y)\in\mathfrak{E}(M^{2n},P^{k})$. As for the surjectivity of the map $G_{2}$, for any point $y\in W_{\sigma}\subset M^{2n}$ and any point $x\in P_{\sigma}^{{}^{\prime}}$ such that $\widehat{\pi}(x)=\mu(y)$, we see that $(x,y)\in\mathfrak{E}(M^{2n},P^{k})$.

• •

The maps $G_{1}$ and $G_{2}$ are obviously continuous. It follows from the fact that $G_{1}^{-1}(U)=(U\times M^{2n})\cap\mathfrak{E}(M^{2n},P^{k})$ and $G_{2}^{-1}(V)=(CQ(M^{2n},P^{k})\times V)\cap\mathfrak{E}(M^{2n},P^{k})$ are open sets in $\mathfrak{E}(M^{2n},P^{k})$ for open sets $U\subseteq CQ(M^{2n},P^{k})$ and $V\subseteq M^{2n}$.

Lemma 9.2**.**

The map $G_{2}$ is injective, while the map $G_{1}$ is not injective.

Proof.

The map $G_{2}$ is injective since the condition $G_{2}(x_{1},y)=G_{2}(x_{2},y)$ implies that $\widehat{\pi}(x_{1})=\widehat{\pi}(x_{2})=\widehat{\mu}(y)$ and $x_{1},x_{2}\in P_{\sigma}^{{}^{\prime}}$. But, $\widehat{\pi}:P_{\sigma}^{{}^{\prime}}\to\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}$ is a homeomorphism, which implies $x_{1}=x_{2}$. The map $G_{1}$ is not injective since for any $y_{1},y_{2}$ which belong to the same non-trivial $T^{k}$-orbit of a stratum $W_{\sigma}$, there exists a point $x\in P_{\sigma}^{{}^{\prime}}$ such that $\widehat{\pi}(x)=\mu(y_{1})=\mu(y_{2})$ and $(x,y_{1}),(x,y_{2})\in\mathfrak{E}(M^{2n},P^{k})$. It implies that $G_{1}(x,y_{1})=G_{1}(x,y_{2})=x$. ∎

Altogether this leads to the following key result:

Theorem 9.3**.**

The space $\mathfrak{E}(M^{2n},P^{k})$ is homeomorphic to the space $M^{2n}$. The homeomorphism $G_{2}:\mathfrak{E}(M^{2n},P^{k})\to M^{2n}$ is given by the map $G_{2}(x,y)=y$.

Proof.

The map $G_{2}:\mathfrak{E}(M^{2n},P^{k})\to M^{2n}$ is a continuous bijection. Since $\mathfrak{E}(M^{2n},P^{k})$ is a compact space and $M^{2n}$ is Hausdorff, it follows that elementary topology arguments lead that $G_{2}$ is a homeomorphism. ∎

We define an action of the torus $T^{k}$ on $CQ(M^{n},P^{k})\times M^{2n}$ using the given $T^{k}$-action on $M^{2n}$. Since the strata as well as the almost moment map $\mu$ are invariant for this torus action, it follows that this action induces an action of the torus $T^{k}$ on $\mathfrak{E}(M^{2n},P^{k})$. We obtain

[TABLE]

Remark 9.4*.*

Since $W_{\sigma}/T^{k}\cong\stackrel{{\scriptstyle\circ}}{{P}}_{\sigma}\times F_{\sigma}$ and $P_{\sigma}^{{}^{\prime}}\cong\stackrel{{\scriptstyle\circ}}{{P}}_{\sigma}$, it follows that the points from the set $\mathfrak{E}(M^{2n},P^{k})/T^{k}$ can be represented as the pairs $(x,c_{\sigma})$, where $x\in P_{\sigma}^{{}^{\prime}}$ and $c_{\sigma}\in F_{\sigma}$.

From Lemma 9.1 it follows

Corollary 9.5**.**

The space $\mathfrak{E}(M^{2n},P^{k})/T^{k}$ is a compact Hausdorff topological space.

Note that the maps $G_{1}$ and $G_{2}$ defined by (10) are $T^{k}$-equivariant, where $CQ(M^{2n},P^{k})$ is considered to be with the trivial $T^{k}$-action. Therefore, they induce the maps of the corresponding orbit spaces $\widehat{G_{1}}:\mathfrak{E}(M^{2n},P^{k})/T^{k}\to CQ(M^{n},P^{k})$ and $\widehat{G_{2}}:\mathfrak{E}(M^{2n},P^{k})/T^{k}\to M^{2n}/T^{k}$.

Combining the $T^{k}$-equaivariance of the map $\widehat{G_{2}}$ and Theorem 9.3 we obtain :

Theorem 9.6**.**

The map $\widehat{G_{2}}:\mathfrak{E}(M^{2n},P^{k})/T^{k}\to M^{2n}/T^{k}$ given by $\widehat{G_{2}}(x,y)=y$ is a homeomorphism.

The map $\widehat{G_{1}}$ has the following important feature:

Proposition 9.7**.**

The space $\widehat{G_{1}}^{-1}(P_{\sigma}^{{}^{\prime}})$ is homeomorphic to the space $P_{\sigma}^{{}^{\prime}}\times F_{\sigma}$, that is to the space $\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}\times F_{\sigma}$ for any $\sigma\in\mathfrak{S}$.

Proof.

For any point $x\in P_{\sigma}^{{}^{\prime}}\in CQ(M^{n},P^{k})$ we have that $\widehat{G_{1}}^{-1}(x)=\{(x,y)|y\in W_{\sigma}/T^{k},\widehat{\mu}(y)=\pi(x)\}$. It follows from Corollary 6.4 that the space $\widehat{G_{1}}^{-1}(x)$ is homeomorphic to the space $F_{\sigma}$. Then Corollary 8.2 implies that the space $\widehat{G_{1}}^{-1}(P_{\sigma}^{{}^{\prime}})$ is homeomorphic to the space $\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}\times F_{\sigma}$. ∎

Remark 9.8*.*

We want to emphasize that, according to Proposition 9.7, the space $\mathfrak{E}(M^{2n},P^{k})/T^{k}$, that is the orbit space $M^{2n}/T^{k}$, is the union of the trivial fiber bundles $\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}\times F_{\sigma}$, where $\sigma$ runs through the set $\mathfrak{S}$. Then, according to Axiom 5, the gluing of fibers $F_{\sigma}$ and $F_{\bar{\sigma}}$ for $\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}$ and $\stackrel{{\scriptstyle\circ}}{{P_{\bar{\sigma}}}}\subset\partial\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}$ respectively, is given by the map $\eta_{\sigma,\bar{\sigma}}:F_{\sigma}\to F_{\bar{\sigma}}$.

10 The gluing of the orbit spaces of strata in $\mathfrak{E}(M^{2n},P^{k})/T^{k}$

In the previous section we proved that the space $\mathfrak{E}(M^{2n},P^{k})/T^{k}$, that is the orbit space $M^{2n}/T^{k}$, is the union of total spaces of the trivial fiber bundles $\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}\times F_{\sigma}$, where $\sigma\in\mathfrak{S}$. In this section we want to describe how the trivial bundles $\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}\times F_{\sigma}$ are glued together, or, in other words, to describe the $\bar{\partial}$-boundary of $\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}\times F_{\sigma}$. Recall that the $\bar{\partial}$- boundary of $\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}\times F_{\sigma}$ is homeomorphic to the $\bar{\partial}$- boundary of the orbit space $W_{\sigma}/T^{k}$. Note that $\overline{W_{\sigma}/T^{k}}\cong\overline{W_{\sigma}}/T^{k}$, which implies that $\bar{\partial}(W_{\sigma}/T^{k})\cong\bar{\partial}W_{\sigma}/T^{k}$. Lemma 3.8 immediately implies:

Corollary 10.1**.**

There is an embedding $\bar{\partial}(\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}\times F_{\sigma})\subset\cup_{\tilde{\sigma}\subset\sigma}\stackrel{{\scriptstyle\circ}}{{P_{\tilde{\sigma}}}}\times F_{\tilde{\sigma}}$, where $\tilde{\sigma}$ runs through all admissible subsets of the set $\sigma$.

Remark 10.2*.*

We want to point that for a general $(2n,k)$-manifold, there might exist an admissible subsets $\tilde{\sigma}$ of a admissible set $\sigma$ for which the polytope $P_{\tilde{\sigma}}$ is not a face of the polytope $P_{\sigma}$. Furthermore, $P_{\tilde{\sigma}}$ might not belong to the boundary of the polytope $P_{\sigma}$. Therefore, we specially denote by $\bar{\sigma}$ those admissible subsets of a admissible set $\sigma$ for which $P_{\bar{\sigma}}$ is a face of the polytope $P_{\sigma}$.

Note that, as we will demonstrate in Section 18, the $\bar{\partial}$-boundary of the orbit space $W_{\sigma}/T^{k}$ can not be, in general, represented as a union of the orbit spaces of some other strata. This implies that, in general, the $\bar{\partial}$-boundary of the space $\stackrel{{\scriptstyle\circ}}{{P_{\sigma}}}\times F_{\sigma}$ in $\mathfrak{E}(M^{2n},P^{k})/T^{k}$ can not be represented as a union of total spaces of some other trivial fiber bundles.

As for the boundary of the polytopes in $CQ(M^{2n},P^{k})$ we have the following:

Lemma 10.3**.**

If $P_{\tilde{\sigma}}^{{}^{\prime}}\subset\partial P_{\sigma}^{{}^{\prime}}$ for some any $P_{\sigma}^{{}^{\prime}},P_{\tilde{\sigma}}^{{}^{\prime}}\subset CQ(M^{2n},P^{k})$ then $\tilde{\sigma}\subset\sigma$.

Proof.

If $P_{\tilde{\sigma}}^{{}^{\prime}}\subset\partial P_{\sigma}^{{}^{\prime}}$ it follows that $f^{-1}(P_{\tilde{\sigma}}^{{}^{\prime}})\subset f^{-1}(\partial P_{\sigma}^{{}^{\prime}})$, where $f:M^{2n}\to CQ(M^{2n},P^{k})$ is a quotient map. It implies that $f^{-1}(\partial P_{\sigma}^{{}^{\prime}})=\bar{\partial}f^{-1}(P_{\sigma}^{{}^{\prime}})=\bar{\partial}W_{\sigma}\subseteq\cup_{\tilde{\sigma}\subset\sigma}W_{\tilde{\sigma}}$ and, thus, $f^{-1}(P_{\tilde{\sigma}}^{{}^{\prime}})=W_{\tilde{\sigma}}$ for some $\tilde{\sigma}\subset\sigma$. ∎

Remark 10.4*.*

Recall that we already remarked that for any $P_{\bar{\sigma}}^{{}^{\prime}},P_{\sigma}^{{}^{\prime}}\subset CQ(M^{2n},P^{k})$, such that $P_{\bar{\sigma}}$ is a face of the polytope $P_{\sigma}$, we have that $P_{\bar{\sigma}}^{{}^{\prime}}\subset\partial P_{\sigma}^{{}^{\prime}}$ in $CQ(M^{2n},P^{k})$.

At the end of this section we derive some results under the additional assumption: if $P_{\tilde{\sigma}}^{{}^{\prime}},P_{\sigma}^{{}^{\prime}}\subset CQ(M^{2n},P^{k})$ and $\partial P_{\sigma}^{{}^{\prime}}\cap P_{\tilde{\sigma}}^{{}^{\prime}}\neq\emptyset$ then $P_{\tilde{\sigma}}^{{}^{\prime}}\subset\partial P_{\sigma}^{{}^{\prime}}$.

Then Lemma 10.3 implies:

Corollary 10.5**.**

The boundary of any area $P_{\sigma}^{{}^{\prime}}\subset CQ(M^{2n},P^{k})$ is the union of some areas $P_{\tilde{\sigma}}^{{}^{\prime}}$ such that $\tilde{\sigma}\subset\sigma$.

From previous results we also have the following direct consequence:

Corollary 10.6**.**

For any point $x\in P_{\sigma}^{{}^{\prime}}$ it holds $\overline{x}=\{y\in P_{\tilde{\sigma}}^{{}^{\prime}}\;|\>P_{\tilde{\sigma}}^{{}^{\prime}}\subset\partial P_{\sigma}^{{}^{\prime}},\;\widehat{\pi}(y)=\widehat{\pi}(x)\}$