text:
\section{Introduction}

The goal of this  paper is to extend our work~\cite{BTO}  and  axiomatize  the class of $(2n,k)$ - manifolds. It is  the 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 develop the theory of such manifolds and propose the tools for effective description of  their equivariant  structure and the structure of their orbit spaces.

This class contains toric and quasitoric manifolds, whose effective description is given by the  combinatorial data  $(P,\Lambda)$, where $P$ is a simple polytope and $\Lambda$ is a characteristic function~\cite{BP}. It contains as well the homogeneous spaces of  compact Lie groups among which in the focus of our interest are the  complex Grassmann manifolds $G_{k+1,q}$  and the complex flag manifolds $F_{k+1}$, with the standard action of the torus $T^{k+1}$ which induces  the effective action of the torus $T^k$.

Any $(2n,k)$-manifold  is equipped with  the so called  almost moment map $\mu : M^{2n}\to \R ^k$, whose image is a convex polytope $P^k$.  The polytope  $P^k$ does not need to be simple  and the orbit space $M^{2n}/T^k$ 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 $P{e}^{k}$.

One of the main tools we introduce is the family of admissible polytopes $P_{\sigma}$,  which are spanned by some subsets $\sigma$ of vertices of $P^k$. In the quasitoric case the family of admissible polytopes coincides with the family of  faces of
a simple polytope $P^k$. To each admissible polytope $P_{\sigma}$ we assign $T^k$-invariant subspace $W_{\sigma}\subset M^{2n},$ called the stratum,   and the subtorus $T^{\sigma}\subseteq T^k$, $\dim T^{\sigma}=\dim P_{\sigma}$,  which acts freely on $W_{\sigma}$. One of our axioms requires  that $\mu (W_{\sigma}) = \stackrel{\circ}{P_{\sigma}}$  and  that the  restriction of the induced almost moment map $\hat{\mu}$ to $W_{\sigma}/T^{\sigma}$ is a locally trivial fiber bundle over   $\stackrel{\circ}{P_{\sigma}}$. It implies that $W_{\sigma}/T^{\sigma} \cong \stackrel{\circ}{P_{\sigma}}\times F_{\sigma}$, for some topological space $F_{\sigma}$, which we call the space of parameters of the stratum $W_{\sigma}$.  The family of spaces $W_{\sigma}$ provides  the stratification of $M^{2n}$,  that is $M^{2n}$ is the disjoint union of all $W_{\sigma}$. In the case of Grassmann manifolds the family  $W_{\sigma}$ coincides with the  strata of  Gelfand, MacPherson, Goresky and Serganova, which they introduced using the action  of the algebraic torus $(\C ^{*})^{k+1}$ on $G_{k+1,q}$.

The union of all admissible polytopes $P_{\sigma}$ is the polytope $P^k$  and to each point $x\in P^k$ we assign the cort\'ege $\sigma (x) = \{ P_{\sigma} : x\in \stackrel{\circ}{P_{\sigma}}\}$. Under assumption that $\mu$ is a smooth map, we prove that $x$ is a regular value for $\mu$ if and only if $\dim P_{\sigma}=k$ for any $P_{\sigma}\in \sigma (x)$.

We introduce the structural data for $(2n,k)$-manifolds with the aim to describe the equivariant topology of the  manifolds $M^{2n}$ and the orbit spaces $M^{2n}/T^k$, that is to obtain the {\it model}  for $M^{2n}/T^k$ in  terms of the structural data. Our structural data include as well the notions of the {\it virtual} spaces of parameters $\tilde{F_{\sigma}}$ together with the projections $p_{\sigma} : \tilde{F}_{\sigma} \to F_{\sigma}$, and the  {\it universal} space of parameters $\mathcal{F}$ such that $\mathcal{F} =\cup _{\sigma} \tilde{F}_{\sigma}$.   More precisely, the orbit space  of the main stratum $W/T^k \cong \stackrel{\circ}{P^{k}}\times F$ is  a dense set in $M^{2n}/T^k$.  We
require that the compactification of  $\stackrel{\circ}{P^{k}}\times F$, which corresponds to the compactification $\overline{W/T^k}= M^{2n}/T^k$,
is realized by some  compactification $\mathcal{F}$  of the space of parameters $F$ of the main stratum. We include the space $\mathcal{F}$, which we call the
universal space of parameters,  into  our structural data and formalizes by axiom its properties. The main property of  the universal space of parameters $\mathcal{F}$  is that it is the
union of so called virtual spaces of parameters $\tilde{F}_{\sigma}$ and  there exist the projections $p_{\sigma} : \tilde{F}_{\sigma}\to F_{\sigma}$,  which are  all  the part of our structural
data.
The idea for the virtual spaces of parameters  $\tilde{F}_{\sigma}$  and the projections $p_{\sigma}$  arises  from the decomposition
$ \cup _{\sigma}W_{\sigma}/T^{\sigma} = \overline{\stackrel{\circ}{P^{k}}\times F}$  and  the structural  result  that $W_{\sigma}/T^{\sigma}\cong \stackrel{\circ}{P_{\sigma}}\times F_{\sigma}$. We realized in detail this approach in~\cite{BT-1} for the Grassmann manifold $G_{4,2}$ and in~\cite{BTN} for the Grassmann manifold $G_{5,2}$.
In this way we obtain the orbit space  $M^{2n}/T^k$ as the quotient space of $\cup _{\sigma} (\stackrel{\circ}{P_{\sigma}}\times \tilde{F}_{\sigma})$ by the equivalence relation which is defined in terms of the structural maps $\tilde{F}_{\sigma}\to \mathcal{F}$ and $p_{\sigma} : \tilde{F}_{\sigma}\to F_{\sigma}$.

The complexity of the $(2n,k)$-manifold is defined to be the number $d=n-k$. There are the other analogous notions of complexity of torus action in algebraic geometry and symplectic geometry.    The  complexity one torus actions is widely known and under some appropriate assumptions these actions are well studied in algebraic geometry and symplectic geometry, ~\cite{T},~\cite{K}.  In the recent  paper~\cite{AA}, inspired by our work~\cite{BT-1},~\cite{BTN},  it is described the approach for  solving the problem of the classification of complexity one torus actions in terms of equivariant topology. The  problem of the torus actions of complexity two or more is still not well understood and, in the literature, it is discussed  to be quite difficult. Our theory leads  to the results in this direction.

The axiomatization and techniques we present here are inspired by the results on algebraic torus action on Grassmann manifolds~\cite{GM},~\cite{GS},~\cite{GGMS}~\cite{Kap}, as well as our results on   the  orbit spaces of  the compact torus action on Grassmann manifolds~\cite{BT-1},~\cite{BTN}.

\section{Definition of  $(2n,k)$-manifolds}\label{jedan}

We assume the following to be given:
\begin{itemize}
\item  a smooth, closed, oriented,  simply connected manifold $M^{2n}$;
\item  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 connected;
\item   an open $\theta$-equivariant  map $\mu : M^{2n}\to \R^k$,  whose image is a $k$-dimensional convex polytope, where $\R ^k$  is considered with trivial $T^k$ action.
\end{itemize}
The polytope which is obtained as an image of $\mu $ we denote by $P^k$. The map $\mu $ we call an {\it almost moment map} for the given $T^k$-action on $M^{2n}$.

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

\subsection{Smooth structure.}

\begin{ax}\label{atlas}
There is a smooth atlas $\mathfrak{M} = \{(M_{i}, \varphi _i)\}_{i\in I}$,  where  $\varphi _i : M_{i}\to \R ^{2n}\approx \C ^n$,  for the fixed identification $\approx$,  are homeomorphisms,  such that any chart  $M_{i}$ is $T^k$-invariant, contains  exactly one fixed point $x_{i}$ with $\varphi _{i}(x_i) = (0,\ldots ,0)$
 and the closure  of $M_{i}$ is  $M^{2n}$.
\end{ax}

Since $M^{2n}$ is  a compact manifold there are finitely many such charts given by Axiom~\ref{atlas} what implies:

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

Denote by $m$ the number of fixed points for $T^k$-action on $M^{2n}$. The charts given by Axiom~\ref{atlas} we enumerate as $(M_1,\varphi _1),\ldots ,(M_{m},\varphi _{m})$.  The sets $Y_{i} = M^{2n}\setminus  M_{i}$ are closed and $T^k$-invariant, since $M_{i}$ are open and $T^k$-invariant. For a subspace $V\subset M^{2n}$ we set $\partial V = \overline{V}\setminus V$.  The fact that  $M_{i}$ is a dense in $M^{2n}$   implies that $Y_{i} = \partial M_{i}$.

Define the spaces $W_{\sigma}$, where $\sigma = \{i_1,\ldots ,i_l\}\subseteq [1,m]$ by :
\[
W_{\sigma} = M_{i_1}\cap \cdots \cap M_{i_l}\cap Y_{i_{l+1}}\cap \cdots \cap Y_{i_{m}},
\]
where $\{i_{l+1},\ldots ,i_{m}\} = [1,m] \setminus \{i_1,\ldots ,i_l\}$.

\begin{defn}
The non-empty space $W_{\sigma}$  is said to be an  admissible space. The  index set $\sigma \subset [1,m]$  of an admissible space $W_{\sigma}$ is   said to be an  admissible set.
\end{defn}

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

\begin{proof}
The admissible space $W_{\sigma}$ is $T^k$-invariant since the spaces  $M_{i}$ 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,  $x\notin Y_{i}$ what gives that $x\notin W_{\sigma _2}$. Also  if $x\in W_{\sigma _2}$ then $x\in Y_{i}$ and thus  $x\notin M_{i}$ what implies $x\notin W_{\sigma _1}$. Since the  charts cover $M^{2n}$   the union of admissible spaces gives $M^{2n}$.
\end{proof}

\begin{ex}\label{main}
The space $W_{[1,m]} = M_1\cap \cdots \cap M_{m}$ is non-empty and, thus, it is  an admissible space  and the set $\sigma = [1,m]$ is an admissible set. The space $W_{[1,m]}$  is a  dense set in $M^{2n}$ because, by Axiom~\ref{atlas} any  chart $M_i$, $1\leq i\leq m$ is an  open  dense in $M^{2n}$. We call $W_{[1,m]}$ the main space and use further the  notation $W= W_{[1,m]}$.
\end{ex}
\begin{ex}\label{one}
The space  $W_{\{i\}} = M_{i}\cap _{j\neq i} Y_{j}$ is an admissible space and the set $\sigma = \{i\}$ is an admissible set  for any $1\leq i\leq m$. The space $W_{i}$ is non-empty since, by Axiom~\ref{atlas}  the fixed point $x_i$ which belongs to
$ M_{i}$ also belongs  to all $Y_j$, $j\neq i$, $1\leq j\leq m$.
\end{ex}

\begin{rem}
Since $M^{2n}=M_1\cup \cdots \cup M_{m}, $ we have that $W_{\emptyset}=Y_1\cap \cdots \cap Y_{m}=\emptyset$, what implies that $W_{\emptyset}$ is not an admissible space.
\end{rem}
\begin{lem}\label{open}
Any  admissible space $W_{\sigma}$ different from the main admissible space $W$ can not be an open set.
\end{lem}
 \begin{proof}
Let $\sigma = \{i_1,\ldots ,i_l\}\subset [1,m]$, $1\leq  l\leq m$  and assume that $W_{\sigma}\neq W$ is an open set in $M^{2n}$. There exists $i\in [1,m]$ such that $i\notin \sigma$, what implies that $W_{\sigma}\subset Y_i$.   Since $M_i$ is everywhere dense in $M^{2n}$, it follows that  $W_{\sigma}$ can not  contain an open set.
\end{proof}

As for the boundary for $W_{\sigma}$ we have the following:
\begin{lem}\label{bound}
The boundary $\partial W_{\sigma}$ of an admissible space $W_{\sigma}$ is contained in the union of the admissible spaces $W_{\tilde{\sigma}}$, where $\tilde{\sigma}$ are the admissible sets such that $\tilde{\sigma}\subset \sigma$.
\end{lem}

\begin{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
\[
\partial W_{\sigma} \subseteq  \cup _{j_1=1}^{l}(\partial M_{i_{j_1}}\cap (\cap _{s\neq j_1} M_{i_s})\cap Y_{i_{l+1}}\cap \ldots \cap Y_{i_m})\cup
\]
\[
\cup _{1\leq j_1<j_2\leq l}(\partial M_{i_{j_1}}\cap \partial M_{i_{j_2}}\cap (\cap _{s\neq j_1,j_2}M_{i_s})\cap Y_{i_{l+1}}\cap \cdots \cap Y_{i_m}) \cup \cdots
\]
\[
\cup (\partial M_{i_1}\cap \cdots \cap \partial M_{i_l}\cap Y_{i_{l+1}}\cap \cdots \cap Y_{i_m})
\]
\[
= \cup _{j_1=1}^{l}(Y_{i_{j_1}}\cap(\cap_{s\neq j_1} M_{i_s})\cap Y_{i_{l+1}}\cap \ldots \cap Y_{i_m})\cup
\]
\[\cup _{1\leq j_1<j_2\leq l}(Y_{i_{j_1}}\cap Y_{i_{j_2}}\cap (\cap _{s\neq j_1,j_2}M_{i_s})\cap Y_{i_{l+1}}\cap \cdots \cap Y_{i_m}) \cup \cdots
\]
\[ \cup (Y_{i_1}\cap \cdots \cap Y_{i_l}\cap Y_{i_{l+1}}\cap \cdots \cap Y_{i_m}).
\]

Therefore $\partial W_{\sigma} \subseteq  \cup W_{\tilde{\sigma}}$, where ${\tilde{\sigma}}$ goes through all  admissible subsets  of $\sigma$.
\end{proof}

\section{Almost moment map, the admissible spaces and polytopes}

\begin{ax}\label{bij}
The map $\mu$ gives the bijection between  the set of fixed points   and the set of vertices of the polytope $P^k$.
\end{ax}

As the $k$ dimensional polytope has at least $k+1$ vertices we obtain:

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

Let $S(P^k)$ be the family of all convex polytopes which are spanned by the vertices of the polytope $P^k$. Denote by $\mathfrak{S}$ the set of all admissible sets.
Using almost moment map $\mu$  we define the  map
$s : \mathfrak{S}\to  S(P^k)$ as follows. Let  $\sigma  = \{i_1,\ldots ,i_l\}\subseteq [1,m]$. By Axiom~\ref{atlas} for any  $i_{j}\in \sigma$  there exists unique fixed point    $x_{i_j}\in M_{i_j}$. Denote by   $v_{i_1},\ldots ,v_{i_l}$  the vertices of the polytope $P^k$ determined by
$v_{i_j} = \mu (x_{i_j})$. We put
\[
s(\sigma) = P_{\sigma},\; \text{where}\; P_{\sigma} =\text{convhull}(v_{i_1},\ldots ,v_{i_l}),
\]

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

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

\begin{ex}\label{adm_P}
The polytope $P^{k}$ is an admissible polytope. The set $\sigma = \{1,\ldots ,m\}$ is an admissible set as it is shown in Example~\ref{main}. In addition $s(\sigma)$ is, by Axiom~\ref{bij}  a convex hull of all vertices of $P^k$, what implies that  $s(\sigma) = P^{k}$.
\end{ex}

\begin{ex}\label{adm_v}
Any vertex $v$ of $P^{k}$ is an admissible polytope.   To see that take, using Axiom~\ref{bij}, the fixed point $x_i$ such that $\mu (x_i) = v$. Then the set $\sigma = \{i\}$ is an admissible set as it is shown in Example~\ref{one} and $s(\sigma ) = v$.
\end{ex}

\begin{rem}
For a general $(2n,k)$-manifold  two  admissible polytopes may intersect.
One can see it, for example, in the case  complex Grassmann manifold $G_{4,2}$ which is  $(8,3)$-manifold, see~\cite{BT-1}.
\end{rem}

\begin{defn}
A point $p\in P^k$ is said to be exceptional  if $p\in \stackrel{\circ}{P}_{\sigma _1}\cap \stackrel{\circ}{P}_{\sigma _2}$ for some admissible polytopes  $P_{\sigma _1}, P_{\sigma _2}$. Otherwise, it is said to be simple.
\end{defn}

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

\section{Stabilizers for the torus action on admissible spaces}

Denote by $S(T^k)$ the set  of all connected subgroups of $T^k$. Note that a connected subgroup of $T^k$ is a torus.
Let us consider the function $\chi : M^{2n}\to S(T^k)$ defined by  $\chi (x)$ is the stabilizer of a point $x$ for the given $T^k$-action
on $M^{2n}$. By the general set up assumption  $\chi (x)$ is  a connected subgroup of $T^k$. We assume the following to be satisfied:

\begin{ax}\label{stab}
 The characteristic function $\chi$ is constant on any admissible space $W_{\sigma}$.
\end{ax}

Using Axiom~\ref{stab} we  define the torus $T^{\sigma} = T^{k}/\chi (W_{\sigma})$ for any admissible space  $W_{\sigma}$.

\begin{cor}
Torus $T^{\sigma}$ acts freely on $W_{\sigma}$, what gives the  principal fibration
\begin{equation}
T^{\sigma}\to W_{\sigma}\to W_{\sigma}/T^{\sigma}.
\end{equation}
\end{cor}

Following the terminology of~\cite{GS} and~\cite{BTN},  the  admissible spaces we call the strata as well.

\section{Orbit spaces of the strata}

By the definition we assume  that the  almost moment map $\mu : M^{2n}\to P^k$ is  $T^k$ - invariant. Therefore,  the almost moment map $\mu$   induces the map  $\widehat{\mu} : M^{2n}/T^k \to P^k$.  We require the following to be satisfied.

\begin{ax}~\label{fiber}
The almost moment map $\mu$:
\begin{itemize}
\item gives the mapping from $W_{\sigma}$ to $\stackrel{\circ}{P_{\sigma}}$,
\item induces locally trivial fiber bundle $ \widehat{\mu} : W_{\sigma}/T^{\sigma}\to \stackrel{\circ}{P_{\sigma}}$,
\item $\dim P_{\sigma} = \dim T^{\sigma}$.
\end{itemize}
\end{ax}

An immediate consequence of this Axiom is:
\begin{cor}
\begin{itemize} It holds
\item
$\widehat{\mu}(W/T^k) = \stackrel{\circ}{P^{k}}$  for the  main space $W = M_1\cap\cdots \cap M_{m}$,
\item  $\mu (W_{\{i\}}) = \{v_i\}$, where  $v_{i}$  is a vertex and $W_{\{i\}} = M_{i}\cap _{j\neq i} Y_{j}$.

\end{itemize}
\end{cor}

Denote by $[F_{\sigma}]$ the homeomorphic type of the fiber of the fibration $ \widehat{\mu} : W_{\sigma}/T^k\to \stackrel{\circ}{P_{\sigma}}$.
\begin{defn}\label{param}
The space $F_{\sigma}$ of the homeomorphic type $[F_{\sigma}]$ we call the space of parameters  of the stratum $W_{\sigma}$.
\end{defn}

Since $\stackrel{\circ}{P_{\sigma}}$ is contractible and the bundle $ W_{\sigma}/T^{k}\to  \stackrel{\circ}{P_{\sigma}}$ is locally trivial we obtain:

\begin{cor}\label{homeom}
The fiber bundle $\widehat{\mu} : W_{\sigma}/T^{k}\to  \stackrel{\circ}{P_{\sigma}}$  is isomorphic to the trivial bundle. That is $W_{\sigma}/T^k$ is homeomorphic to $\stackrel{\circ}{P_{\sigma}} \times F_{\sigma}$ by the fiber wise homeomorphism
\begin{equation*}\begin{CD}
 W_{\sigma}/T^{k} @>>>  \stackrel{\circ}{P_{\sigma}} \times F_{\sigma}\\
  @VV{\widehat{\mu}}V  @VV{}V\\
\stackrel{\circ}{P_{\sigma}} @= \stackrel{\circ}{P_{\sigma}}.
\end{CD}\end{equation*}
\end{cor}

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

Let $\overline{W_{\sigma}/T^{\sigma}}$ denotes the closure of $W_{\sigma}/T^{\sigma}$. It is a compact set in $M^{2n}/T^k$ since we assume $M^{2n}$ to be compact. Therefore:

\begin{cor}\label{closure}
$\widehat{\mu}(\overline{W_{\sigma}/T^{\sigma}}) = P_{\sigma}$.
\end{cor}
\begin{proof}
Since $\widehat{\mu} : M^{2n}/T^k \to P^{k}$ is continuous and $\widehat{\mu}(W_{\sigma}/T^k) = \stackrel{\circ}{P_{\sigma}}$ it implies that $\stackrel{\circ}{P_{\sigma}} \subset \widehat{\mu}(\overline{W_{\sigma}/T^k}) \subseteq P_{\sigma}$. Since $\widehat{\mu}(\overline{W_{\sigma}/T^k})$ is a compact set it follows  $\widehat{\mu}(\overline{W_{\sigma}/T^k}) = P_{\sigma}$.
\end{proof}

The trivialization $h_{\sigma} : W_{\sigma}/T^{\sigma} \to \stackrel{\circ}{P}_{\sigma}$ induces the trivialization $\xi _{\sigma} :  W_{\sigma}/T^{\sigma} \to F_{\sigma}$. For any   point $c_{\sigma}\in F_{\sigma}$ define the subspace $W_{\sigma}[\xi _{\sigma},c_{\sigma}]$ of  $W_{\sigma}$ by
\begin{equation}\label{leaf}
W_{\sigma}[\xi _{\sigma},c_{\sigma}] = (\pi _{\sigma} ^{-1}\circ \xi _{\sigma}^{-1})(c_{\sigma}),
\end{equation}
where $\pi _{\sigma} : W_{\sigma}\to W_{\sigma}/T^{\sigma}$ is the projection.

We say that the space $W_{\sigma}[\xi _{\sigma},c_{\sigma}]$ is the {\it leaf} of the stratum $W_{\sigma}$ given by the trivialization $\xi _{\sigma}$.
Note that, by its definition, the leaf $W_{\sigma}[\xi _{\sigma},c_{\sigma}]$ is invariant under the action of $T^k$ and  $W_{\sigma} = \cup _{c_{\sigma}\in F_{\sigma}}W_{\sigma}[\xi _{\sigma},c_{\sigma}]$.

The definition of the leaf also  implies:
\begin{lem}\label{leaf}
Let $\widehat{\mu}_{\xi _{\sigma},c_{\sigma}}$ denotes the restriction of $\widehat{\mu} : M^{2n}/T^k\to P^k$  to $W_{\sigma}[\xi _{\sigma},c_{\sigma}]/T^{\sigma}$.
Then $\widehat{\mu}_{\xi _{\sigma},c_{\sigma}} : W_{\sigma}[\xi _{\sigma},c_{\sigma}]/T^{\sigma} \to \stackrel{\circ}{P_{\sigma}}$ is a homeomorphism for any $c_{\sigma}\in F_{\sigma}$.
\end{lem}

Moreover we obtain:
\begin{lem}\label{leaf-bound}
$\widehat{\mu} (\overline{W_{\sigma}[\xi _{\sigma}, c_{\sigma}]/T^{k}}) = P_{\sigma}$.
\end{lem}

\begin{proof}
Since $\widehat{\mu}$ is a continuous map, it follows from Lemma~\ref{leaf}  that $\widehat{\mu} (\overline{W_{\sigma}[\xi _{\sigma}, c_{\sigma}]/T^{k}}) \subseteq P_{\sigma}$. On the other hand $\overline{W_{\sigma}[\xi _{\sigma}, c_{\sigma}]/T^{k}}$ is a closed subset in the compact space $M^{2n}/T^k$ and,  thus,  it is compact as well. It implies that $\widehat{\mu} (\overline{W_{\sigma}[\xi _{\sigma}, c_{\sigma}]/T^{k}})$ is a  compact set in $P_{\sigma}$ which contains the interior of $P_{\sigma}$ what gives that $\widehat{\mu} (\overline{W_{\sigma}[\xi _{\sigma}, c_{\sigma}]/T^{k}}) = P_{\sigma}$.
\end{proof}

We use the description of the strata  as the spaces consisting of leafs to formulate the properties which allow to describe the gluing of the strata. Recall that we have structurally fixed  trivialization trivialization $\xi _{\sigma} :  W_{\sigma}/T^{\sigma} \to F_{\sigma}$.

\begin{ax}\label{leaf}
 For any $c_{\sigma}\in F_{\sigma}$ the boundary $\partial W_{\sigma}[\xi _{\sigma}, c_{\sigma}]$ of the leaf  $W_{\sigma}[\xi _{\sigma},c_{\sigma}]$ of $W_{\sigma}$  is  the union of the 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  $\sigma\in \mathfrak{S}$.
\end{ax}

\begin{rem}
Axiom~\ref{leaf}, as it can be seen in Section~\ref{gr}, is motivated by the results of Atiyah, Guillemin-Sternberg and Gel'fand-MacPherson about $(\C ^{*})^{n}$-action on the complex Grassmann manifold $G_{n,k}$.
\end{rem}

We deduce the following important consequence of Axiom~\ref{leaf}.

\begin{prop}\label{face}
A face of any admissible polytope is an admissible polytope.
\end{prop}
\begin{proof}
Let us fix some  face $P_{\tilde{\sigma}}$ of an admissible polytopes $P_{\sigma}$ and consider a point $p\in P_{\tilde{\sigma}}$.   By  Lemma~\ref{leaf-bound}  we see  that $\widehat{\mu}^{-1}(p)\cap \partial W_{\sigma}[\xi _{\sigma}, c_{\sigma}]/T^{k}\neq \emptyset$. Then there exists  a point $x$ from the boundary of $W_{\sigma}[\xi _{\sigma}, c_{\sigma}]$ such that $\pi (x) \in \widehat{\mu}^{-1}(p)$. It implies that $\mu (x) = p$. By Axiom~\ref{leaf} 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  $\mu (x)=p \in  P_{\bar{\sigma}}$. It follows that $p\in  P_{\tilde{\sigma}}\cap P_{\bar{\sigma}}$, and since $P_{\bar{\sigma}}$ and $P_{\tilde{\sigma}}$ are the faces of the same polytope $P_{\sigma}$  this  implies that $P_{\tilde{\sigma}} = P_{\bar{\sigma}}$. Therefore  $P_{\tilde{\sigma}}$ is an admissible polytope.
\end{proof}

\begin{rem}
Note that the Axiom~\ref{leaf} together with the proof of the Proposition~\label{face} implies that  if   $P_{\tilde{\sigma}}$  is a face of $P_{\sigma}$ then  there exists the leaf $W_{\tilde{\sigma}}[\xi _{\tilde{\sigma}}, c_{\tilde{\sigma}}]$ which is contained in the boundary of the leaf $W_{\sigma}[\xi _{\sigma}, c_{\sigma}]$.
\end{rem}

It follows that the Axiom~\ref{leaf} can be strengthen:

\begin{cor}\label{all}
For any $c_{\sigma}\in F_{\sigma}$, the boundary $\partial W_{\sigma}[\xi _{\sigma}, c_{\sigma}]$ of the leaf $W_{\sigma}[\xi _{\sigma},c_{\sigma}]$ of the stratum  $W_{\sigma}$  is  the union of the 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  the {\bf all} faces for  $P_{\sigma}$.
\end{cor}

Corollary~\ref{all} directly implies that Lemma~\ref{leaf-bound} can be strengthen:
\begin{cor}\label{cl-leaf}
The map $\widehat{\mu} : \overline{W_{\sigma}[\xi _{\sigma}, c_{\sigma}]/T^{\sigma}} \to P_{\sigma}$ is a homeomorphism.
\end{cor}

Let $P_{\bar{\sigma}}$ be a face of $P_{\sigma}$. Then Axiom~\ref{leaf} implies the existence of the map
\begin{equation}\label{eta}
\eta _{\sigma, \bar{\sigma}} : F_{\sigma} \to F_{\bar{\sigma}}
\end{equation}
defined by: $\eta _{\sigma, \bar{\sigma}}(c_{\sigma})$ is the unique point $c_{\bar{\sigma}}$ from $F_{\bar{\sigma}}$ such that $W_{\bar{\sigma}}[\xi _{\bar{\sigma}}, c_{\bar{\sigma}}] \subset \partial W_{\sigma}[\xi _{\sigma}, c_{\sigma}]$.

Let now $P_{\bar{\bar{\sigma}}}$ be a face of $P_{\bar{\sigma}}$ and $P_{\bar{\sigma}}$ be a face of $P_{\sigma}$. Then by Axiom~\ref{leaf} there are trivializations $\xi _{\bar{\bar{\sigma}}} : W_{\bar{\bar{\sigma}}}/T^k\to F_{\bar{\bar{\sigma}}}$, $\xi _{\bar{\sigma}} : W_{\bar{\sigma}}/T^k\to F_{\bar{\sigma}}$ and  $\xi _{\sigma} : W_{\sigma}/T^k\to F_{\sigma}$ such that
\[
W_{\bar{\bar{\sigma}}}[\xi _{\bar{\bar{\sigma}}}, c_{\bar{\bar{\sigma}}}] \subset \partial W_{\bar{\sigma}}[\xi _{\bar{\sigma}}, c_{\bar{\sigma}}] \subset \partial \partial W_{\sigma}[\xi _{\sigma}, c_{\sigma}] = \partial W_{\sigma}[\xi _{\sigma}, c_{\sigma}].
\]

All together this implies:
\begin{cor}
For any pair $P_{\bar{\sigma}}\subset P_{\sigma }$ there exists the map  $\eta _{\sigma ,\sigma ^{'}} : F_{\sigma}\to F_{\bar{\sigma}}$ such that if $P_{\bar{\bar{\sigma}}}\subset P_{\bar{\sigma}}\subset P_{\sigma}$ then
$\eta _{\bar{\sigma} ,\bar{\bar{\sigma}}}\circ \eta _{\sigma ,\bar{\sigma}} = \eta _{\sigma ,\bar{\bar{\sigma}}}$.
\end{cor}

\section{On singular and regular values of the almost moment map}
We characterize the  singular and regular values of the almost moment map $\mu : M^{2n} \to P^k$,  under the assumption that it is a smooth map.
\begin{defn}
The cort\'ege $\sigma (x)$ of the  point $x\in P^{k}$ is defined by:
\[
\sigma(x) = \{ P_{\sigma} \in P_{\mathfrak{S}}: x\in \stackrel{\circ}{P}_{\sigma}\}.
\]
\end{defn}

Obviously, $\sigma(x)\neq \emptyset$ for any $x\in P^k$ since $\cup _{\sigma\in \mathfrak{S}}\stackrel{\circ}{P}_{\sigma}=P^k$.
Moreover,  $P^k\in \sigma(x)$ for any $x\in \stackrel{\circ}{P^{k}}$ .
\begin{defn}
The point $x\in P^k$ is said to be  regular  if $\dim P_{\sigma} =k$ for any $P_{\sigma}\in \sigma (x)$.
\end{defn}

\begin{thm}
The point $x\in P^k$ is the regular value for the almost  moment map $\mu : M^{2n}\to P^k$ if and only if $\dim P_{\sigma}=k$ for any $P_{\sigma}\in \sigma (x)$.
\end{thm}
\begin{proof}
Let  $x\in P^k$, then $ \mu ^{-1}(x)\subset \cup _{\sigma} W_{\sigma}$, where  the union goes through all admissible sets $\sigma$ such that $P_{\sigma}\in \sigma (x)$. The point $x$  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 the rank equal to $k$.  The rank of the differential for $\mu$ at the point $y\in W_{\sigma} $ is, by the general slice theorem~\cite{BR} and Axiom~\ref{fiber},   equal to the dimension of $P_{\sigma}$. It follows that $y$ is a regular point for $\mu $ if and only if $\dim P_{\sigma}=k$, which proves the statement.
\end{proof}
\begin{cor}
If the point $x\in P^k$ is a regular value of the almost moment map  $\mu$ then $x\in \stackrel{\circ}{P^{k}}$.
\end{cor}
\begin{cor}
For any regular point $x\in \stackrel{\circ}{P}^{k}$ the preimage $\mu ^{-1}(x)$ is a closed submanifold in $M^{2n}$ of the dimension $2(n-k)$.
\end{cor}
Note that the manifold $M_{x}^{2(n-k)}=\mu ^{-1}(x)$ is a compactification of the space of parameters $F$ of the main stratum,   for a regular point $x\in P^k$.
\subsection{The main stratum and its  orbit space}\label{main-sp-orb}
 As noted in Example~\ref{main} the main stratum $W$ is everywhere dense in $M^{2n}$,  what implies that $W/T^k$ is everywhere dense space  in the orbit space $M^{2n}/T^k$. Moreover $\widehat{\mu}(W/T^k) = \stackrel{\circ}{P^{k}}$ and $W/T^k \cong \stackrel{\circ}{P^{k}}\times F$, where  $F$ denotes the space of parameters of the stratum  $W$ introduced by Definition~\ref{param}. It follows that there exists compactification of the product $\stackrel{\circ}{P^{k}}\times F$ which is homeomorphic to $M^{2n}/T^k$.

The first result in this direction is the following.

\begin{prop}\label{point}
If $F$ is a point then the orbit space $M^{2n}/T^k$ is homeomorphic to $P^k$.
\end{prop}
\begin{proof}
If $F$ is a point, we have that $W$ consists of one leaf. Then Lemma~\ref{cl-leaf} implies that the closure $\overline{W/T^k}$ is homeomorphic to $P^k$, what implies that $M^{2n}/T^{k}$ is homeomorphic to $P^k$.
\end{proof}

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

Assume now that the space of parameters $F$ of the main stratum  is not a point. Let $x\in  \stackrel{\circ}{P^{k}}$ and denote by $\mathfrak{S}(x) = \{\sigma \in \mathfrak{S} | x\in \stackrel{\circ}{P_{\sigma}}\}$, 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
\[
\widehat{\mu}^{-1}(x) =  \cup _{\sigma \in \mathfrak{S}(x)} \{y\in W_{\sigma}/T^k : \widehat{\mu}(y) = x\} \cong \cup _{\sigma \in \mathfrak{S}(x)} F_{\sigma}.
\]
Let $\overline{F_{x}}$ denotes the closure of $F_{x} = \widehat{\mu}^{-1}(x)\cap W/T^k \cong F$. Since $\overline{W/T^k} = M^{2n}/T^k$ we obtain that for any $x\in \stackrel{\circ}{P^{k}}$ it  holds
\begin{equation}\label{param-cl}
 \overline{F_{x}} = \widehat{\mu}^{-1}(x)\cong \cup _{\sigma \in \mathfrak{S}(x)} F_{\sigma}.
 \end{equation}

We first point the following:
\begin{lem}
The space of parameters $F$ of the main stratum  is a compact space if and only if all points of $\stackrel{\circ}{P^{k}}$ are simple.
\end{lem}
\begin{proof}
If all  points from  $\stackrel{\circ}{P^{k}}$ are simple, then  $P^k$ is the only admissible polytope that intersects interior of $P^k$. It implies that $\widehat{\mu}^{-1}(x) \cong F$ for any $x\in \stackrel{\circ}{P^{k}}$.  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 $x\in \stackrel{\circ}{P^k}$, it follows that     $F\cong  \widehat{\mu}^{-1}(x)$  what means that all points from $\stackrel{\circ}{P^{k}}$ are simple.
\end{proof}
\

\begin{rem}
We want to note that although $F_{x}$ is homeomorphic to $F$ for any $x\in \stackrel{\circ}{P^{k}}$, there is no argument to claim that their  closures $\overline{F_{x}}$ are homeomorphic. But, since there are finitely many admissible sets it follows from~\eqref{param-cl} that among all $\overline{F_{x}}$ it can be only finitely many non homeomorphic spaces.
\end{rem}

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

Let $P_{\sigma_{0}}$ be a face of $P^{k}$ and denote by $\mathfrak{S}(\sigma_{0})$ the set of those $\sigma \in \mathfrak{S}$  for which  $P_{\sigma _{0}}$ is a face of $P_{\sigma}$ and $P_{\sigma}$ is not a face of $P^{k}$.
\begin{prop}\label{paramet-bound}
Assume that all points from $\partial P^k$ are simple,  that the spaces $\overline{F_{x}}$ and $\overline{F_{y}}$ are homeomorphic for all  $x,y \in \stackrel{\circ}{P^{k}}$ and denote their  homeomorphic type by   $\overline{F}$.  Then for any $\sigma _{0}\in \mathfrak{S}$ such that  $P_{\sigma_{0}}$ is  a face of $P^{k}$ it holds
\[
\cup_{\sigma \in \mathfrak{S}(\sigma_{0})} F_{\sigma} \cong \overline{F}.
\]
\end{prop}

\begin{proof}
We first note  that the assumption that the points from $\partial P^k$ are simple implies that  if $P_{\sigma _{0}}$ is not a face of $P_{\sigma}$ for 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 it belongs   to some face $P_{\bar{\sigma}}$ of $P_{\sigma}$,  which is an  admissible polytope.  Since $z$ is a simple point it implies that  $P_{\bar{\sigma}} = P_{\sigma _{0}}$. Therefore, since there are only finitely many admissible polytopes,  there exists a neighborhood $U$ of $P_{\sigma _{0}}$ in $P^k$ such that $U\cap P_{\sigma}=\emptyset$ for any admissible polytope $P_{\sigma}$ such that $P_{\sigma _{0}}$ is not a face of $P_{\sigma}$. Further $U\cap (\cap _{\sigma \in \mathfrak{S}(\sigma _{0})}P_{\sigma}) \neq \emptyset$ since $P_{\sigma _{0}}$ belongs to all of them, and for a point $x\in  U\cap (\cap _{\sigma \in \mathfrak{S}(\sigma _{0})}P_{\sigma})$ we have that $\overline{F}\cong \widehat{\mu}^{-1}(x) \cong \cup_{\sigma \in \mathfrak{S}(\sigma_{0})} F_{\sigma}$.
\end{proof}

\begin{thm}\label{main-main}
Assume that all points from $\partial P^k$ are simple and that  the spaces $\widehat{\mu}^{-1}(x) \subset M^{2n}/T^k$ are homeomorphic for all $x\in \stackrel{\circ}{P^{k}}$. Then the orbit space $M^{2n}/T^k$ is homeomorphic to
\[
P^{k}\times \overline{F}/\approx ,
\]
\[
(x,f_1)\approx (y,f_2) \Leftrightarrow x=y\in P_{\sigma _{0}}\subset \partial P^k\; \text{and}\; \eta _{\sigma _{1} ,\sigma _{0}}(f_1) = \eta _{\sigma _{2} ,\sigma _{0}}(f_2),
\]
where $\sigma _1, \sigma _{2} \in \mathfrak{S}(\sigma _{0})$ 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~\eqref{eta}.
\end{thm}
\begin{proof}
Let us denote $\overline{F} = \widehat{\mu}^{-1}(x)$, where  $x\in \stackrel{\circ}{P^{k}}$. Then $\widehat{\mu} : \widehat{\mu}^{-1}(\stackrel{\circ}{P^{k}})\to  \stackrel{\circ}{P^{k}}$ is a fibration with a fiber $\overline{F}$. Since $\stackrel{\circ}{P^{k}}$ is contractible it implies that $\widehat{\mu}^{-1}(\stackrel{\circ}{P^{k}})$ is homeomorphic to $\stackrel{\circ}{P^{k}}\times \overline{F}$.

Let $P_{\sigma _{0}}$ be a face of $P^{k}$. It is an admissible polytope.  Since any point from $\partial P^{k}$ is simple, it follows that
$W_{\sigma_{0}}/T^k = \widehat{\mu}^{-1}(P_{\sigma _{0}})$, and, thus, $\widehat{\mu}^{-1} : \widehat{\mu}^{-1}(P_{\sigma _{0}})\to P_{\sigma _{0}}$ is a fibration with a fiber $F_{\sigma _{0}}$. It implies that $\widehat{\mu}^{-1}(P_{\sigma _{0}})$ is homeomorphic to $P_{\sigma _{0}}\times F_{\sigma _{0}}$.  We also have that $F_{\sigma _{0}}\subseteq \overline{F}$ since $\widehat{\mu}^{-1}(\stackrel{\circ}{P^{k}})$ is everywhere dense set in $M^{2n}/T^k$ and $\overline{F}$ is closed. Together with Proposition~\ref{paramet-bound} we obtain that
$F_{\sigma _{0}} \subseteq \cup_{\sigma \in \mathfrak{S}(\sigma_{0})} F_{\sigma}$. Therefore, the family of maps $\eta _{\sigma, \sigma_{0}} : F_{\sigma}\to F_{\sigma _{0}}$ gives the map $\eta : \overline{F}\to F_{\sigma _{0}}$ which is surjective. It implies that $P_{\sigma _{0}}\times F_{\sigma _{0}}$ is homeomorphic to the quotient of $P_{\sigma _{0}}\times \overline{F}$ by the relation $(x, c_1)\approx (x, c_2)$ if and only if
$\eta (c_1) = \eta (c_2)$, which  is equivalent to $\eta _{\sigma _{1}, \sigma _{0}}(c_1) = \eta _{\sigma _{2}, \sigma _{0}}(c_2)$, 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.
\end{proof}

\begin{thm}\label{join}
Assume that all points from $\partial P^k$ are simple and that  the spaces $\widehat{\mu}^{-1}(x) \subset M^{2n}/T^k$ are homeomorphic for all $x\in \stackrel{\circ}{P^{k}}$. If $F_{\sigma _{0}}$ is a point for any $\sigma _{0}\in \mathfrak{S}$ such that $P_{\sigma _{0}}$ is a face of $P^{k}$ then:
\[
M^{2n}/T^k \cong S^{k-1}\ast \overline{F} .
\]
\end{thm}
\begin{proof}
Recall that the joint $X\ast Y$ for a topological spaces $X$ and $Y$ is defined to be quotient space of $CX\times Y$ by the relation $(x_1,0,y_1)\approx (x_2,0,y_2) \Leftrightarrow x_1=x_2$, where $CX$ is the cone over $X$. In our case Theorem~\ref{main-main} gives that
\[
M^{2n}/T^k \cong P^{k}\ast \times \overline{F}/\approx, \;\; (x,f_1)\approx (y,f_2) \Leftrightarrow x=y\in \partial P^{k}.
\]
We identify $P^{k}$ with $\overline{D^{k}}$ and further  $\overline{D^{k}}$  with $CS^{k-1}$. It implies that
\[
M^{2n}/T^k \cong CS^{k-1}\times \overline{F}/\approx,\;\; (x,0, f_1)\approx (x,0,f_2),
\]
what proves the statement.
\end{proof}

\section{Complex of  admissible polytopes}
Let $P_{\mathfrak{S}}$ denotes  the  family of admissible polytopes. By Example~\ref{adm_P}, Example~\ref{adm_v} and Proposition~\ref{face} we have
\begin{itemize}
\item $P^{k}\in P_{\mathfrak{S}}$ and $v\in P_{\mathfrak{S}}$ for any vertex $v$.
\item  If $P_{\sigma} \in P_{\mathfrak{S}}$ and $P_{\bar{\sigma}}$ is a face of $P_{\sigma}$, then  $P_{\bar{\sigma}} \in P_{\mathfrak{S}}$.
\end{itemize}

Consider the complex  $C(M^{2n}, P^k)$,  which is obtained as the formal disjoint union of  all admissible polytopes:

\begin{equation}\label{complex}
 C(M^{2n}, P^k) = \cup _{P_{\sigma}\in P_{\mathfrak{S}}}P_{\sigma}.
\end{equation}

Since by the construction of  $C(M^{2n}, P^k), $ there is the bijection between the set of polytopes  of $C(M^{2n}, P^k)$  and the  set of admissible polytopes $P_{\mathfrak{S}}$ we have:

\begin{lem}
There is the canonical map $\widehat{\pi} : C(M^{2n}, P^k)\to P^k$.
\end{lem}
\begin{proof}
For  any  polytope $P_{\sigma}^{'}$ in $C(M^{2n}, P^k)$  there is the unique admissible polytope  $P_{\sigma}\in P_{\mathfrak{S}}$ which corresponds to the polytope $P_{\sigma}^{'}$ and  we define $\hat{\pi}(P_{\sigma}^{'})=P_{\sigma}$.
\end{proof}
\begin{cor}\label{homeom2}
For any $P_{\sigma}\in P_{\mathfrak{S}}$ there is the unique  polytope $P_{\sigma}^{'}$ in $C(M^{2n}, P^k)$  such that  the map $\widehat{\pi} : P_{\sigma}^{'}\to P_{\sigma}$ is a homeomorphism.
\end{cor}

\begin{cor}\label{quot}
The canonical map $\widehat{\pi} : C(M^{2n}, P^k)\to P^k$ is a quotient map.
\end{cor}
\begin{proof}
Define the relation $\rho$ on $C(M^{2n}, P^k)$ by $x\rho y$ if and only if $\widehat{\pi} (x) =\widehat{\pi} (y)$. It is obvious that $P^{k} = C(M^{2n}, P^k)/\rho$ and that  $\widehat{\pi} : C(M^{2n}, P^k)\to P^k$ is a quotient map.
\end{proof}

\begin{cor}\label{can-bij}
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.
\end{cor}

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

Taking into account the existence of an almost moment map $\mu : M^{2n}\to P^{k}$ we prove more:

\begin{lem}\label{canonic}
There is the canonical map $f : M^{2n}\to C(M^{2n}, P^k)$ such that $\mu = \widehat{\pi} \circ f$.
\end{lem}
\begin{proof}
For any  $x\in M^{2n}$ there exists unique  stratum  $W_{\sigma}$ such that  $x\in W_{\sigma}$. Then  $\mu(x)\in \stackrel{\circ}{P_{\sigma}}$ and by Corollary~\ref{homeom2}  there exists  unique $P_{\sigma}^{'}\subseteq C(M^{2n}, P^k)$ such that $\widehat{\pi} :\stackrel{\circ}{ P_{\sigma}^{'}}\to \stackrel{\circ}{P_{\sigma}}$ is a homeomorphism. It  implies that there exists unique $y\in \stackrel{\circ}{P_{\sigma}^{'}}$ such that $\widehat{\pi} (y) = \mu (x)$. In this way it is defined the map $f : M^{2n}\to C(M^{2n}, P^k)$ by $f(x)=y$.
\end{proof}

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

Denote by $S(P_{\mathfrak{S}})$ the partitive set of $P_{\mathfrak{S}}$. It is  defined the operator
\[
d : P_{\mathfrak{S}}\to S(P_{\mathfrak{S}})\;
\text{by}\; dP_{\sigma}\; \text{is disjoint union of the faces of}\; P_{\sigma}.
\]
The operator $d$  induces the operator  $d_{C}$ from the complex  $C(M^{2n}, P^k)$ to its partitive set $S(C(M^{2n}, P^k))$ defined by
\[
d_{C}(P_{\sigma}^{'} )  = dP_{\sigma}^{'},
\]
where  $ dP_{\sigma}^{'}$ denotes the union   of the faces of $P_{\sigma}$.

It is important to emphasize the following. One can try to ask if it is possible similarly to define the boundary operator $\tilde{d} : W_{\mathfrak{S}}\to S(W_{\mathfrak{S}})$. where  $W_{\mathfrak{S}}$  denotes the set of all strata and $S(W_{\mathfrak{S}})$ its partitive set,  by    $\tilde {d}(W_{\sigma}) = \overline{W_{\sigma}} - W_{\sigma}$.  The answer is negative in general. Namely, as it is shown in~\ref{Grassmann}, the Grassmann manifold $G_{7,3}$  is an example of a manifold that belongs to our class but which contains  the stratum  $W_{\sigma}$ whose boundary is not the union of the strata. Therefore, the operator $\tilde{d}$ is  not  defined for $G_{7,3}$.

We still want to note that there are important examples of $(2n,k)$-manifolds  for which the operator $\tilde{d}$ is correctly defined. Such examples are among the others Grassmann manifold $G_{4,2}$ and complex projective space $\C P^5$ with the canonical action of $T^4$ studied in~\cite{BT-1}. In these cases the operator $\tilde{d}$  produces the complex of  admissible sets $C(W_{\mathfrak{S}})$.  Moreover, the canonical map $f : M^{2n}\to C(M^{2n}, P^k)$  gives the covering of the  complex $C(M^{2n}, P^k)$ by the complex of the strata  $W_{\mathfrak{S}}$.
More precisely:
\begin{lem}
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 bijection between the complexes.
\end{lem}
\begin{proof}
Let $W_{\sigma}$ be a stratum. Then  $\mu (W_{\sigma}) = \stackrel{\circ}{P_{\sigma}}$ and Lemma~\ref{canonic} implies that $\pi(f(W_{\sigma})) =  \stackrel{\circ}{P_{\sigma}}$. By Corollary~\ref{homeom} we conclude that $f(W_{\sigma}) =\stackrel{\circ}{ P_{\sigma}^{'}}$,  which is a polytope 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 since there is bijection between the set of the strata   $W_{\mathfrak{S}}$ and the set of admissible polytopes $P_{\mathfrak{S}}$. Moreover, it follows from Corollary~\ref{all} that $d\circ f_{\mathfrak{S}} = f_{\mathfrak{S}}\circ \tilde{d}$ meaning that $f_{\mathfrak{S}}$ is a  map between the complexes.
\end{proof}

\subsection{CW-topology on $C(M^{2n},P^k)$}\label{topCW}

As $C(M^{2n},P^k)$ is the complex of all  admissible polytopes, it is natural  to define  the  topology on $C(M^{2n},P^k)$,  such that $C(M^{2n},P^k)$ becomes CW-complex  which we denote by $CW(M^{2n},P^k)$, that is:
\begin{itemize}
\item  the cells of these complex are $\stackrel{\circ}{P_{\sigma}}$ for $P_{\sigma}\in P_{\mathfrak{S}}$,
\item these cells are glued by induction using the operator $d_{C}$, which confirms that the cell axiom is satisfied,
\item $CW(M^{2n}, P^{k})$ has the weak topology related to the cell decomposition.
\end{itemize}

Then the following is satisfied:
\begin{lem}
\begin{itemize}
 \item The canonical map $\widehat{\pi} : CW(M^{2n}, P^k)\to P^k$ is  a continuous map.
\item  The map $\widehat{\pi} : CW(M^{2n}, P^k)\to P^k$  is a  cell map  for the standard cell decomposition of $P^{k}$ if and only if it is a homeomorphism.
\end{itemize}
\end{lem}
\begin{proof}
To prove that the map $\widehat{\pi}$ is a continuous map it is enough to notice that if $U\subseteq P^{k}$ is a closed set in $P^{k}$ it implies that $U\cap P$ is a closed set in $P$ , for any polytope $P$ over some subsets of vertices of $P^k$. It will be as well true for an admissible polytopes what implies that $\widehat{\pi}^{-1}(U)$ is a closed set in $CW(M^{2n}, P^k)\to P^k$.
The second part follows directly from Corollary~\ref{can-bij}.
\end{proof}

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

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

\begin{proof}
Let $P_{\sigma} \in  P_{\mathfrak{S}}$, $\dim P_{\sigma} =k$ and $P_{\sigma}\neq P^k$.  Then $\stackrel{\circ}{P_{\sigma}^{'}} \cong \stackrel{\circ}{P_{\sigma}}$ is an open cell in $CW (M^{2n}, P^k)$.  Consider the corresponding stratum  $W_{\sigma}$, that is $\mu (W_{\sigma}) =\stackrel{\circ}{ P_{\sigma}}$. Since $\mu = \widehat{\pi}\circ f$, it follows that $f^{-1}(\stackrel{\circ}{P_{\sigma}^{'}}) = W_{\sigma}$, which is not an open set in $M^{2n}$ by Lemma~\ref{open}
\end{proof}

\begin{rem}
Note that under the same assumption as in  Proposition~\ref{canon-ncont} the canonical map  $\widehat{f} : M^{2n}\to CW (M^{2n}, P^k)$ is not continuous as well.
\end{rem}

\begin{ex}\label{G42}
We demonstrate Proposition~\ref{canon-ncont} in the case of the  Grassmann $G_{4,2}$ endowed with the canonical action of $T^4$.
Following~\cite{BT-1}, any four sided pyramid $P$  in $\Delta _{4,2}$ is an admissible polytope, whicht implies that $\stackrel{\circ}{P^{'}}\cong \stackrel{\circ}{P}$ is an open cell in $CW(G_{4,2}, \Delta _{4,2})$. On the other hand $f^{-1}(\stackrel{\circ}{P^{'}})$ is $(\C ^{*})^3$-orbit in the eight-dimensional manifold $G_{4,2}$ , and it can not be an open set.
\end{ex}

\subsection{Quotient topology on $C(M^{2n},P^k)$}\label{topQ}
As we have seen from Proposition~\ref{canon-ncont}, the CW-topology on  $C(M^{2n},P^k)$ does not reflect the topology of $M^{2n}$.  Therefore,  if we want to use the complex  $C(M^{2n},P^k)$ as a  tool for the description of  the topology of the orbit space $M^{2n}/T^k$, the CW-topology  on  $C(M^{2n},P^k)$ is not an appropriate one.

 We define the topology on  the complex $C(M^{2n},P^k)$    such that $CQ(M^{2n},P^k)= C(M^{2n},P^k)$  becomes the  quotient space of $M^{2n}$ by the canonical map $f : M^{2n}\to C(M^{2n}, P^k)$.  More precisely,  $U\subseteq C(M^{2n},P^k)$  is open if and only if $f^{-1} (U)\subseteq M^{2n}$ is open.

Note that this is equivalent to say that  $CQ(M^{2n},P^k)$  is the quotient space of $M^{2n}/T^k$ by the canonical map
map $\widehat{f} : M^{2n}/T^k\to C(M^{2n}, P^k)$.

We immediately have:
\begin{lem}
The canonical maps $\widehat{\pi} : C(M^{2n}, P^{k})\to P^{k}$,  $f: M^{2n}\to C(M^{2n}, P^{k})$ and  $\widehat{f} : M^{2n}/T^k \to C(M^{2n}, P^k)$ are continuous maps.
\end{lem}

\begin{itemize}
\item Axiom~\ref{leaf} implies that the face  of any polytope is $\stackrel{\circ}{P}\in CQ(M^{2n}, P^{k})$ is contained in its boundary related to the quotient  topology.
\item Note also that $\stackrel{\circ}{P^k}$ is everywhere dense set in  $CQ(M^{2n}, P^k)$ since $f^{-1}( \stackrel{\circ}{P^k}) = W$, that is  the main stratum  which is everywhere  dense in $M^{2n}$.
\end{itemize}

Recall that a topological space $X$ is said to be $T_{0}$-topological space if for any two points in $X$ there exists an open set containing one of these two points but not containing the other, while $X$ is said to be $T_{1}$-topological space  if for any two points    $x,y\in  X$ there is an open set containing $x$ but not $y$ and there is an open set containing $y$ but not $x$. A topological space  $X$ is said to be Hausdorff if for any two points in $X$ there are an open disjoint sets which contain these points.

\begin{lem}\label{T0}
The space $CQ(M^{2n}, P^k)$ is $T_{0}$-topological space.
\end{lem}

\begin{lem}\label{HF}
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 Hausdorff topological space.
\end{lem}

\begin{lem}\label{T1}
If the set of admissible polytopes $P_{\mathfrak{S}}$ contains  a  polytope different from $P^{k}$ and its faces, then the space
$CQ(M^{2n}, P^k)$ is not $T_1$-topological space.
\end{lem}
\begin{proof}
Let $P_{\sigma}$ be   an  admissible polytope different from $P^{k}$ and its faces.  Then $\widehat{\pi}(\stackrel{\circ}{P_{\sigma}^{'}})\subset \stackrel{\circ}{P^{k}}$ and  for any point $x\in \stackrel{\circ}{P_{\sigma}}\subset CQ(M^{2n}, P^{k})$ there exists the point  $y\in \stackrel{\circ}{P^{k}}\subset CQ(M^{2n}, P^k)$   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   of all points from $W_{\sigma}/T^k$  which map to $\widehat{\pi}(x)$ by the map $\widehat{\mu}$.  On the other hand by~\eqref{param-cl} there exists a point $m\in \widehat{\mu}^{-1}(\pi (x))\cap W/T^k$ such that $m\in  \widehat{f}^{-1}(V)$. Note that $\widehat{f}(\widehat{\mu}^{-1}(\widehat{\pi}(x)\cap W/T^k) = y$ what implies that $y\in V$. Thus every neighborhood of $x$ in $CQ(M^{2n}, P^k)$ contains the point $y$.
\end{proof}
It implies:
\begin{cor}
If the set of admissible polytopes $P_{\mathfrak{S}}$ contains   a polytope  different from $P^{k}$ and its faces, then the space
$CQ(M^{2n}, P^k)$  is not  Hausdorff  topological space.
\end{cor}

As for the relations between these CW  and CQ topologies on   $C(M^{2n},P^k)$   we first note:

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

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

 As the direct consequence of Proposition~\ref{canon-ncont} we  also deduce the following:

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

Example~\ref{G42} demonstrates 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 CW topology is not contained in the CQ topology on   $C(M^{2n}, P^{k})$.

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

\subsubsection{Example of a closed set  in  $CQ(G_{4,2}, \Delta_{4,2})$ which is not closed in   $CW(G_{4,2}, \Delta_{4,2})$}

To provide such example we follow notations and methods from~\cite{BT-1}.
Consider in $G_{4,2}$ the set $\mathcal{C}$ given by
\[
\mathcal{C}=\left(\begin{array}{cc}
1 & 0\\
0 & 1\\
c & 1\\
1 & 1
\end{array}\right) ,\;\; c\neq 0,1.
\]
It belongs to the main stratum since all Pl\"ucker coordinates are non zero. The closure of this set we obtain when
$c\rightarrow 0,1, \infty.$
\begin{itemize}
\item When $c\rightarrow 0$ we obtain the point
\[
C_{0}=\left(\begin{array}{cc}
1 & 0\\
0 & 1\\
0 & 1\\
1 & 1
\end{array}\right).
\]
\item When $c\rightarrow 1$ we obtain the point
\[
C_{1}=\left(\begin{array}{cc}
1 & 0\\
0 & 1\\
1 & 1\\
1 & 1
\end{array}\right) .
\]
\item When $c\rightarrow \infty$ to see which point we obtain we can proceed as follows. Since the set $\mathcal{C}$ belongs to all charts we write down its expression in the chart $M_{23}$. The points in the chart $M_{12}$ and $M_{23}$ are given by:
\[
\left(\begin{array}{cc}
1 & 0\\
0 & 1\\
z_1 & z_3\\
z_2 & z_4
\end{array}\right)  \;\; \text{and}\;\;
\left(\begin{array}{cc}
w_1 & w_3\\
1 & 0\\
0 & 1\\
w_2 & w_4
\end{array}\right) ,
\]
what implies that the coordinates $(z_1,z_2,z_3,z_4)$ and $(w_1, w_2,w_3, w_4)$ in these charts are related by
\[
w_1= -\frac{z_3}{z_1},\; w_2 = z_4-\frac{z_2z_3}{z_1},\; w_3 = \frac{1}{z_1},\; w_4=\frac{z_2}{z_1}.
\]
It implies that the set $\mathcal{C}$ writes in the chart $M_{23}$ as
\[
\mathcal{C}=\left(\begin{array}{cc}
-\frac{1}{c} & \frac{1}{c}\\
1 & 0\\
0 & 1\\
1-\frac{1}{c} & \frac{1}{c}
\end{array}\right) , \;\; c\neq 0,1.
\]
Thus,  when $c\rightarrow \infty$ we obtain the point
\[
C_{\infty}=\left(\begin{array}{cc}
0 & 0\\
1 & 0\\
0 & 1\\
1 & 0
\end{array}\right).
\]
\end{itemize}

Consider the closure $\overline{\mathcal{C}} = \mathcal{C}\cup \{C_{0}, C_{1}, C_{\infty}\}.$  Its image in the complex $C(G_{4,2}, \Delta _{4,2})$ under the canonical map $f : G_{4,2}\to C(G_{4,2}, \Delta _{4,2})$ will be as follows:
\begin{itemize}
\item $f(\mathcal{C})\subset \stackrel{\circ}{\Delta _{4,2}}$ since $\mathcal{C}$ belongs to the main stratum.
\item $f(C_{0})\in \stackrel{\circ}{P_{23}}$, where $P_{23}$ is a four sided pyramid which does not contain the vertex $P_{23}$.
\item $f(C_{1})\in \stackrel{\circ}{P_{34}}$, where $P_{34}$ is a four sided pyramid which does not contain the vertex $P_{34}$.
\item $f(C_{\infty})\in (\delta _{23}, \delta _{34})$, where $(\delta _{23}, \delta _{34})$ is an edge with the vertices $\delta _{23}$ and $\delta _{34}$.
\end{itemize}

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

On the other hand, $f(\overline{\mathcal{C}})$   is obviously  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 we have the an open in one topology which is not open in the other.

Note that in general, when the operator $\tilde{d}$ is defined on the set of strata   $W_{\mathfrak{S}}$, it can be also defined  the 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 coming from $M^{2n}$ since the underlying space $W_{\mathfrak{S}}$ is  $M^{2n}$.

\subsection{Induced partial ordering on $CQ(M^{2n}, P^k)$}
There is the canonical way~\cite{E} to introduce the preorder on  $CQ(M^{2n}, P^{k})$ using the quotient topology. More precisely
for $x,y\in CQ(M^{2n}. P^{k})$ one defines
\[
x\leq y \; \text{iff}\; x\in \overline{y}.
\]
where $\overline{y}$ denotes the closure of $y$ in the quotient topology.  This preorder is even partial  order since $CQ(M^{2n}, P^{k})$ is $T_{0}$ space according to Lemma~\ref{T0}.  It is often called  specialization preorder. Note that in general on $T_{1}$-spaces 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 we have:
\[
 x\leq y \rightarrow \widehat{\pi}(x) = \widehat{\pi}(y).
\]

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

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

Recall that in the 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 standard fact that related to the specialization preorder on a topological space $X$ every open set in $X$  is an upper set  and every closed set is a lower set.  Recall also that a topological space $X$  is said to be 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  related to the specialization preorder is an open set.

As for the space $CQ(M^{2n}, P^k)$ we prove :

\begin{lem}
The space $CQ(M^{2n}, P^k)$ is an   Alexandrov space if and only if  the set of admissible polytopes $P_{\mathfrak{S}}$ does not  contain  a  polytope different from $P^{k}$ and its faces, that is if and only if $CQ(M^{2n}, P^k)$ is Hausdorff.
\end{lem}

\begin{proof} If  the set of admissible polytopes $P_{\mathfrak{S}}$ does not  contain  a  polytope different from $P^{k}$ and its faces, it is straightforward to see that  $CQ(M^{2n}, P^k)$ is an   Alexandrov space. Now assume that   $P_{\mathfrak{S}}$  contains a polytope $P_{\sigma}$ different from $P^k$ and its faces.  Then there exists a face $P_{\bar{\sigma}}$ of $P_{\sigma}$ such that $\stackrel{\circ}{P_{\bar{\sigma}}} \subset \stackrel{\circ}{P^k}$.  Let $V\subset CQ(M^{2n}, P^k)$  be the smallest,  under inclusion,  upper set which contains $\stackrel{\circ}{P^k}$ and   $\stackrel{\circ}{P_{\bar{\sigma}}}$. Then $\stackrel{\circ}{P_{\sigma}}\not\subset V$ since there is no $x\in \stackrel{\circ}{P_{\bar{\sigma}}}$ such that $x\leq y$ for some $y\in \stackrel{\circ}{P_{\sigma}}$ 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 $\stackrel{\circ}{P_{\bar{\sigma}}}$  but it does not contain the stratum
over  $\stackrel{\circ}{P_{\sigma}}$, while  from Axiom~\ref{leaf} it follows  that there are the points in the stratum  over $\stackrel{\circ}{P_{\bar{\sigma}}}$,  which are in closure of the stratum  over $\stackrel{\circ}{P_{\sigma}}$.
\end{proof}

\begin{ex}
For the sake of clearness we provide  the  explicit  example of the 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 $\stackrel{\circ}{G_{4,2}}$ and $\stackrel{\circ}{P_{12, 34}}$ where $P_{12,34}$ is a square in $\Delta _{4,2}$  which does not contain the vertices $\delta _{12}$ and $\delta _{34}$, see~\cite{BT-1}.   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  four-dimensional $(\C ^{*})^3$-orbit which maps to the square $\stackrel{\circ}{P_{12,34}}$. This orbit is, on the other hand,  contained in the closure of two six-dimensional $(\C ^{*})^3$-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 $G_{4,2}$, what 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 $x,y\in V$ is $x\leq y$ it implies that $x\in \stackrel{\circ}{P_{12,34}}$ and $y\in \stackrel{\circ}{\Delta _{4,2}}$ and $\widehat{\pi}(x) =\widehat{\pi}(y)$.
\end{ex}

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

Using the results from the previous sections we can define the set  $\mathfrak{E}(M^{2n}, P^{k})$ over the complex $C(M^{2n}, P^{k})$ by:
\begin{equation}
\mathfrak{E}(M^{2n}, P^{k}) = \{(x,y)\in C(M^{2n}, P^{k})\times M^{2n}  : x\in \stackrel{\circ}{P}_{\sigma}^{'}, y\in W_{\sigma},  \widehat{\pi} (x)=\mu (y)\}.
\end{equation}

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

Proposition~\ref{point}  immediately implies:
\begin{cor}
 If $F$ is a point then $\mathfrak{E}(M, P^{k})/T^k = CQ(M, P^{k})$ and $CQ(M, P^{k})$ is homeomorphic to $P^{k}$.
 \end{cor}

\begin{lem}\label{E-CH}
The space $\mathfrak{E} (M^{2n}, P^{k})$ is a compact  Hausdorff topological space.
\end{lem}

\begin{proof} Since $\mathfrak{E} (M^{2n}, P^{k})\subset CQ(M^{2n}, P^k)\times M^{2n}$ and the manifold $M^{2n}$ is a Hausdorff topological space it implies that $\mathfrak{E}(M^{2n}, P^{k})/T^k$ is Hausdorff space as well. As for the compactness  consider the covering $E_{i}, i\in I$ of   $\mathfrak{E}(M^{2n}, P^{k})$ by an open sets. Then $E_i = (U_i\times V_i) \cap \mathfrak{E}(M^{2n}, P^{k})$ 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 $y\in M^{2n}$ there exists $x\in CQ(M^{2n}, P^{k})$ such that $(x,y)\in \mathfrak{E}(M^{2n}, P^{k})$, what implies that $V_{i}, i\in I$ is an open covering for $M^{2n}$.  In addition,  for any $ x\in CQ(M^{2n}, P^{k})$,  there exists  $y\in M^{2n}$  such that $(x,y) \in \mathfrak{E}(M^{2n}, P^{k})$,  what 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 compact,  which  implies that the quotient space $CQ (M^{2n}, P^k)$ of $M^{2n}$  is 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 $M^{2n}/T^k$, what implies that $X_{ij} = (U_i\times V_j )\cap \mathfrak{E} (M^{2n}, P^{k})$ is the finite sub-covering of $E_{i}$, $i\in I$ for $\mathfrak{E}(M^{2n}, P^{k})$.
\end{proof}

There are two natural  projection maps
\begin{equation}\label{pr}
G :  \mathfrak{E} (M^{2n}, P^{k}) \to CQ(M^{2n}, P^k)\; \text{and} \; F :  \mathfrak{E} (M^{2n}, P^{k})\to M^{2n}.
\end{equation}

\begin{itemize}
\item  The  maps $G$ and $F$  are obviously surjective. For the map $G$ it follows from the observation that   for any $x\in P_{\sigma}^{'}\subset  CQ(M^{2n}, P^k)$ there exists
$y\in W_{\sigma}\subset M^{2n}$,  such that $\widehat{\pi}(x) = \mu (y)$ since $\mu (W_{\sigma}) = \stackrel{\circ}{P_{\sigma}} = \widehat{\pi}(P_{\sigma}^{'})$, what implies that $(x,y)\in  \mathfrak{E}(M^{2n}, P^{k})$. As for the surjectivity of the map $F$, for any $y\in W_{\sigma}\subset  M^{2n}$ we see that $(x,y)\in  \mathfrak{E}(M^{2n}, P^{k})$, where $x\in P_{\sigma}^{'}$ such that $\widehat{\pi}(x) = \mu (y)$.
\item The  maps $G$ and $F$ are obviously continuous. Namely $G^{-1}(U) =( U\times M^{2n})\cap  \mathfrak{E}(M^{2n}, P^{k})$  and $F^{-1}(V) = (CQ(M^{2n}, P^k) \times V) \cap  \mathfrak{E}(M^{2n}, P^{k})$ , which  are open sets in  $\mathfrak{E} (M^{2n}, P^{k})$  for an open sets $U\subseteq CQ(M^{2n}, P^k)$ and $V\subseteq M^{2n}$.
\end{itemize}

\begin{lem}
The map $F$ is injective, while the map $G$ is not injective.
\end{lem}
\begin{proof} The map  $F$ is injective since $F(x_1,y)=F(x_2,y)$ implies $\widehat{\pi} (x_1) = \widehat{\pi} (x_2) = \widehat{\mu}(y)$ and $x_1,x_2\in \stackrel{\circ}{P_{\sigma}^{'}}$. But, $\widehat{\pi} : P_{\sigma}^{'}\to \stackrel{\circ}{P_{\sigma}}$ is a homeomorphism,  which  implies $x_1=x_2$. The map  $G$ is not injective since  for any $y_1,y_2$ from the same non-trivial $T^k$-
orbit in some $W_{\sigma}$,   we have that $(x,y_1), (x,y_2)\in   \mathfrak{E}(M^{2n}, P^{k})$, where $x\in\stackrel{\circ}{ P_{\sigma}^{'}}$,  such that $\widehat{\pi}(x) = \mu (y_1) = \mu (y_2)$. It implies that $G(x, y_1)=G(x,y_2)=x$.
\end{proof}

All togetherthis  leads to the following important result:

\begin{thm}\label{ME}
The  space $\mathfrak{E}(M^{2n}, P^{k})$ is homeomorphic to the space $M^{2n}$.
The homeomorphism  $F: \mathfrak{E}(M^{2n}, P^{k}) \to M^{2n}$ is given by $F(x,y)=y$.
\end{thm}

\begin{proof}
The map  $F: \mathfrak{E}(M^{2n}, P^{k}) \to M^{2n}$ is a continuous bijection. Since $\mathfrak{E}(M^{2n}, P^{k})$ is compact  and $M^{2n}$ is Hausdorff, the  elementary topology argumenst lead that $F$ is a  homeomorphism.
\end{proof}

We define the action of torus $T^{k}$ on $CQ(M^{n}, P^{k})\times M^{2n}$ by its given action on $M^{2n}$. Since the strata  as well as the almost moment map $\mu$ are invariant under torus  action,  this action  induces the action of $T^k$ on $\mathfrak{E}(M^{2n}, P^{k})$. We obtain that
\[
\mathfrak{E}(M^{2n}, P^{k})/T^k = \{(x,y)\in CQ(M^{n}, P^{k})\times M^{2n}/T^k  :  x\in \stackrel{\circ}{P_{\sigma}^{'}}, y\in W_{\sigma}/T^k,\widehat{\pi} (x)=  \widehat{\mu} (y)\}.
\]
\begin{rem}
Since $W_{\sigma}/T^k\cong \stackrel{\circ}{P}_{\sigma}\times F_{\sigma}$ and $\stackrel{\circ}{P_{\sigma}^{'}}\cong \stackrel{\circ}{P}_{\sigma}$, it follows that the elements from $\mathfrak{E}(M^{2n}, P^k)/T^k$ can be written as the pairs $(x, c_{\sigma})$, where $x\in \stackrel{\circ}{P}_{\sigma}^{'}$ and $c_{\sigma}\in F_{\sigma}$.
\end{rem}

From  Lemma~\ref{E-CH} it follows

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

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

The $T^k$-equaivariance of the map $\widehat{F}$ together with Theorem~\ref{ME} implies:

\begin{thm}
The orbit space $M^{2n}/T^k$ is homeomorphic to the  space $\mathfrak{E}(M^{2n}, P^{k})/T^k$ by the map $\widehat{F}: \mathfrak{E}(M^{2n}, P^{k})/T^k \to M^{2n}/T^k$  given by $\widehat{F}(x,y)=y$.
\end{thm}

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

\begin{prop}\label{cfblocal}
$\widehat{G}^{-1}(P_{\sigma}^{'})$ is homeomorphic to   $P_{\sigma}^{'}\times F_{\sigma}$,  that is to $\stackrel{\circ}{P_{\sigma}}\times F_{\sigma}$ for any $\sigma \in \mathfrak{S}$.
\end{prop}
\begin{proof}
For any point  $x\in P_{\sigma}^{'}\in CQ(M^{n}, P^{k})$ we have that $\widehat{G}^{-1}(x) = \{ (x, y) | y \in W_{\sigma}/T^k, \widehat{\mu} (y) = \pi (x)\}$. It follows from Corollary~\ref{homeom} that $\widehat{G}^{-1}(x)$ is homeomorphic to $F_{\sigma}$ what together with Corollary~\ref{homeom2} implies that $\widehat{G}^{-1}(P_{\sigma}^{'})$ is homeomorphic to $\stackrel{\circ}{P_{\sigma}}\times F_{\sigma}$.
\end{proof}

\begin{rem}
We want to emphasize that Proposition~\ref{cfblocal} proves that the space $\mathfrak{E}(M^{2n}, P^{k})/T^k$,  that is the orbit space $M^{2n}/T^k$, is   the  collection of trivial fiber bundles $\stackrel{\circ}{P_{\sigma}}\times F_{\sigma}$ for any $\sigma \in \mathfrak{S}$. Then,  further,  according to Axiom~\ref{leaf}  the gluing of the fibers for  $\stackrel{\circ}{P_{\sigma}}$ and the fibers for $\stackrel{\circ}{P_{\overline{\sigma}}} \subset \partial \stackrel{\circ}{P_{\sigma}}$ is given by the map $\eta _{\sigma, \overline{\sigma}} : F_{\sigma}\to F_{\overline{\sigma}}$.
\end{rem}

\section{The gluing of the  orbit spaces of the strata  in $\mathfrak{E}(M^{2n}, P^k)/T^k$}
We proved in the previous Section that the space  $\mathfrak{E}(M^{2n}, P^{k})/T^k$, that is the orbit space $M^{2n}/T^k$, is  a collection of trivial fiber bundles $\stackrel{\circ}{P_{\sigma}}\times F_{\sigma}$, where  $\sigma \in \mathfrak{S}$.  We want to describe how  the trivial bundles $\stackrel{\circ}{P_{\sigma}}\times F_{\sigma}$  are gluing together, or,  in other words,   to describe the  boundary of $\stackrel{\circ}{P_{\sigma}}\times F_{\sigma}$. Recall that the boundary of $\stackrel{\circ}{P_{\sigma}}\times F_{\sigma}$ is homeomorphic to the boundary of the orbit space $W_{\sigma}/T^k$.
Lemma~\ref{bound} immediately implies:
\begin{cor}
$\partial(\stackrel{\circ}{P_{\sigma}}\times F_{\sigma})\subset \cup _{\tilde{\sigma}\subset \sigma}\stackrel{\circ}{P_{\tilde{\sigma}}}\times F_{\tilde{\sigma}}$, where $\tilde{\sigma}$ runs through all admissible subsets of $\sigma$.
\end{cor}

Note that, as we demonstrated in Section~\ref{Grassmann},
 the boundary of $W_{\sigma}/T^k$ can not be, in general, represented as the union of the orbit spaces of some other strata. This implies that, in general, the boundary of   $\stackrel{\circ}{P_{\sigma}}\times F_{\sigma}$ in  $\mathfrak{E}(M^{2n}, P^{k})/T^k$ can not be represented as the union of some other trivial fiber bundles.

As for the boundary of the polytopes in $CQ(M^{2n}, P^k)$ we have the following:
\begin{lem}\label{subset}
If  $ P_{\tilde{\sigma}}^{'}\subset \partial P_{\sigma}^{'}$ for some any $P_{\sigma}^{'},  P_{\tilde{\sigma}}^{'} \subset CQ(M^{2n}, P^k)$  then  $\tilde{\sigma}\subset \sigma$.
\end{lem}
\begin{proof}
If  $P_{\tilde{\sigma}}^{'} \subset \partial P_{\sigma}^{'}$  it follows that  $f^{-1}( P_{\tilde{\sigma}}^{'})\subset f^{-1}(\partial P_{\sigma}^{'})$, where $f: M^{2n}\to CQ(M^{2n},P^k)$ is a quotient map. It implies that $f^{-1}(\partial P_{\sigma}^{'}) = \partial f^{-1}( P_{\sigma}^{'}) = \partial W_{\sigma}\subseteq \cup _{\tilde{\sigma}\subset \sigma}W_{\tilde{\sigma}}$ and, thus,  $f^{-1}( P_{\tilde{\sigma}}^{'}) = W_{\tilde{\sigma}}$ for some $\tilde{\sigma}\subset \sigma$.
\end{proof}

\begin{rem}
We want to recall that we already remarked that for any $P_{\bar{\sigma}}^{'}, P_{\sigma}^{'}\subset CQ(M^{2n}, P^{k})$  such that $P_{\bar{\sigma}}$ is a face of $P_{\sigma}$, we have that  $P_{\bar{\sigma}}^{'} \subset \partial  P_{\sigma}^{'}$ in $CQ(M^{2n}, P^{k})$.
\end{rem}

We  derive some consequences of the additional assumption that
if  $\partial P_{\sigma}^{'}\cap P_{\tilde{\sigma}}^{'}\neq \emptyset$ then  $P_{\tilde{\sigma}}^{'}\subset \partial P_{\sigma}^{'}$ for  any $P_{\tilde{\sigma}}^{'}, P_{\sigma}^{'}\subset CQ(M^{2n}, P^k)$, which we will assume to  the end of this Section.

Then  Lemma~\ref{subset} implies:
\begin{cor}
The  boundary of any  $ P_{\sigma}^{'}\subset CQ(M^{2n}, P^k)$ is the union of some  $P_{\tilde{\sigma}}^{'}$ such that $\tilde{\sigma}\subset \sigma$.
\end{cor}

We also have the following direct consequence:

\begin{cor}
For any $x\in P_{\sigma}^{'}$ we have that  $\overline{x} = \{y \in  P_{\tilde{\sigma}}^{'} \; | \:   P_{\tilde{\sigma}}^{'}\subset \partial P_{\sigma}^{'},\; \widehat{\pi}(y) =\widehat{\pi}(x)\}$.
\end{cor}

As for the closures of the spaces of parameters $F_{\sigma}$  for  $\stackrel{\circ}{P_{\sigma}^{'}}$ we obtain:

\begin{lem}
For any $x\in P_{\sigma}^{'}$ we have that
\[
\overline{x\times F_{\sigma}}  = \overline{x}\times \overline{F_{\sigma}}\subseteq \cup _{y\in \overline{x}}(y\times F_{\tilde{\sigma}}).
\]
\end{lem}
\begin{proof}
Let  $(y, c) \in \overline{x\times F_{\sigma}}$. . Then  $\widehat{F}(y,c) = a \in M^{2n}/T^k$ such that $a\in
\overline{W_{\sigma}}/T^k$ and $\widehat{f}(a) = y \in P_{\tilde{\sigma}}^{'} $, where  $P_{\tilde{\sigma}}^{'}\subset \partial P_{\sigma}^{'}$ and $y\in \overline{x}$.
\end{proof}

\begin{prop}\label{boundtr}
$\partial (\stackrel{\circ}{P_{\sigma}}\times F_{\sigma})\subset \cup \stackrel{\circ}{P_{\tilde{\sigma}}}\times F_{\tilde{\sigma}}$,
where $\tilde{\sigma}$ runs through all admissible sets such that  $P_{\tilde{\sigma}}^{'}\subset \partial P_{\sigma}^{'}$
\end{prop}
\begin{proof}
Let, as before, $\widehat{f} : M^{2n}/T^k\to CQ(M^{2n}, P^k)$ be a quotient map and let a point $a\in  \mathfrak{E}(M^{2n}, P^{k})/T^k$ belongs to the boundary of  $\stackrel{\circ}{P_{\sigma}}\times F_{\sigma}$. It means that a point $b=\widehat{F}^{-1}(a)$ belongs to the boundary of $W_{\sigma}/T^k$. It implies that $\widehat{f}(b)$ belongs to the boundary of
$P_{\sigma}^{'}$ in $CQ(M^{2n}, P^k)$.
\end{proof}

We can say more, that is which points form the union $\cup \stackrel{\circ}{P_{\tilde{\sigma}}}\times F_{\tilde{\sigma}}$ given by Proposition~\ref{boundtr} are for sure contained in the boundary of  $\stackrel{\circ}{P_{\sigma}}\times F_{\sigma}$.

Let $\bar{\sigma}\subset \sigma$ such that $P_{\bar{\sigma}}^{'}$ is a face of $P_{\sigma}^{'}$ and let $\eta _{\sigma, \bar{\sigma}} : F_{\sigma} \to F_{\bar{\sigma}}$ is a map introduced by~\eqref{eta}.  Denote by $F_{\sigma, \bar{\sigma}} = \eta _{\sigma, \bar{\sigma}}(F_{\sigma})$.

\begin{lem}
$\stackrel{\circ}{P_{\bar{\sigma}}}\times F_{\sigma, \bar{\sigma}}\subset \partial (\stackrel{\circ}{P_{\sigma}}\times F_{\sigma})$ for any $\bar{\sigma}\subset \sigma$ such that $P_{\bar{\sigma}}^{'}$ is a face of $P_{\sigma}^{'}$.
\end{lem}
\begin{proof}
If  $P_{\bar{\sigma}}\times  c_{\bar{\sigma}}\subset  \stackrel{\circ}{P_{\bar{\sigma}}}\times F_{\sigma, \bar{\sigma}}$ then there exists $c_{\sigma}\in F_{\sigma}$ such that $\eta _{\sigma, \bar{\sigma}}(c_{\sigma}) = c_{\bar{\sigma}}$, meaning that the boundary of the leaf    $W_ {[\xi_{\sigma}, c_{\sigma}]}$  contains the leaf $W_ {[\xi_{\bar{\sigma}}, c_{\bar{\sigma}}]}$. It implies that  $W_ {[\xi_{\bar{\sigma}}, c_{\bar{\sigma}}]}/T^k \subset \partial W_ {[\xi_{\sigma}, c_{\sigma}]}/T^k$ what further implies  that
$\stackrel{\circ}{P_{\bar{\sigma}}}\times c_{\bar{\sigma}}\subset \partial (\stackrel{\circ}{ P_{\sigma}}\times c_{\sigma})$.
\end{proof}

As for the points from $\stackrel{\circ}{P_{\tilde{\sigma}}}\times F_{\tilde{\sigma}}$, where $P_{\tilde{\sigma}}^{'}\subset \partial P_{\sigma}^{'}$
we prove the following:
\begin{lem}
 Let $y\in P_{\tilde{\sigma}}^{'}$ where $P_{\tilde{\sigma}}^{'}\subset \partial P_{\sigma}^{'}$. Then there exists $c_{\tilde{\sigma}}\in F_{\tilde{\sigma}}$ such that $(y, c_{\tilde{\sigma}})$ belongs to the boundary of $\stackrel{\circ}{P_{\sigma}}\times F_{\sigma}$.
\end{lem}
\begin{proof}
If  $P_{\tilde{\sigma}}^{'}\subset \partial P_{\sigma}^{'}$ then $\widehat{f}^{-1}(P_{\tilde{\sigma}}^{'}) = \cap \partial \widehat{f}^{-1}( P_{\sigma}^{'}) = \partial  W_{\sigma}/T^k)\neq \emptyset$.  It implies that  $W_{\tilde{\sigma}}/T^k \cap = \partial  W_{\sigma}/T^k \neq \emptyset$ and moreover that $P_{\tilde{\sigma}}^{'}\subset \widehat{f} (W_{\tilde{\sigma}}/T^k \cap = \partial  W_{\sigma}/T^k)$, what means that for any $y\in P_{\tilde{\sigma}}^{'}$ there exists $c_{\tilde{\sigma}}\in F_{\tilde{\sigma}}$ such that $(y, c_{\tilde{\sigma}})\in \partial (\stackrel{\circ}{P_{\sigma}}\times F_{\sigma})$.
\end{proof}

 \section{Complex Grassmann manifolds $G_{k+1,q}$ as $(2q(k-q+1),k)$-manifolds}\label{gr}

The complex Grassmann manifold $G_{k+1,q}$ consists of  $q$-dimensional complex subspaces in complex vector space $\C^ {k+1}$. The canonical action of $T^{k+1}$ on $\C ^{k+1}$ induces the action of $T^{k+1}$ on $G_{k+1,q}$.
This action is not effective, namely the diagonal circle $H =\{(t,\ldots ,t), t\in S^1\}$ acts trivially on $G_{k+1,q}$. Therefore 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. Fix the basis in $\C ^{k+1}$.  Then any $q$-dimensional subspace $L$ in $\C ^{k+1}$, up to choice of a basis in this subspace $L$, can be    represented by $(k+1)\times q$ matrix $A$ and $rang A=l$. For any subset $J\subset \{1,\ldots ,k+1\}$ consisting of $q$ elements, $|J|=q$, denote by $A_{J}$ the matrix of dimension  $q\times q$ given by the rows of  the matrix $A$ which are indexed by $J$. Note 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 there are  defined the numbers
\[
P(A)= (P^{J}(A)) = (\det A_{J}),
\]
which are called the Pl\"ucker coordinates for $G_{k+1,q}$.

The Pl\"ucker coordinates depend on the fixed base for $L$ and they are, up to constant, uniquely defined. More precisely given two bases $f_1,\ldots, f_q$ and $e_1,\ldots ,e_q$  for $L$ and $f_j=\sum _{i=1}^{q}\alpha _i^je_i$, $1\leq j\leq q$ then the Pl\"ucker coordinates for $L$ in these two bases are related with
\begin{equation}\label{plucker}
P^{J}_{f}(L) = (\sum_{\sigma \in S_k}\sg (\sigma) \alpha _{\sigma (1)}^1\cdots \alpha _{\sigma (k)}^k)P^{J}_{e}(L).
\end{equation}
In this way Pl\"ucker coordinates give the embedding of the Grassmann manifold $G_{k+1,q}$ into $\C P^{{k+1\choose q}-1}$.

The standard moment map on $\C P^{{k+1\choose q}-1}$  gives the moment map $\mu : G_{k+1,q} \to \R ^{k+1}$, see~\cite{Kir}, which is defined by
\begin{equation}\label{gr-moment}
\mu (L) = \frac{\sum _{J}|P^{J}(L)|^2\delta _{J}}{\sum _{J}|P^{J}(L)|^2},
\end{equation}
where $\delta _{J}\in \R ^{k+1}$ is given by
\[
(\delta _{J})_{i}=1,\;\; i\in J,\;\; (\delta _{J})_{i}=0,\;\;  i\notin J.
\]
and $J$ runs through the $q$-element  subsets of ${1,\ldots, k+1}$.

The map $\mu$ is $T^{k+1}$-invariant and it follows from its definition that its image is a convex hull over the points $\delta _{J}$.  The convex polytope obtained in this way is known as hypersimplex  $\Delta _{k+1,q}$, see~\cite{Z}.

Using the notion of the Pl\"ucker coordinates and the canonical action of $T^{k+1}$ on the Grassmann manifold $G_{k+1,q}$ we prove:

\begin{thm}
$G_{k+1,q}$ has the canonical structure of $(2q(k-q+1), k)$-manifold.
\end{thm}

The detailed proof that the Axioms 1 - 4  are satisfied can be found in~\cite{BTN}. We are just left to verify the fifth axiom.

{\bf Axiom~\ref{leaf}}. It follows from the definition of the leaf and the description of the strata for $G_{k+1,q}$ given in~\cite{BTN}, that any leaf $W_{\sigma}[\xi _{\sigma},c_{\sigma}]$ is the    $(\C ^{*})^{k+1}$-orbit of a point from $W_{\sigma}$. On the other hand by the result of~\cite{AT} the boundary of any $(\C ^{*})^{k+1}$ -orbit in $G_{k+1,q}$ consists of $(\C ^{*})^{k+1}$-orbits of the smaller dimensions and the moment map gives the bijection between these orbits and the faces of  $P_{\sigma}$. In this  way we verify that the Axiom~\ref{leaf} is satisfied for the Grassmann manifolds $G_{k+1,q}$ as well.

\begin{rem} The strata $W_{\sigma}$ coincides with the matroid strata defined in~\cite{GGMS}.  Moreover,  in the papers~\cite{GS} and~\cite{GGMS} see also~\cite{BT-1} for the summary,  there are defined three more  stratifications of the Grassmann manifolds $G_{k+1,q}$: by "soft Schubert cells", by the moment map and  by the arrangements
of planes, and it is showed that all these four stratifications are  equivalent.
\end{rem}
\begin{rem}
The closure of the strata  $W_{\sigma}$ on Grassmann manifolds are known in the literature as matroid varieties. These varieties do not behave nicely related to the matroids assigned to them. In that context these varieties were recently studied in~\cite{FNR},~\cite{F}.
\end{rem}

As an application of the results obtained in~\ref{main-sp-orb}, let us consider the Grassmann manifold $G_{4,2}$ of complex $2$-planes in $\C ^4$ as an example of $(8,3)$-manifold. The canonical action of the compact torus $T^4$, which induces effective action of $T^3$  on $G_{4,2}$ is analyzed in detail in ~\cite{BT-1}.
In the notation of~\ref{main-sp-orb},   it is proved in~\cite{BT-1} that the set of parameters of the main stratum   for $G_{4,2}$ is $F=\C -\{0,1\}$.  Moreover, from what was done on $G_{4,2}$ in~\cite{BT-1} it  directly follows that all points from the boundary of octahedron $\Delta _{4,2}$ are simple and that $\widehat{\mu}^{-1}(x)$ is homeomorphic to $\C P^1$ for any $x\in \Delta _{4,2}$. Then Theorem~\ref{join} implies that $G_{4,2}/T^4$ is homeomorphic to $\C P^1\ast S^2$ as it is shown in~\cite{BT-1}.

In~\cite{BT-1} it is also studied in detail the action of the torus $T^4$ on $\C P^5$ given as the composition of the embedding of $T^4$ into $T^6$ by the second symmetric power and the canonical action of $T^6$ on $\C P^6$. This action is not effective, but it induces the effective action of $T^3$, the main admissible polytope is again octahedron and the main admissible set of parameters is $\{(c_1,c_2)\in \C ^{2} | c_1c_2\neq 0\}$. All points from the boundary of the octahedron are simple and  $\widehat{\mu}^{-1}(x)$ is homeomorphic to $\C P^2$ for any $x\in \Delta _{4,2}$.  Therefore Theorem~\ref{join} implies that $\C P^5/T^4$ is homeomorphic to $\C P^2\ast S^2$ which is also directly proved in~\cite{BT-1}.

\subsection{Gel'fand-Serganova example}~\label{Grassmann}
We use what we said about Grassmann manifolds to demonstrate that the  the inclusion $\partial W_{\sigma} \subseteq  \cup W_{\overline{\sigma}}$, where ${\overline{\sigma}}$ runs  through all  admissible  subsets of  $\sigma$ given,  by Lemma~\ref{bound} in Section~\ref{jedan},   is a strict inclusion in general. Moreover,  $\partial W_{\sigma}$ in general can not be obtained as the union of some strata . One can see that using  the example given in the paper of Gel'fand-Serganova~\cite{GS}. They provided  the example of the stratum $W_{\sigma}$  on the Grassmann manifold for which
$\partial W_{\sigma}$ is not the union of the strata.
\
Consider  the point $L\in G_{7,3}$ given by a matrix
\[
\left(\begin{array}{ccccccc}
1 & 0 & 0 & 0 & b_1 & c_1 & d_1\\
0 & 1 & 0 & a_2 & 0 & c_2 & d_2\\
0 & 0 & 1 & a_3 & b_3 & 0 & d_3
\end{array}\right),
\]
such that
\begin{equation}\label{cond}
d_1d_2d_3\neq 0, \;\; d_1c_2=d_2c_1,\;\; d_1b_3=d_3b_1, \;\; d_2a_3=a_2d_3.
\end{equation}
It implies that
\[ a_3b_1c_2=a_2b_3c_1.
\]
Since the charts of the Grassmann manifold are determined and enumerated by the non-zero Pl\"ucker coordinates,  it follows that  all points which satisfy~\eqref{cond} form  the stratum  $W$ obtained as the intersection of the sets $X_{ijk}$, $1\leq i<j<k\leq 7$, where $X_{ijk} = Y_{ijk}$ or $ijk = 126, 135, 234, 147, 257, 367, 456$ and $X_{ijk}=M_{ijk}$ for the others.

On the other hand the $(\C ^{*})^{6}$-orbit of such point is given by
\[
(0,\frac{t_5}{t_1}b_1, \frac{t_6}{t_1}c_1, \frac{t_7}{t_1}d_1, \frac{t_4}{t_2}a_2, 0,  \frac{t_6}{t_2}c_2, \frac{t_7}{t_2}d_2, \frac{t_4}{t_3}a_3, \frac{t_5}{t_3}b_3, 0, \frac{t_7}{t_3}d_3).
\]
It can be further written as
\begin{equation}\label{orb}
(0, \tau _{1}b_1, \tau _{2}c_1, \tau _{3}d_1, \tau _{4}a_2, 0, \tau _{5}c_2, \frac{\tau _{3}\tau _{5}}{\tau _{2}}d_2, \tau _{6}a_3, \frac{\tau _{1}\tau_{5}\tau _{6}}{\tau _{2}\tau _{4}}b_3, 0, \frac{\tau _{3}\tau _{5}\tau _{6}}{\tau _{2}\tau _{4}}d_3).
\end{equation}
Therefore all points of the given form  belong to the one $(\C ^{*})^{6}$-orbit. It implies that the stratum $W$ consists of the one  $(\C ^{*})^{6}$-orbit.

Consider the point $L_1\in G_{7,3}$ given by the matrix
\begin{equation}\label{point}
\left(\begin{array}{ccccccc}
1 & 0 & 0 & 0 & b_1 & c_1 & 0\\
0 & 1 & 0 & a_2 & 0 & c_2 & 0\\
0 & 0 & 1 & a_3 & b_3 & 0 & 0
\end{array}\right),
\end{equation}
such that $a_3b_1c_2=a_2b_3c_1$.  Let   $W^{'}$ be the stratum  which contains this point. It is obtained as the intersection of the sets $X_{ijk}$ where $X_{ijk}= M_{ijk}$ for $ijk =123, 124, 125$, $135, 136, 235,236, 456$ and $X_{ijk}=Y_{ijk}$ for the others. It implies that the stratum  $W^{'}$ consists of the points that can be represented by the matrices
\[
\left(\begin{array}{ccccccc}
1 & 0 & 0 & 0 & b_1 & c_1 & 0\\
0 & 1 & 0 & a_2 & 0 & c_2 & 0\\
0 & 0 & 1 & a_3 & b_3 & 0 & 0
\end{array}\right),
\]
such that $a_3b_1c_2\neq - a_2b_3c_1$.

On the one hand,  it is obvious that the point $X$ as well as its $(\C ^{*})^{6}$-orbit,  which is given by
$(0, \tau _{1}b_1, \tau _{2}c_1, 0, \tau _{3}a_2, 0, \tau _{4}c_2, 0, \tau _{5}a_3, \frac{\tau _{1}\tau_{4}\tau _{5}}{\tau _{2}\tau _{3}}b_3, 0, 0)$,  belong to the boundary of the stratum  $W$.

On the other hand,  the stratum  $W^{'}$ does not coincide with  $(\C ^{*})^{6}$-orbit of the point $X$. Namely, this  $(\C ^{*})^{6}$-orbit is contained in $W^{'}$,  but its points  satisfy $a_3b_1c_2= a_2b_3c_1$, while the points from $W^{'}$ are defined by the  weaker relation  $a_3b_1c_2\neq - a_2b_3c_1$.

It implies that
\[
\partial W \cap W^{'}\neq \emptyset \;\; \text{and}\;\; W^{'} \not\subset \partial W .
\]

\section{$(2n,1)$-manifolds}

We first observe the following:
\begin{prop}
Any $(2n, 1)$-manifold is homeomorphic to the standard sphere $S^{2n}$
\end{prop}
\begin{proof}
For a $(2n, 1)$ manifold $M^{2n}$ by the definition there exists an almost moment map $\mu : M^{2n} \to [-1,1]$ which is $S^1$-invariant. Then  Axiom 2 implies that $S^1$ action on $M^{2n}$ has exactly two fixed points $A_1$, $A_2$ and Axiom 1 implies that $M^{2n}$ has an atlas consisting of two charts $(M_1,u_1), (M_{2},u_2)$ each of them containing exactly one fixed point. By Axiom~\ref{stab} and Axiom~ \ref{bij}, we have that $A_1 = W_1=M_1\cap Y_2$ and $A_2= W_2= Y_1\cap M_2$, where $Y_i=M^{2n}\setminus M_i$, $i=1,2$. It implies that  $M_1 = M^{2n}-A_2$ and $M_2=M^{2n}-A_1$. Since $u_i : M_{i}\to \R ^{2n}$ are homeomorphisms, we obtain that $M^{2n}$ is the one-point compactification of $\R ^{2n}$ and therefore it is homeomorphic to $S^{2n}$.
\end{proof}

The vice versa is true as well:
\begin{thm}
The standard sphere $S^{2n}$ has the structures of $(2n, 1)$-manifold for any $n$.
\end{thm}
\begin{proof}
Represent  the sphere $S^{2n}$ as the hypersurface
\[
 |z_1|^2 +\cdots + |z_n|^2 +r^2= 1\; \text{in}\; \R ^{2n+1}\approx \C ^{n}\times \R
\]
and consider  the action of the circle $S^1$ on $S^{2n}$  defined by
\[
t(z_1,\ldots ,z_n, r) = (t^{\epsilon _1}z_1,\ldots ,t^{\epsilon _{n}}z_n, r)\;\; \text{where}\;\; \epsilon _{k}=\pm 1.
\]
The fixed point for this action are $A_{1} = (0,\ldots ,0,1)$ and $A_{-1}=(0,\ldots ,0,-1)$. The almost moment map we define   by
\[ \mu : S^{2n} \to P= [-1, 1],\;\;  \mu (z_1,\ldots ,z_n,r)=r. \]
It is straightforward to see that the Axiom~\ref{bij} is satisfied.

Consider the  atlas consisting of two charts  $(M_1,u_1), (M_{-1},u_{-1})$, where
\[
M_{1}=\{ (z,r), r\neq 1\}\; \text{and} \;  u_{1}(z,r)= \frac{1}{1-r}z,
\]
\[
M_{-1}\{ (z,r), r\neq -1\}\to \C ^{n}\; \text{and}\;  u_{-1}(z,r)=\frac{1}{1+r}z.
\]
The charts $M_{1}$ and $M_{-1}$ are $S^1$-equivariant, each of them contains exactly one fixed point and $\overline{M_{1}}=\overline{M_{-1}}= S^{2n}$, so Axiom~\ref{atlas} is satisfied. The only non point stratum  is $W_{\{-1,1\}}=M_{1}\cap M_{-1} = \{(z,r)\in S^{2n} | r\neq -1,1\}$. The induced map $\widehat{\mu} : W_{\{-1,1\}}/S^1\to (-1,1)$ and
it is a locally trivial fiber bundle with the   fiber $\C P^{n-1}$, so Axiom~\ref{fiber} is satisfied and $W_{\{-1,1\}}/S^1$ is homeomorphic to $\C P^{n-1}\times (-1, 1)$.  The circle $S^1$ acts freely on $W_{\{-1,1\}}$,  so the projection $\pi : W_{\{-1,1\}}\to W_{\{-1,1\}}/S^1$ is a fibration. Therefore the leaf $W_{\{-1,1\}}[\xi, c]$, where $\xi : W_{\{-1,1\}}/S^1 \to \C P^{n-1}$ is a fixed trivialization and $c\in \C P^{n-1}$,   defined by~\eqref{leaf} is given as  $S^1\cdot c \times (-1,1)$. It  implies that its boundary consists of two fixed points $A_{1}$ and $A_{-1}$, which  are the leafs over the vertices of $P$. In this way we see that Axiom~\ref{leaf} is satisfied as well.
 \end{proof}
Since the only admissible polytopes for $P$ are $-1$, $1$ and $(-1,1)$, Theorem~\ref{join} directly applies:
\begin{thm}
\[ S^{2n}/T^1 = \partial P \ast \C P^{n-1},\; \text{where}\; P=[-1, 1].
\]
\end{thm}

\section{Quasitoric manifolds $M^{2n}$ as $(2n,n)$-manifolds}
Recall~\cite{BP} that a quasitoric manifold is a closed manifold $M^{2n}$ equipped with the smooth action of the torus $T^n$ such that:
\begin{itemize}
\item the action of the torus $T^n$ on $M^{2n}$ is locally standard;
\item the orbit space $M^{2n}/T^n$ is diffeomorphic to $P^n$ as a manifold with corners, where $P^n$ is a simple polytope.
\end{itemize}
The second condition gives that the natural projection $\pi : M^{2n}\to P^n$ induces the smooth map $\mu : M^{2n}\to P^{n}$, which is $T^n$-invariant and which plays a role of almost moment map of our definition. The action is free on $\pi ^{-1}(\stackrel{\circ}{P^{n}}), $ which is everywhere dense in set in $M^{2n}$ and the vertices of $P^{n}$ correspond to the fixed points of $T^n$-action. Let $\mathcal{F}$ be the set of faces for $P^n$. The points from $\pi ^{-1}(F)\subset M^{2n}$ have the same stabilizer  for any $F\in \mathcal{F}$.  Therefore,  it is  defined the characteristic function $\Lambda : \mathcal{F}\to S(T^n)$  by $\Lambda (F)$ is the stationary subgroup of $\pi ^{-1}(F)$.  A quasitoric manifold can be recovered, up to diffeomorphism,  using characteristic pair $(P^n, \Lambda )$ by
\begin{equation}\label{recov}
M \cong (T^n \times P^{n})/\approx,\;\; (t_1,p_1)\approx (t_2, p_2)\; \text{iff}\; p_1=p_2,\; t_1t_2^{-1}\in \Lambda (F(p_1)),
\end{equation}
where $F(p_1)$ is the smallest face of $P^{n}$ which contains $p_1$.
\begin{thm}
A quasitoric manifold has the structure of $(2n,n)$-manifold.
\end{thm}
\begin{proof}
In order to check that the Axiom~\ref{atlas} is satisfied, let $P_{v}\subset P^n$ denote the open cone in $P^{n}$ with a fixed vertex $v$ of $P^{n}$. It is an open subset in $P^n$ and $M_{v} = \mu ^{-1}(P_{v})$ will be an open subset in $M^{2n}$. The set $M_{v}$ is $T^n$-invariant, then it contains exactly one fixed point $x_v$ and  $\mu (x_v) = v$ and it is everywhere dense in $M^{2n}$. Moreover, by~\eqref{recov} it follows that $M_v$ is homeomorphic to $(T^n\times P_{v})/\approx$ which is further homeomorphic to $(T^n\times \R ^{n}_{+})\approx \cong \C ^n$.  It is obvious that Axiom~\ref{bij} is satisfied. The admissible polytopes $P_{\sigma}$ are the faces of $P^n$ and the strata   $W_{\sigma}$ are given by  $\sigma$, which  runs through the set of vertices of faces for  $P^n$. It directly follows that $\widehat{\mu} : W_{\sigma}\to \stackrel{\circ}{P_{\sigma}}$ and $W_{\sigma}/T^n \cong \stackrel{\circ}{P_{\sigma}}$, so Axion~\ref{fiber} is satisfied. We see that any stratum  $W_{\sigma}$ consists of one leaf and its boundary is the union of the strata  over the faces of the corresponding admissible polytope $P_{\sigma}$, so Axiom~\ref{leaf} is satisfied as well.
\end{proof}

\section{Complete complex flag manifolds $F_{k+1}$ as $(k(k+1),k)$-manifolds}

The complete flag manifold $F_{k+1}$ consists of flags of $k$ complex subspaces $L =(L_1\subset L_{2}\subset \ldots \subset L_{k})$ in $\C ^{k+1}$. It is a homogeneous space represented by $F_{k+1} = U(k+1)/T^{k+1}$.  As in the case of complex Grassmann manifolds the canonical action    of $T^{k+1}$ on $\C ^{k+1}$ induces the effective action  of $T^{k}$ on $F_{k+1}$. This action extends to the action of the corresponding algebraic torus. The moment map $\mu : F_{k+1} \to \R ^{k+1}$ is defined by
\begin{equation}\label{momflag}
\mu (L) = \mu _{1}(L_1) + \mu _{2}(L_2)+\ldots +\mu _{k}(L_k),
\end{equation}
where $L=(L_1\subset L_2\subset \ldots \subset L_{k})$ and
\[
\mu _{i}(L_i) = \frac{\sum _{J}|P^{J}(L_i)|^2\delta _{J}}{\sum _{J}|P^{J}(L_i)|^2},
\]
where $J \subset \{1,\ldots k+1\}$, $|J|=i$ and $\delta _{J}\in \R ^{k+1}$ is given by
\[
(\delta _{J})_{i}=1,\;\; i\in J,\;\; (\delta _{J})_{i}=0,\;\;  i\notin J.
\]

The image of any $\mu _{i}$ is a hypesimplex $\Delta _{k+1,i}$  as it is the image by the moment map of the Grassmann manifold $G_{k+1,i}$. Therefore the image of $\mu$ is a Minkowski sum of the hypersimplices $\Delta _{k+1,i}$, $1\leq i\leq k$ which  is known to be the   standard permutahedron $P_{k+1}$.  Recall that the standard permutahedron $P_{k+1}$  is a convex hull over $(k+1)!$ points given by $\sigma(0,1,2,\ldots k)$, where $\sigma$ runs through the symmetric group $S_{k+1}$.

The algebraic torus $(\C ^{*})^{k+1}$ acts on $F_{k+1}$.  In the analogous way as for the complex Grassmann manifolds we prove:

\begin{thm}
The complete complex flag manifold $F_{k+1}$  has the structure of $(k(k+1), k)$-manifold.
\end{thm}

We  provide the proof  for the case $k=2$:

\begin{prop}\label{F3}
$F_{3}/T^2 \cong S^1 \ast \C P^1\cong S^4$
\end{prop}
\begin{proof}
 We consider the atlas for  $F_{3}$ whose charts are given by
\[
M_{i,jk} = \{ L =(L_1\subset L_2) \in F_{3}\; |\: P^{i}(L_1)\neq 0, P^{jk}(L_2)\neq 0\}, \; 1\leq i,j,k\leq 3, \; j\neq k
\]
and the homeomorphisms $u_{i,jk} : M_{i,jk}\to \C ^3$  are  defined as follows.  Any $L\in M_{i,jk}$ can be represented by the matrix $A_{L} = (a_{sl})$, $1\leq s\leq 3$, $1\leq l\leq 2$  such that $a_{i1}=1$, $a_{i2}=0$ and  $a_{j2}=1$ or $a_{k2}=1$. Then
$u_{i,jk}(L) = (a_{sl})$, where $sl\neq i1, i2$ and $sl\neq j2$ or $sl\neq k2$.  This atlas is invariant under the canonical action of the algebraic torus $(\C ^{*})^{3}$.
Consider the chart $M_{1,12}$. Any $L\in M_{1,12}$ represents by a matrix
\[
\left(\begin{array}{cc}
1 & 0\\
a_1 & 1\\
a_2 & a_3
\end{array}\right)
\;\; \text{and}\;\; u_{1,12}(L) = (a_1,a_2,a_3).
\]
The moment map~\eqref{momflag} in this chart writes as
\[
\mu (L) = \frac{1}{1+|a_1|^2 + |a_2|^2}((1,0,0)  + |a_1|^2(0,1,0)+|a_2|^2(0,0,1)) +\]
\[ + \frac{1}{|a_1|^2 + |a_3|^2+|a_1a_3-a_2|^2}(|a_1|^2(1,1,0)+|a_3|^2(1,0,1)+|a_1a_3-a_2|^2(0,1,1)).
\]
The action of the algebraic torus $(\C ^{*})^{2}$ on $\C ^{3}$ is  induced by the action of $(\C ^{*})^{3}$ on $F_3$, is, in the chart $M_{1,12}$,  given by
\[
(t_1,t_2)\cdot (a_1,a_2,a_3) = (t_1a_1, t_2a_2, \frac{t_2}{t_1}a_3).
\]
Thus, the orbits  for this action   are as follows:
\begin{enumerate}
\item  if $a_i\neq 0$, $1\leq i\leq 3$ and $a_1a_3-a_2\neq 0$  it is a hypersurface $\frac{z_2z_3}{z_1}=c$, where $c\in \C -\{0,1\}$;
\item if all $a_i\neq 0$ and $a_1a_3-a_2=0$ it is the hypersurface $\frac{z_2z_3}{z_1}=1$;
\item if $a_i=0$ and $a_s, a_l\neq 0$  it is the coordinate subspace $\C _{ls}$;
\item if $a_i, a_s=0$ and $a_l\neq 0$ it is the coordinates axis $\C_{l}$;
\item if all $a_i=0$ is the point $(0,0,0)$.
\end{enumerate}

Note that the orbits 1. form the main stratum and they are parametrized by $c\in \C -\{0,1\}$. All other strata consist of one orbit. Since the main stratum is everywhere dense in $F_{3}$ all other orbits can be parametrized using this  parametrization of the main stratum.

The admissible polytopes for the chart $M_{1,12}$ are:
\begin{enumerate}
\item the interior of the hexagon $P_{3}$;
\item the quadrilateral $Q$  over the vertices $(2,1,0), (2,0,1), (1,2,0), (1,0,2)$;
\item the  interiors of three quadrilaterals $Q_i$, $1\leq i\leq 3$  defined by the vertices:
$Q_1 -  (2,1,0), (2,0,1), (1,0,2), (0,1,2)$; $Q_2 - (2,1,0), (2,0,1), (1,2,0), (0,2,1)$; $Q_3 - (2,1,0), (1,2,0), (0,2,1), (0,1,2)$;
\item the interiors of three  segments $I_{i}$, $1\leq i\leq 3$   defined by the vertices: $I_1 - (2,1,0), (0,1,2)$; $I_2 - (2,1,0), (2,0,1)$, $I_3  - (2,1,0), (1,2,0)$;
\item the vertex $(0,1,2)$.
\end{enumerate}
We see that the orbit over $Q$ can be parametrized by $c=1$, the orbits over $Q_1$ by $c=\infty$, while the orbits over $Q_2$ and $Q_3$ by $c=0$. Also  the orbits over the intervals $I_1$ and $I_2$ can be parametrized by any $c\in \C \cup \{\infty\}$, while the orbit over $I_3$ by $c=0$ and the fixed point by any $c\in \C \cup \{\infty\}$. Note that $Q_2$, $Q_3$ and $I_3$ glue together and give the interior of $P_{3}$ and they are all parametrized by $c=0$.

By considering the other charts it is easy to generalize this case and conclude  that all admissible polytopes are given by  the hexagon and its faces and  six quadrilaterals in the hexagon and their faces. The quadrilateral complementary to $Q$ as well as its edge which belongs to the interior of the  hexagon can be  parametrized by $c=1$, while  the quadrilateral complementary to $Q_1$ as well as its edge which belongs to the interior of hexagon can be parametrized  by $c=\infty$. In this way we prove that $\widehat{\mu}^{-1}(x) \cong \C P^1$ for all $x$ from the interior of the hexagon. It follows from Theorem~\ref{join}  that $F_{3}/T^3\cong S^1\ast \C P^1$.

\end{proof}

\section{The universal space of parameters}
In the theory of $(2n,k)$-manifolds happens Gel'fand-Serganova effect, which means that there exists  $(2n,k)$-manifold $M^{2n}$ and the stratum $W_{\sigma}\subset M^{2n}$ such that $W_{\sigma ^{'}}\cap \partial W_{\sigma}\neq \emptyset$,  but $W_{\sigma^{'}}\not\subset \partial W_{\sigma}$. This is demonstrated  in~\ref{Grassmann} for $M^{24} = G_{7,3}$. Note that in~\cite{GS} it is proved that this effect does not appear for $(4(n-2), n-1)$-manifolds $G_{n,2}$, although they are manifolds of complexity  $n-3$. Gel'fand-Seganova effect shows that  the description of the equivariant structure of $(2n,k)$-manifolds is a difficult problem. In~\cite{BTN} we proposed the approach for the   solution of this problem for the  Grassmann manifolds, which is   based on the new notion of the universal space  of parameters. This notion formalizes  for $(2n,k)$-manifolds by the following axiom.

\begin{ax}\label{universal}
There exists the topological space $\mathcal{F}$ and the subspaces $\tilde{F}_{\sigma}\subset\mathcal{F}$ such that:
\begin{itemize}
\item $\mathcal{F}$ is the compactification of the space of parameters $F$ of the main stratum,
\item $\mathcal{F} = \cup _{\sigma}\tilde{F}_{\sigma}$,
\item there exists the projection $p_{\sigma} : \tilde{F}_{\sigma}\to F_{\sigma}$ for any $\sigma\in \mathfrak{S}$,
\item the map $H : \mathcal{E} = \cup _{\sigma}\stackrel{\circ}{P^{'}_{\sigma}}\times \tilde{F}_{\sigma} \to \mathfrak{E}(M^{2n}, P^k)/T^k$ defined by $H(x_{\sigma}, \tilde{c}_{\sigma}) = (x_{\sigma}, p_{\sigma}(\tilde{c}_{\sigma}))$ is a continuous map, where the topology on $\mathcal{E}$ is defined by the embedding $\mathcal{E} \to CQ(M^{2n}, P^k)\times \mathcal{F}$.
\end{itemize}
\end{ax}
\begin{defn}
The space $\mathcal{F}$ is said to be the universal space of parameters and the spaces $\tilde{F}_{\sigma}$, $\sigma\in \mathfrak{S}$ are said to be the virtual spaces of parameters.
\end{defn}

It follows from Axiom~\ref{universal} that the orbit space $M^{2n}/T^k$ can be described in terms of the structure elements of $(2n,k)$-manifolds defined by our axioms.

\begin{thm}
The  orbit space $M^{2n}/T^k$ of a $(2n,k)$-manifold $M^{2n}$ is homeomorphic to the quotient space of the space $\mathcal{E}$ by the map $H$.
\end{thm}

\begin{proof}
The orbit space $M^{2n}/T^k$ is homeomorphic to $\mathfrak{E}(M^{2n},P^k)/T^k$ and the map $H : \mathcal{E} = \cup _{\sigma}\stackrel{\circ}{P}_{\sigma}^{'}\times \tilde{F}_{\sigma} \to \mathfrak{E}(M^{2n}, P^k)/T^k$ is onto. Since $ \mathfrak{E}(M^{2n}, P^k)/T^k$  is a Hausdorff space, the statement follows.
\end{proof}

It directly follows that for  $(2n,k)$-manifolds which satisfy Theorem~\ref{main-main} or Theorem~\ref{join},  we have that $\mathcal{F} = \overline{F}$. Moreover,  for such manifolds the  virtual space of parameters for any polytope,  which  belong to the interior of $P^k$, coincides with its   space of parameters.  The manifolds $G_{4,2}$, $\C P^{5}$ and $F_3$ are examples of such manifolds.
It is proved in~\cite{BT-1} that the universal spaces of parameters for $G_{4,2}$ and $\C P^{5}$ are $\C P^1$ and $\C P^2$,  respectively, while  Proposition~\ref{F3} proves that the universal space of parameters for $F_3$ is $\C P^1$.  The first non-trivial example in this direction is  the Grassmannian  $G_{5,2}$.   In~\cite{BTN} it is proved that the universal  space of parameters for $G_{5,2}$ is homeomorphic to the blow up of $\C P^2$ at four  points. Moreover, the  case  $G_{5,2}$ shows that the virtual spaces of parameters are, in  general, wider then the spaces of parameters.  As it is shown in~\cite{BTN}, the spaces of parameters of the strata over pyramids $K_{ij}(7)$ consist of a point, while their virtual spaces of parameters are homeomorphic to $\C P^1$.

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