---
abstract: |
    We construct small covers and quasitoric manifolds over $n$-dimensional simple polytopes which allow proper colorings of facets with $n$ colors. We calculate Stiefel-Whitney classes of these manifolds as obstructions to immersions and embeddings into Euclidean spaces. The largest dimension required for embedding is achieved in the case $n$ is a power of two.

    **2010 Mathematics Subject Classification**: 57N35, 52B12.

    **Keywords**: immersions, quasitoric manifolds, simple polytopes, colorings, Stiefel-Whitney classes.
author:
- |
    [ore Baralić]{}\
    [Mathematical Institute of Serbian Academy of Sciences and Arts]{}\
    [Belgrade, Serbia]{}\
    [djbaralic@mi.sanu.ac.rs]{}
- |
    Vladimir Grujić\
    [Faculty of Mathematcs]{}\
    [University of Belgrade, Serbia]{}\
    [vgrujic@matf.bg.ac.rs]{}
title: 'Quasitoric manifolds and Small covers over properly colored polytopes: Immersions and Embeddings'
---

Introduction
============

Colorings of simple polytopes
-----------------------------

An $n$-dimensional convex polytope is *simple* if the number of facets which are meeting at each vertex is equal to $n$. The *proper coloring* into $k$ colors of a simple polytope $P^n$ is a map $$h: \{F_1, \dots, F_m\}\rightarrow
\{1,\ldots,k\}$$ of its set of facets such that $h (F_i)\neq h
(F_j)$ for each two intersecting facets. The *chromatic number* $\chi (P^n)$ of a simple polytope $P^{n}$ is the least $k$ for which there exists a proper coloring of $P^n$ into $k$ colors. The chromatic numbers of the simplex $\Delta^n$, the cube $I^n$ and the permutahedron $\Pi^n$ (Figure \[bojenja\]) are $$\chi (\Delta^n)=n+1, \ \ \ \chi (I^n)=n, \ \ \ \chi
(\Pi^n)=n.$$

![The colorings of the cube and the permutahedron[]{data-label="bojenja"}](bojenja.pdf){width="\textwidth"}

Obviously, $\chi (P^n)\geq n$ for any simple polytope $P^n$. The chromatic number of a polygon is clearly equal to $2$ or $3$, depending on the parity of the number of its faces. By famous Four Color Theorem chromatic numbers of $3$-dimensional simple polytopes are equal to $3$ or $4$. In general case, for $n\geq 4$ it does not hold $\chi (P^n)\leq n+1$. Moreover, there are simple polytopes such that their chromatic numbers are equal exactly to numbers of their facets. The polars of the cyclic polytopes with $m$ vertices $C^n (m)$ are examples of such type (see [@5 Example 0.6]).

We consider the class of $n$-dimensional simple polytopes with the chromatic number equal to $n$. Denote this class by $\mathcal{C}$. The class $\mathcal{C}$ is closed with respect to products (see [@2 Construction 1.12]) and connected sums (see [@2 Construction 1.13]). Also from any given simple polytope $P^n$ by truncation over all its faces we obtain a simple polytope $Q^n$ which belongs to the class $\mathcal{C}$. The complete description of the class $\mathcal{C}$ is obtained in [@9]. A simple $n$-polytope $P^n$ admits a proper coloring in $n$ colors if and only if every its $2$-face has an even number of edges (see [@9 Theorem 16]).

Small covers and quasitoric manifolds
-------------------------------------

Quasitoric manifolds and they real analogues called small covers are introduced by Davis and Januszkiewicz in [@3]. Their geometric and algebraic topological properties are closely related to combinatorics of simple polytopes. The following definitions and notions are extensively elaborated in [@2]. Denote by $$G_d=\left\{\begin{array}{rl}
S^{0}, & d=1,\\
S^{1},  & d=2,\end{array} \right. \ R_d=\left\{\begin{array}{rl}
\mathbb{Z}_2, & d=1,\\
\mathbb{Z},  & d=2,\end{array}\right. \
\mathbb{K}_d=\left\{\begin{array}{rl}
\mathbb{R}, & d=1,\\
\mathbb{C},  & d=2,\end{array} \right.$$ where $S^{0}=\{-1,+1\}$ and $S^{1}=\{z\mid |z|=1\}$ are multiplicative subgroups of real and complex numbers, $\mathbb{Z}$ is the ring of integers and $\mathbb{Z}_2=\{0,1\}$ is the ring of integers modulo $2$.

The group $G_d^{n}$ acts standardly on $\mathbb{K}_d^{n}$ by $(t_1,\ldots,t_n)\cdot(x_1,\ldots,x_n)=(t_1x_1,\ldots,t_nx_n)$. The action of $G_d^{n}$ on a smooth $dn$-dimensional manifold $M^{dn}$ is locally standard if for any point of $M^{dn}$ there is a $G_d^{n}$-invariant neighborhood which is weakly equivariantly diffeomorphic to some open $G_d^{n}$-invariant subset of $\mathbb{K}_d^{n}$ with the standard action of $G_d^{n}$. Recall that two $G_d^{n}$-manifolds are weakly equivariantly diffeomorphic if there is an automorphism $\omega:G_d^n\rightarrow
G_d^n$ and a diffeomorphism $f:M_1^{dn}\rightarrow M_2^{dn}$ such that $f(g\cdot x)=\omega(g)\cdot f(x)$ for any $g\in G_d^n$ and $x\in M_1^{dn}$. A smooth $dn$-dimensional $G_d^{n}$-manifold $M^{dn}$ with a locally standard action of the group $G_d^{n}$ such that the orbit space $M^{dn}/G_d^{n}$ is diffeomorphic, as a manifold with corners, to a simple $n$-dimensional polytope $P^{n}$ is called [*small cover*]{} over $P^{n}$ if $d=1$, correspondingly [*quasitoric manifolds*]{} over $P^{n}$ if $d=2$. Let $\pi:M^{dn}\rightarrow P^{n}$ be the projection map and $\{F_1,\ldots,F_m\}$ be the set of facets of the polytope $P^{n}$. The inverse images $M^{d(n-1)}_j=\pi^{-1}(F_j)$ are $G_d^{n}$-invariant submanifolds of codimension $d$ called [*characteristic submanifolds*]{}. To each characteristic submanifold $M_j^{d(n-1)}$ corresponds the isotropy subgroup $G(F_j)=G_{\lambda_j}$ of the rank one, where $G_{\lambda_j}=\{(1,\ldots,1),((-1)^{\lambda_{1j}},\ldots,(-1)^{\lambda_{nj}})\}$ for some $\lambda_j=(\lambda_{1j},\ldots,\lambda_{nj})\in\mathbb{Z}_2^{n}\setminus\{0\}$ if $d=1$ and $G_{\lambda_j}=\{(e^{2\pi
i\lambda_{1j}t},\ldots,e^{2\pi i\lambda_{nj}t})\mid
t\in\mathbb{R}\}$ for some primitive vector $\lambda_j=(\lambda_{1j},\ldots,\lambda_{nj})\in\mathbb{Z}^{n}$ defined up to the sign, if $d=2$. In this way, the action of the group $G_d^{n}$ on $M^{dn}$ defines the [*characteristic map*]{} $$l_d:\{F_1, \dots, F_m\}\rightarrow R^n_d,$$ which to a facet $F_j$ of the polytope $P^{n}$ assigns a primitive vector $\lambda_j=(\lambda_{1j},\ldots,\lambda_{nj})\in R^n_d$. Denote by $\Lambda= (\lambda_1, \dots, \lambda_m)$ the matrix formed by these vectors. Then $\det \Lambda_{{(V)}}=\pm 1$ for any vertex $V\in P^{n}$, where $\Lambda_{(V)}$ is the submatrix of the matrix $\Lambda$ formed by vectors corresponding to facets which are intersecting in that vertex. A pair $(P^{n}, \Lambda)$ satisfying this condition on submatrices is called a [*characteristic pair*]{}. The manifold $M^{dn}$ is reconstructible from the characteristic pair $(P^{n}, \Lambda)$ up to weak equivariant diffeomorphism in the following way (see [@3] and [@2 Construction 6.18]). Let $F_{i_1}\cap\ldots\cap F_{i_k}$ be the minimal face containing a point $q\in P^{n}$. To the point $q$ we associate the subgroup $G(q)=G_{\lambda_{i_1}}\times\cdots\times G_{\lambda_{i_k}}$. Define $M(P^{n}, \Lambda)=G_d^{n}\times P^{n}/\sim$, where $(t_1,p)\sim(t_2,q)$ if and only if $p=q$ and $t_1t_2^{-1}\in
G(q)$.

Though not all simple polytopes allow characteristic maps (see [@3 Example 1.15]), to considered polytopes $P^{n}\in\mathcal{C}$ in a simple way can be assigned a characteristic matrix $\Lambda$ if each color $i\in\{1,\ldots,n\}$ is identified with the vector $e_i$ of the standard basis in $R_d^{n}$.

The cohomology ring of a small cover and a quasitoric manifold $M^{dn}, d=1,2$ is described in the following way. Let $v_j=D[M_j^{d(n-1)}], j=1,\ldots,m$ be cohomology classes which are Poincare duals to fundamental classes of characteristic manifolds. The characteristic matrix $\Lambda=(\lambda_1, \dots,
\lambda_m)$ defines following linear forms $$\theta_i:=\sum_{j=1}^m \lambda_{i j} v_j, i=1,\ldots,n,$$ where $\lambda_j=(\lambda_{1 j}, \dots, \lambda_{n j})^t \in R_d^n,
j=1,\ldots,m$. Let $\mathcal{J}$ be the ideal in $R_d[v_1, \dots,
v_m]$ generated by elements $\theta_1,\ldots,\theta_n$ and $\mathcal{I}$ be the Stanley-Reisner ideal of the polytope $P^{n}$, which is generated by monomials $v_{i_1}\cdots v_{i_k}$ whenever $F_{i_1}\cap\ldots F_{i_k}=\emptyset$ in $P^{n},
i_1<\ldots<i_k$. Then for the cohomology ring the following isomorphism holds (see [@3])

$$\label{djf} H^\ast (M^{dn},R_d)\simeq R_d[v_1, \dots,
v_m]/(\mathcal{I}+\mathcal{J}).$$

The same formula $(\ref{djf})$ holds with $\mathbb{Z}_2$ coefficients if $d=2$. The total Stiefel-Whitney class is determined by the following Davis-Januszkiewicz’s formula

$$\label{djfswc} w (M^{dn})= \prod_{i=1}^m (1+v_i)\in H^\ast
(M^{dn}; \mathbb{Z}_2),$$

where in the case $d=2$ all classes $v_i$ are regarded as $\mathbb{Z}_2$-restrictions of corresponding integral classes.

Immersions and Embeddings
-------------------------

Immersions and embeddings of manifolds are a classical topic in algebraic topology and differential topology. We consider immersions and embeddings in the smooth category. For a manifold $M^{n}$ define the numbers $imm (M^n)$ and $em(M^n)$ as the smallest dimensions of Euclidean spaces in which this manifold is immersed and embedded, correspondingly. In accordance with Whitney’s theorem for each smooth manifold $M^{n}$ is satisfied $imm(M^n)\leq 2n-1$ and $em(M^n)\leq 2n$. On the other hand, the dual Stiefel-Whitney classes $\overline{w}_i$ serve as obstructions to immersions and embeddings of manifolds in Euclidean spaces. Recall that dual Stiefel-Whitney classes $\overline{w}_i(M^{n})$ are characteristic classes of the stable normal bundle of a manifold $M^{n}$.

\[imeem\] For a smooth manifold $M^{n}$ let $k:=\max \{i\mid
{\overline{w}}_i(M^n)\neq 0\}$. Then ${\operatorname{imm}}(M^n)\geq n+k \, \mbox{\,and\,}\,
{\operatorname{em}}(M^n)\geq n+k+1.$

The Stiefel-Whitney classes are also obstructions to so called totally skew embeddings introduced by Ghomi and Tabachnikov in [@6]. A manifold $M^{n}$ is totally skew embedded in the Euclidean space $\mathbb{R}^{N}$ if any two tangent lines in different points of $M^{n}$ are skew lines in $\mathbb{R}^{N}$. It is proved in the paper [@6] that the number $N(M^{n})$ defined as the smallest dimension of Euclidean spaces for which there is a totally skew embedding of $M^{n}$, satisfies $$2n+2\leq N(M^n)\leq 4n+1.$$ The better lower bound is obtained in [@7].

\[skewteo\] If $$k=\max \{i\mid \overline{w}_i(M^n)\neq 0\}$$ then $N(M^n)\geq
2n+2k+1.$

The immersions and embeddings of quasitoric manifolds over cubes are studied in [@1].

In this paper we prove the following results.

\[main\] Let $n$ be a power of two and $P^n$ be a simple convex $n$-dimensional polytope which allows a proper coloring in $n$ colors.

-   There is a small cover $M^{n}$ over the polytope $P^{n}$ which satisfies following identities $imm(M^{n})=2n-1$ and $em(M^{n})=2n$.

-   There is a quasitoric manifold $M^{2n}$ over the polytope $P^{n}$ which satisfies following inequalities $imm(M^{2n})\geq 4n-2$ and $em(M^{2n})\geq 4n-1$. Moreover, for $n\geq 3$ in both relation the equality holds.

We give an explicit construction of manifolds $M^{n}$ and $M^{2n}$ from Theorem \[main\]. By using combinatorial properties of the simple polytope $P^n$ and Davis-Januszkiewicz formula (\[djf\]) and (\[djfswc\]), we describe the cohomology rings $H^*(M^{dn},
\mathbb{Z}_2), d=1,2$ and calculate Stiefel-Whitney classes $w_k(M^{dn}), d=1,2$. When $n$ is a power of two, we prove that classes ${\overline{w}}_{d(n-1)}( M^{dn}), d=1,2$ are nontrivial, the claim that implies Theorem \[main\]. Also from this claim and Theorem \[skewteo\] for totally skew embeddings of constructed manifolds immediately follows

If $n$ is a power of two then $$N(M^{n})\geq 4n-1, \ \ \ N(M^{2n})\geq 8n-3.$$

The obtained small covers $M^{n}$ are a new class of manifolds for which the numbers $N(M^{n})$ are equal to $4n-1, 4n,$ or $4n+1$. So far, only known examples with this property were real projective spaces (see [@7]).

Cohen [@8] in 1985 resolved positively the famous [*Immersion Conjecture*]{}, by showing that each compact smooth $n$-manifold for $n>1$ can be immersed in $\mathbb{R}^{2n-\alpha(n)}$, where $\alpha(n)$ is the number of 1’s in the binary expansion of $n$. The products of real projective spaces are examples of manifolds that achieve the upper bounds. We construct new examples of this type in the class of small covers.

\[exist\] For every positive integer $n$ there is a small cover $M^{n}$ and a quasitoric manifold $M^{2n}$ over some simple $n$-dimensional polytope which allows a proper coloring in $n$ colors such that $$\begin{aligned}
imm(M^{n})=2n-\alpha(n), \ \ \ imm(M^{2n})\geq 4n-2\alpha(n) \\
em(M^{n})=2n-\alpha(n)+1, \ \ \ em(M^{2n})\geq 4n-2\alpha(n)+1, \\
N(M^{n})\geq 4n-2\alpha(n)+1, \ \ \ N(M^{n})\geq 8n-4\alpha(n)+1.\end{aligned}$$

We conjecture that the statement of Theorem \[exist\] is satisfied for any simple $n$-dimensional polytope which is properly colored by $n$ colors.

\[conj1\] Let $P^n$ be a simple convex $n$-dimensional polytope which allows a proper coloring in $n$ colors. Then there exist a small cover $M^{n}$ and a quasitoric manifold $M^{2n}$ over the polytope $P^n$ which satisfy

$$\begin{aligned}
imm(M^{n})= 2n-\alpha(n),\;\;\;\;em(M^{n})= 2n-\alpha(n)+1,\\
imm(M^{2n})\geq  4n-2\alpha(n),\;\;\;\;em(M^{2n})\geq 4n-2
\alpha(n)+1.\end{aligned}$$

Manifolds $M^{dn}$
==================

Construction {#cons}
------------

Let $P^n$ be a simple polytope such that $\chi (P^n)=n$ and $h:\{F_1, \dots,F_m\}\rightarrow\{1,\ldots,n\}$ be its proper coloring. Denote by $\mathcal{F}_j$ the set $h^{-1} (j)$. Every vertex $V\in P^n$ is the intersection of $n$ differently colored facets. Take an arbitrary vertex $V=H_1\cap\cdots\cap H_n$, where the facet $H_i$ is colored by color $i$. Assign to each facet $H_i$ the vector $\lambda_i=(\underbrace{0,\dots,0}_{i-1},\underbrace{1,\dots,1}_{n-i+1})^t$ and vectors $\lambda_F=(\underbrace{0,\dots,0}_{i-1}, 1,
\underbrace{0,\dots,0}_{n-i})^t$ to remaining facets $F\in
\mathcal{F}_i\backslash \{H_i\}$. The corresponding matrix $\Lambda$ clearly induces the characteristic map, since $\det
{\Lambda}_{(v)}=1$ for every vertex $V\in P^n$. Define $M^{dn}=M(P^{n}, \Lambda)$ as the manifold constructed from the characteristic pair $(P^{n}, \Lambda)$.

Cohomology ring
---------------

Let $u_1,\dots,u_n$ be Poincaré duals to characteristic submanifolds over the facets $H_1,\dots,H_n$ respectively. For every facet $F$ of $P^n$ distinct from $H_1,\dots,H_n$, let $v_F$ denotes the Poincaré dual to the characteristic submanifold over $F$.

The cohomology ring of the manifold $M^{dn}$ is determined by the Stanley-Reisner ideal $\mathcal{I}$ of $P^n$ and the ideal $\mathcal{J}$ which is generated by linear forms $$\label{j1}
\theta_i:=\sum_{j=1}^i u_j+\sum_{F\in
{\mathcal{F}_i\setminus\{H_i\}}} v_F, i=1, \dots, n.$$ Recall that $$H^*(M^{dn}; \mathbb{Z}_2)=\mathbb{Z}_2[u_1, \dots,
u_n, v_F |F\in {\mathcal{F}\setminus\{H_1,\dots,
H_n\}}]/(\mathcal{I}+\mathcal{J}).$$

From the coloring of $P^n$ we easily deduce

\[p1\] For facets $F$, $G\in {\mathcal{F}_i\setminus\{H_i\}}$, $F\neq G$ corresponding classes satisfy the following relation in the cohomology ring $H^*(M^{dn}; \mathbb{Z}_2)$ $$v_F v_G=u_i v_F=u_i v_G=0.$$

Proposition \[p1\] and (\[j1\]) together imply

\[p2\] The following equalities hold in $H^*(M^{dn};
\mathbb{Z}_2)$ $$\label{r1}
u_1^2=0, \, u_2^2=u_1 u_2, \dots, u_n^2=u_1 u_n+\cdots+u_{n-1}
u_n.$$

Let $F$ be an arbitrary facet of $P^n$ and $v_F$ the corresponding class over $F$, even if $F$ is one of $H_1$, $\dots$, $H_n$.

\[lema1\] Let $k$ be a positive integer and $h$ a proper coloring map of the polytope $P^n$. Then the class $v_F^k$ is either trivial or equal to the degree $k$ homogenous polynomial $Q_k^F(u_1,\ldots,u_{h(F)-1},v_F)$ whose monomials are square free.

Let $h(F)=i$. We prove the claim by induction on $k$. For $k=1$ it is trivial. Assume that $v_F^k=Q_k^F (u_1,\ldots,u_{i-1},
v_F)$. By multiplying with $v_F^k$ the relation $$\sum_{j=1}^{i-1} u_j + \sum_{F\in
{\mathcal{F}_i\setminus\{H_i\}}} v_{F}=0$$ and by using Proposition \[p1\] we get $$v_F^{k+1}=(u_1+\cdots+u_{i-1}) Q_k^F (u_1, \dots, u_{i-1},
v_F).$$ If $v_F^{k+1}=0,$ the claim follows directly. In the opposite case it is obvious that $(u_1+\cdots+u_{i-1}) Q_k^F (u_1,
\dots, u_{i-1}, v_F)$ is the degree $k+1$ homogenous polynomial in variables $u_1$, $\dots$, $u_{i-1}$, $v_F$. However, from relations (\[r1\]) follows $(u_1+\cdots+u_{i-1}) Q_k^F (u_1,
\dots, u_{i-1}, v_F)=Q_{k+1}^F (u_1, \dots, u_{i-1}, v_F)$, where all monomials of the polynomial $Q_{k+1}^{F}$ are square free.

From lemma \[lema1\] immediately follows

\[posl1\] Each class $v_{F_{i_1}}^{r_1}\cdots v_{F_{i_k}}^{r_k}$ is either trivial or equal to the degree $r_1+\dots+r_k$ homogenous polynomial in variables $u_1$, $\dots$, $u_{n}$, $v_{F_{i_1}}$, $\dots$, $v_{F_{i_k}}$ whose monomials are square free.

\[p3\] The class $u_1 \dots u_n$ is the fundamental cohomology class in the ring $H^{dn} (M^{dn}; \mathbb{Z}_2)$.

Denote by $\mathcal{F}_V$ the set of facets containing a vertex $V\in
P^{n}$. Let $V^{\ast}=\prod_{F\in\mathcal{F}_V}v_F$. We show that $V^{\ast}=V'^{\ast}$ for each two vertices $V,V'\in P^{n}$. For this it is sufficient to suppose that vertices are connected by the edge $VV'$. In this case we have $\mathcal{F}_{V'}=\mathcal{F}_V\setminus\{G\}\cup\{G'\}$ for unique facets $G$ and $G'$ which are colored by the same color. If $h(G)=h(G')=i$, we multiply by $\prod_{F\in\mathcal{F}_V\cap\mathcal{F}_{V'}}v_F$ the relation $$\sum_{j=1}^{i-1} u_j + \sum_{F\in \mathcal{F}_i} v_{F}=0.$$ From Proposition \[p1\] we obtain $$(u_1+\ldots+u_{i-1})\prod_{F\in\mathcal{F}_V\cap\mathcal{F}_{V'}}v_F+V^{\ast}+V'^{\ast}=0.$$ In the case $i=1$ we get the required equality $V^{\ast}=V'^{\ast}$. If $i>1$, for each $1\leq j<i$, we have by Propositions \[p1\] and \[p2\] $$u_j\prod_{F\in\mathcal{F}_V\cap\mathcal{F}_{V'}}v_F=\left\{\begin{array}{cc} 0,
&\mbox{if} \ H_j\notin\mathcal{F}_V\cap\mathcal{F}_{V'} \\
(u_1+\cdots
u_{j-1})\prod_{F\in\mathcal{F}_V\cap\mathcal{F}_{V'}}v_F, &
\mbox{if} \
H_j\in\mathcal{F}_V\cap\mathcal{F}_{V'}\end{array}\right..$$ Then by induction follows $u_j\prod_{F\in\mathcal{F}_V\cap\mathcal{F}_{V'}}v_F=0,
j=1,\ldots,i-1$ which again gives $V^{\ast}=V'^{\ast}$.

Suppose now that $u_1\cdots u_n=0$. From the so far proven part follows that $V^{\ast}=0$ for each vertex $V\in P^{n}$. Moreover in this case all $dn$-dimensional classes vanish since by Corollary \[posl1\] square free monomials of degree $n$ linearly generate $H^{dn} (M^{dn}; \mathbb{Z}_2)$. But it contradicts the known fact that $H^{dn} (M^{dn}; \mathbb{Z}_2)\simeq\mathbb{Z}_2$.

Stiefel-Whitney classes
-----------------------

Proposition \[p1\] implies that the total Stiefel-Whitney class could be expressed as $$w(M^{dn})=\prod_{i=1}^{n} \prod_{F\in \mathcal{F}_i} (1+v_F)=\prod_{i=1}^{n}(1+\sum_{F\in \mathcal{F}_i}v_F).$$ By applying (\[j1\]) we obtain

$$w (M^{dn})=(1+u_1)(1+u_1+u_2)\cdots(1+u_1+u_2+\cdots+u_{n-1}).$$

In order to prove the main theorem \[main\], we are going to use another set of generators $t_1,\dots,t_n$ which are defined by $$\begin{aligned}
\label{tgen}
t_i=\sum_{j=1}^{i}u_j, i=1,\ldots,n.\end{aligned}$$ Consequently we have $$\label{sw3} w(M^{dn})=(1+t_1)\cdots(1+t_{n-1}).$$ By Proposition \[p2\] the classes $t_1^2$, $t_2^2+t_1
t_2,\ldots,t_n^2+t_{n-1}t_n$ vanish. Let $\mathcal{T}_n$ be the ideal generated by these classes. By Proposition \[p3\] it is easily seen that the class $t_1 t_2 \cdots t_n$ also represents the fundamental class.

Proof of Theorem \[main\]
=========================

In order to prove Theorem \[main\] it is sufficient to show that the top dual Stiefel-Whitney class $\overline{w}_{d(n-1)}(M^{dn})$ is nontrivial.

Dual Stiefel-Whitney classes
----------------------------

Stiefel-Whitney classes and dual Stiefel-Whitney classes are related by $$w(M^{dn})\cdot \overline{w}(M^{dn})=1.$$ From the relation (\[sw3\]) we obtain

\[swd\] The total Stiefel–Whitney class $\overline{w}(M^{dn})$ is expressed by $$\overline{w}
(M^{dn})=(1+t_1)(1+t_2+t_2^2)\cdots(1+t_{n-1}+\cdots+t_{n-1}^{n-1}).$$

The statement follows from the fact that $t_k^{k+1}=0$ for all $k=1,\ldots,n-1$. In fact $t_k^{k+1}=(u_1+\cdots+u_k)^{k+1}$ is the sum of monomials of the form $\sum u_{i_1}^{r_1}\cdots
u_{i_j}^{r_j},$ where $1\leq i_1<\cdots<i_j\leq k$ and $r_1+\cdots
r_k=k+1$. In its turn, by Corollary \[posl1\] and Proposition \[p2\] this is a sum of homogeneous polnomials of the degree $k+1$ of the form $\sum Q_{k+1}(u_1,\ldots,u_k)$. Since monomials of each polynomial in the sum is square free, we have $Q_{k+1}(u_1,\ldots,u_k)=0$.

We want to determine the highest nontrivial dual $\overline{w}_k
(M^{dn})$. For small $n$, we could calculate $\overline{w}
(M^{dn})$ directly.

-   $\overline{w} (M^{d \cdot 2})=1+t_1$,

-   $\overline{w} (M^{d\cdot 3})=1+(t_1+t_2)$,

-   $\overline{w} (M^{d\cdot 4})=1+(t_1+t_2+t_3)+t_1 t_3+t_1 t_2 t_3$,

-   $\overline{w} (M^{d\cdot 5})=1+(t_1+t_2+t_3+t_4)+(t_1
    t_3+t_1 t_4+t_2 t_4)+(t_1 t_2 t_3+t_2 t_3 t_4)$.

The class $\overline{w}_{d (n-1)}(M^{dn})$
------------------------------------------

Consider arbitrary two manifolds $M^{dn}$ and $M^{d(n+1)}$ constructed as in subsection \[cons\] over simple polytopes $P_1^{n}$ and $P_2^{n+1}$, properly colored in $n$ and $n+1$ colors, respectively.

\[sub\] The ring $\mathbb{Z}_2[t_1,\ldots,t_n]/\mathcal{T}_n$ is a subring of the cohomology ring $H^{\ast}(M^{dn};
\mathbb{Z}_2)$ which is generated by elements $t_1,\ldots,t_n$.

It is necessary to prove that $\mathcal{T}_n$ is the ideal of all relations among elements $t_1,\ldots,t_n$ in $H^{\ast}(M^{dn}; \mathbb{Z}_2)$. The rings $\mathbb{Z}_2 [t_1,
\dots, t_n]/\mathcal{T}_n$ and $\mathbb{Z}_2 [u_1, \dots,
u_n]/\mathcal{U}_n$ are isomorphic, where $\mathcal{U}_n$ is the ideal generated by elements $u_1^{2}$ and $u_i^{2}+(u_1+\ldots+u_{i-1})u_i, i=2,\ldots,n$ from Proposition \[p2\]. It is sufficient to show that $\mathcal{U}_n$ is the ideal of all relations among $u_1,\ldots,u_n$ in $H^{\ast}(M^{dn};
\mathbb{Z}_2)$.

By Corollary \[posl1\] any homogeneous polynomial in $H^{\ast}(M^{dn}; \mathbb{Z}_2)$ is expressed as a sum of square free monomials. So let we have a relation $$\sum_{j=1}^{m}u_{I_j}=0,$$ where $I_j=\{a_{j1}<\ldots<a_{jk}\}\subset\{1,\ldots,n\}, j=1,\ldots,m$ and $u_{I}=\prod_{i\in I}u_i$. It is easy to convince that $u_1\cdots u_{i-1}u_i^{2}=0, i=1,\ldots,n$. Let us order sets lexicographically $I_1<\ldots<I_m$. Let $I_m=\{a_1<\ldots<a_k\}$ and $d_i=a_i-a_{i-1}, i=1,\ldots,k$ where $a_0=0$. Define following elements $$U_i=\left\{\begin{array}{cc} u_{a_{i-1}+1}\cdots u_{a_i-1}, & d_i>1
\\ 1, & d_i=1\end{array}\right.,$$ where $i=1,\ldots,k$. By multiplying the relation with $U_1\cdots U_k,$ we get $u_1\cdots u_{a_k}=0,$ which contradicts the fact that $u_1\cdots u_n$ is the fundamental class.

By abbreviating all terms in the identity in Lemma \[swd\] we obtain that each class $\overline{w}_{dk}(M^{dn})$ is expressed in the ring $\mathbb{Z}_2[t_1,\dots,t_n]/\mathcal{T}_n$ by square free homogeneous polynomials $\overline{W}_{k}(t_1,\dots,t_n)$ of the degree $k$. Note that the inclusion $$i:\mathbb{Z}_2[t_1, \dots, t_n]\rightarrow\mathbb{Z}_2[t_1,
\dots, t_n, t_{n+1}]$$ induces a natural monomorphism

$$i^*:\mathbb{Z}_2[t_1, \dots, t_n]/\mathcal{T}_n\rightarrow
\mathbb{Z}_2[t_1, \dots, t_n, t_{n+1}]/\mathcal{T}_{n+1},$$ which by Lemma \[sub\] allows us to consider the total Stiefel-Whitney class $\overline{w}(M^{dn})$ as an element of the ring $H^*(M^{d(n+1)}; \mathbb{Z}_2)$. Thw total Stiefel-Whitney classes $\overline{w} (M^{dn})$ and $\overline{w} (M^{d(n+1)})$ satisfy the following relation in $H^*(M^{d(n+1)}; \mathbb{Z}_2)$ $$\overline{w} (M^{d(n+1)})=\overline{w}(M^{dn})(1+t_n+\dots+t_n^n).$$ Explicitly $$\label{jed2}
\overline{w}_{dk}(M^{d(n+1)})=\overline{w}_{dk}(M^{dn})+t_n\overline{w}_{d(k-1)}(M^{dn})+\dots+t_n^k,
k=0,\ldots, n.$$ Recall that $\overline{w}_{dn}(M^{dn})=0$ (see [@10]), which implies $$\label{jed3}
\overline{w}_{dn}(M^{d(n+1)})= t_n
\overline{w}_{d(n-1)}(M^{dn})+\cdots+t_n^n=t_n\overline{w}_{d(n-1)}(M^{d(n+1)}).$$

We use the same trick as in [@1]. Define numbers $\sigma^k_n,
0\leq k\leq n-1$ as follows $$\sigma^k_n=\overline{W}_{dk}(\underbrace{1,\dots,1}_n)\pmod 2.$$ By (\[jed2\]) and (\[jed3\]), we have $\sigma^k_{n+1}=\sum_{i=0}^{k} \sigma^i_n $ for every $k=1,\ldots,n-1$ and $\sigma^{n}_{n+1}=\sigma^{n-1}_{n+1}$. By definition of $\sigma^k_n$, if $\sigma^k_n=1$, then $\overline{w}_{dk}$ is the sum of an odd number of linearly independent square free monomials, which implies $\overline{w}_{dk}(M^{dn})\neq 0. $

An easy mathematical induction shows that $$\sigma^k_n\equiv{\tbinom}{n+k}{k}\pmod{2}.$$ Particularly, if $n=2^r$, we have $$\sigma^{n-1}_n
\equiv{\tbinom}{2^r+(2^r-1)}{2^r-1}\equiv{\tbinom}{2^{r+1}-1}{2^r-1}\equiv
1\pmod{2}.$$ Consequently, $$\overline{w}_{d(n-1)}(M^{dn})=t_1 t_2\cdots t_{n-1}\neq 0.$$ Therefore by Theorem \[imeem\] we obtain the required bounds $$imm(M^{dn})\geq d(2n-1), em(M^{dn})\geq d(2n-1)+1.$$ For the small cover $M^{n}$, Whitney’s theorem implies $imm(M^{n})=2n-1$ and $em(M^{n})=2n$. The quasitoric manifold $M^{2n}$ is orientable, so it can be embedded into $\mathbb{R}^{4n-1}$. From Lemma \[swd\] follows $\overline{w}_2(M^{2n})=t_1+t_2+\cdots+t_{n-1},$ which implies that the characteristic class $\overline{w}_2(M^{dn})\cdot
\overline{w}_{2n-2}(M^{2n})$ vanishes. If $n\geq 3$, the result of Massey [@11 Theorem V] yields $$imm(M^{2n})=4n-2 ,$$ which finishes the proof of Theorem \[main\].

Proof of Theorem \[exist\]
==========================

Let $n=2^{r_1}+2^{r_2}+\dots+2^{r_t}$, $r_1>r_2>\dots>r_t\geq 0$ be the binary representation of $n$ and let $m_i=2^{r_i}$ for $i=1,\dots,t$ and $m_0=0$. Let $P^n$ be a simple $n$-polytope such that $$P^n= P_1^{2_{r_1}}\times \cdots\times P_t^{2^{r_t}},$$ where each $P_i^{2_{r_i}}$ is $2^{r_i}$-colored simple $2^{r_i}$-polytope. It is obvious that polytope $P^n$ is $n$-colored.

In subsection \[cons\] we constructed manifolds $M^{d 2^{r_i}}$ over polytopes $P_i^{2_{r_i}}$. It follows from [@2 Proposition 4.7] that $M^{dn}=M^{d2^{r_1}}\times\dots\times M^{d 2^{r_t}}$ is a $G_d^{n}$-manifold over the polytope $P^n= P_1^{2_{r_1}}\times
\cdots\times P_t^{2^{r_t}}$. The total Stiefel–Whitney class of the manifold $M^{dn}$ can be easily determined using the following formula (see [@4 pp.27,54]) $$w(M^{dn})=w(M^{d 2^{r_1}})\cdots w(M^{d 2^{r_t}})\in
H^*(M^{dn})\cong H^*(M^{d 2^{r_1}})\otimes\dots\otimes H^*(M^{d
2^{r_t}}).$$ The corresponding dual total Stiefel-Whitney class is expressed as $$\label{dpr} \overline{w}(M^{dn})=\overline{w}(M^{d 2^{r_1}})\cdots
\overline{w}(M^{d 2^{r_t}}).$$ Let ${\operatorname{rank}}w M):=\max
\{k | w_k (M)\neq 0\}$. Thus, from formula (\[dpr\]) we have

$$\mathrm{rank} \ \overline{w}(M^{dn})=\sum_{i=1}^t\mathrm{rank} \ \overline{w}(M^{d2^{r_i}})=\sum_{i=1}^t d(2^{r_i}-1)=nd -\alpha (n) d.$$ In this way, Theorem \[exist\] is a consequence of Theorems \[imeem\] and \[skewteo\].

Aknowledgements {#aknowledgements .unnumbered}
===============

The authors express their special thanks to T. E. Panov and A. A. Gaifullin for attention to this work, as well as to reviewers for helpful comments.

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