# Manifolds realized as orbit spaces
of non-free $\mathbb{Z}_{2}^{k}$-actions on real moment-angle manifolds

Nikolai Erokhovets  Steklov Mathematical Institute of Russian Academy of
Sciences &Department of Mechanics and Mathematics, Moscow State University
<EMAIL_ADDRESS>
###### Abstract.

We consider (non-necessarily free) actions of subgroups
$H\subset\mathbb{Z}_{2}^{m}$ on the real moment-angle manifold
$\mathbb{R}\mathcal{Z}_{P}$ corresponding to a simple convex $n$ polytope $P$
with $m$ facets. The criterion when the orbit space
$\mathbb{R}\mathcal{Z}_{P}/H$ is a topological manifold (perhaps with a
boundary) can be extracted from results by M.A. Mikhailova and C. Lange. For
any dimension $n$ we construct series of manifolds
$\mathbb{R}\mathcal{Z}_{P}/H$ homeomorphic to $S^{n}$ and series of manifolds
$M^{n}=\mathbb{R}\mathcal{Z}_{P}/H$ admitting a hyperelliptic involution
$\tau\in\mathbb{Z}_{2}^{m}/H$, that is an involution $\tau$ such that
$M^{n}/\langle\tau\rangle$ is homeomorphic to $S^{n}$. For any $3$-polytope
$P$ we classify all subgroups $H\subset\mathbb{Z}_{2}^{m}$ such that
$\mathbb{R}\mathcal{Z}_{P}/H$ is a $3$-sphere or a rational homology
$3$-sphere. For any $3$-polytope $P$ and any subgroup
$H\subset\mathbb{Z}_{2}^{m}$ we classify all hyperelliptic involutions
$\tau\in\mathbb{Z}_{2}^{m}/H$ acting on $\mathbb{R}\mathcal{Z}_{P}/H$. As a
corollary we obtain that a $3$-dimensional small cover has $3$ hyperelliptic
involutions in $\mathbb{Z}_{2}^{3}$ if and only if it is a rational homology
$3$-sphere and if and only if it correspond to a triple of Hamiltonian cycles
such that each edge of the polytope belongs to exactly two of them.

###### Key words and phrases:

Non-free action of a finite group, polytope, real moment-angle manifold,
hyperelliptic manifold, rational homology sphere, Hamiltonian cycle

###### 2010 Mathematics Subject Classification:

57S12, 57S17, 57S25, 52B05, 52B10, 52B70, 57R18, 57R91

This work was supported by the Russian Science Foundation under grant no.
23-11-00143, https://rscf.ru/en/project/23-11-00143/

###### Contents

  1. 1 Real moment-angle manifolds and their factor spaces
  2. 2 A complex $\mathcal{C}(P,c)$ defined by a coloring $c$
  3. 3 A weakly equivariant classification of spaces $N(P,\Lambda)$
  4. 4 A weakly equivariant classification of spaces defined by affine colorings
  5. 5 A criterion when $N(P,\Lambda)$ is a manifold
  6. 6 Manifolds with torus actions
  7. 7 Boolean simplices and simplicial prisms
  8. 8 Special hyperelliptic manifolds $N(P,\Lambda)$
  9. 9 A structure of the complex $\mathcal{C}(P,c)$ for $3$-polytopes
    1. 9.1 Basic facts from the graph theory
    2. 9.2 A characterization of complexes $\mathcal{C}(P,c)$ of $3$-polytopes
  10. 10 A criterion when $N(P,\Lambda)$ is a sphere for $3$-polytopes
  11. 11 Hyperelliptic manifolds $N(P,\lambda)$ over $3$-polytopes
  12. 12 Rational homology spheres $N(P,\lambda)$ over $3$-polytopes
  13. 13 Simple $3$-polytopes with $3$ consistent Hamiltonian cycles
    1. 13.1 General facts
    2. 13.2 Polytopes without $3$ consistent Hamiltonian cycles
    3. 13.3 Reductions
    4. 13.4 Fullerenes
  14. 14 Acknowledgements

## Introduction

Toric topology (see [BP15, DJ91]) assigns to each $n$-dimensional convex
polytope $P$ with $m$ facets $F_{1},\dots,F_{m}$ an $n$-dimensional real
moment-angle manifold $\mathbb{R}\mathcal{Z}_{P}$ with an action of a finite
group $\mathbb{Z}_{2}^{m}$ and an $(n+m)$-dimensional moment-angle manifold
$\mathcal{Z}_{P}$ with an action of a compact torus $\mathbb{T}^{m}$ such that
$\mathbb{R}\mathcal{Z}_{P}/\mathbb{Z}_{2}^{m}=\mathcal{Z}_{P}/\mathbb{T}^{m}=P$
and the equivariant topology of these spaces depends only on combinatorics of
$P$. This construction allows one to build large families of manifolds for
which deep mathematical results can be proved in a more efficient and explicit
form. For example, the problem of classification of $3$-dimensional manifolds
and $6$-dimensional simply-connected manifold by their algebraic topology
invariants can be explicitly solved for the large families of small covers and
quasitoric manifolds over $3$-dimensional right-angled hyperbolic polytopes
[BEMPP17]. The Thurston’s problem of existence of a geometric decomposition of
any orientable $3$-manifold was finally solved by G. Perelman. For all
$3$-dimensional manifolds obtained as orbit spaces of free actions of
subgroups in $\mathbb{Z}_{2}^{m}$ on $\mathbb{R}\mathcal{Z}_{P}$ this
decomposition can be described explicitly and constructively [E22M].

In this paper we consider the specification of the following general question
to the case of real moment-angle manifolds and subgroups
$H\subset\mathbb{Z}_{2}^{m}$:

######  Question 1.

When is the orbit space of a smooth action of a finite group on a smooth
manifold a topological manifold (perhaps with a boundary)?

For manifolds $\mathbb{R}\mathcal{Z}_{P}/H$ we consider the following
questions.

######  Question 2.

When is $\mathbb{R}\mathcal{Z}_{P}/H$ homeomorphic to $S^{n}$?

######  Question 3.

To classify all hyperelliptic involutions in the group $\mathbb{Z}_{2}^{m}/H$
acting on the manifold $\mathbb{R}\mathcal{Z}_{P}/H$, that is involutions with
the orbit space homeomorphic to $S^{n}$.

######  Question 4.

When is $\mathbb{R}\mathcal{Z}_{P}/H$ a manifold with the same rational
homology as $S^{n}$?

The exhaustive answer to Question 1 was obtained in the works by M.A.
Mikhailova and C. Lange [M85, LM16, L19]. For an abelian group this is
equivalent to the fact that for any point the representation of its stabilizer
in the tangent space in the invariant metric is generated by reflexions and
rotations, where the presence of a rotation indicates the presence of a
boundary in the manifold. In our particular case in Theorem 5.1 we give an
effective explicit answer in terms of the polytope and the matrix defining a
subgroup and a short proof. Namely, a subgroup $H$ of rank $m-r$ is defined by
a vector-coloring of rank $r$, that is a mapping
$\Lambda\colon\\{F_{1},\dots,F_{m}\\}\to\mathbb{Z}_{2}^{r}$ such that
$\langle\Lambda_{1},\dots,\Lambda_{r}\rangle=\mathbb{Z}_{2}^{r}$. Usually in
toric topology one considers freely acting subgroups. This is equivalent to
the fact that the coloring is linearly independent, that is the vectors
$\Lambda_{i_{1}}$, $\dots$, $\Lambda_{i_{k}}$ are linearly independent if
$F_{i_{1}}\cap\dots\cap F_{i_{k}}\neq\varnothing$. In this case the orbit
space $N(P,\Lambda)=\mathbb{R}\mathcal{Z}_{P}/H$ is automatically a (smooth)
manifold.

In the general case $N(P,\Lambda)$ is a pseudomanifold, possibly with a
boundary, where the boundary is glued of facets $F_{i}$ with
$\Lambda_{i}=\boldsymbol{0}$. We prove that $N(P,\Lambda)$ is a topological
manifold if and only if different nonzero vectors among $\Lambda_{i_{1}}$,
$\dots$, $\Lambda_{i_{k}}$ are linearly independent if $F_{i_{1}}\cap\dots\cap
F_{i_{k}}\neq\varnothing$.

We prove (Corollary 1.14) that the pseudomanifold $N(P,\Lambda)$ is closed and
orientable if and only if all the vectors $\Lambda_{1}$, $\dots$,
$\Lambda_{m}$ in $\mathbb{Z}_{2}^{r}$ lie in an affine hyperplane
$\boldsymbol{c}\boldsymbol{x}=1$ not containing $\boldsymbol{0}$ (this
generalizes the criterion of orientability of small covers from [NN05]). We
call such colorings affine colorings of rank $r-1$ and denote them $\lambda$.
In some coordinate system $\Lambda_{i}=(1,\lambda_{i})$.

A coloring $c\colon\\{F_{1},\dots,F_{m}\\}\to\\{1,\dots,r\\}$ defines a
complex $\mathcal{C}(P,c)$ with facets $G_{j}$ the connected components of
unions $\bigcup_{i}F_{i}$ corresponding to the same color and faces the
connected components of intersections of facets. The complexes
$\mathcal{C}(P,c_{P})$ and $\mathcal{C}(Q,c_{Q})$ are equivalent
($\mathcal{C}(P,c_{P})\simeq\mathcal{C}(Q,c_{Q})$) if there is a homeomorphism
$P\to Q$ mapping facets to facets. In Corollary 2.7 we prove that any two
colorings of the simplex $\Delta^{n}$ in $r$ colors produce equivalent
complexes. We denote this equivalence class $\mathcal{C}(n,r)$. It turns out
that any affine coloring $\lambda$ of rank $r$ of a polytope $P$ with
$\mathcal{C}(P,\lambda)\simeq\mathcal{C}(n,r+1)$ produces a sphere
$N(P,\lambda)\simeq S^{n}$ (see Construction 5.8). Our main result concerning
Question 2 is that in dimension $n=3$ this construction exhausts all
$3$-spheres among $N(P,\lambda)$ (Theorem 10.1). The $1$-skeleton
$\mathcal{C}^{1}(3,1)$ is empty, $\mathcal{C}^{1}(3,2)$ is a circle without
vertices, $\mathcal{C}^{1}(3,3)$ is a theta-graph – a graph with two vertices
connected by three multiple edges, and $\mathcal{C}^{1}(3,4)$ is the complete
graph $K_{4}$. Thus, subgroups in $\mathbb{Z}_{2}^{m}$ producing spheres
$\mathbb{R}\mathcal{Z}_{P}/H$ bijectively correspond to the empty set, simple
cycles, theta-subgraphs and $K_{4}$-subgraphs in the $1$-skeleton of $P$.

Question 3 is motivated by papers [M90, VM99M, VM99S2] by A.D. Mednykh and
A.Yu. Vesnin who constructed examples of hyperelliptic $3$-manifolds in five
Thurston’s geometries $\mathbb{R}^{3}$, $\mathbb{H}^{3}$, $\mathbb{S}^{3}$,
$\mathbb{H}^{2}\times\mathbb{R}$, and $S^{2}\times\mathbb{R}$. These examples
were build using a right-angled $3$-polytope $P$ equipped with a Hamiltonian
cycle, a Hamiltonian theta-subgraph, or a Hamiltonian $K_{4}$-subgraph, where
a subgraph is Hamiltonian if it contains all the vertices. We call an
involution $\tau\in\mathbb{Z}_{2}^{m}/H$ acting on the manifold $N(P,\lambda)$
defined by an affine coloring of rank $r$ special if the complex
$\mathcal{C}(P,\lambda_{\tau})$ corresponding to the orbit space
$N(P,\lambda)/\langle\tau\rangle$ is equivalent to $\mathcal{C}(n,r)$. By
Construction 5.8 any special involution is hyperelliptic. We introduce
Construction 8.6 producing any special hyperelliptic manifold from a coloring
$c\colon\\{F_{1},\dots,F_{m}\\}\to\\{1,\dots,r\\}$ such that
$\mathcal{C}(P,c)\simeq\mathcal{C}(n,r)$ and a $0/1$-coloring
$\chi\colon\\{F_{1},\dots,F_{m}\\}\to\\{0,1\\}$. In Theorem 8.14 we classify
all special hyperelliptic involutions $\tau\in\mathbb{Z}_{2}^{m}/H$. For $n=3$
Theorem 10.1 implies that any hyperelliptic involution in
$\mathbb{Z}_{2}^{m}/H$ is special. Our main result concerning Question 3 is
the classification of all hyperelliptic involutions in $\mathbb{Z}_{2}^{m}/H$
for $n=3$. In particular, any Hamiltonian empty set ($r=1$), cycle ($r=2$),
theta-subgraph $(r=3)$ or $K_{4}$-subgraph ($r=4$) $\Gamma$ in
$\mathcal{C}^{1}(P,c)$ induces an affine coloring $\lambda_{\Gamma}$ of rank
$r$ by the following rule. The facets of $P$ lying in the same facet $G_{i}$
of $\Gamma$ can be colored in two colors in such a way that adjacent facets
have different colors. Assign to one color the point $\boldsymbol{a}_{i}$ and
to the other color $\boldsymbol{b}_{i}$, where the points
$\boldsymbol{a}_{1}$, $\boldsymbol{a}_{2}$, $\dots$, $\boldsymbol{a}_{r}$,
$\boldsymbol{b}_{1}$ are affinely independent and the vector
$\tau=\boldsymbol{a}_{i}+\boldsymbol{b}_{i}$ does not depend on $i$. We obtain
an affine coloring $\lambda_{\Gamma}$ and the hyperelliptic involution $\tau$
on $N(P,\lambda_{\Gamma})$ induced by $\Gamma$. In Theorem 11.5 we prove that
hyperelliptic involutions in $\mathbb{Z}_{2}^{m}/H(\lambda)$ bijectively
correspond to Hamiltonian subgraphs of the above type inducing $\lambda$. Also
in Theorem 11.7 we classify all pairs $(P,\lambda)$ admitting more than one
hyperelliptic involution. In particular, $3$-dimensional small covers
$N(P,\Lambda)$ with three hyperelliptic involutions correspond to triples of
Hamiltonian cycles on a simple $3$-polytope $P$ such that any edge of $P$
belongs to exactly two cycles.

To study Question 4 one can use the description of the cohomology
$H^{*}(N(P,\Lambda),\mathbb{Q})$ obtained by A. Suciu and A. Trevisan [ST12,
T12], and S. Choi and H. Park [CP17]. Using this description in Theorem 12.5
we describe all $3$-dimensional rational homology $3$-spheres among manifolds
$N(P,\lambda)$. Namely for $n=3$ the manifold $N(P,\lambda)$ corresponding to
an affine coloring of rank $r$ is a rational homology sphere if and only if
for any affine hyperplane $\pi$ in $\mathbb{Z}_{2}^{r}$ the union
$\bigcup_{\lambda_{i}\in\pi}F_{i}$ is a disk. In particular, $3$-dimensional
small covers are exactly small covers with three hyperelliptic involutions
described above. In Example 12.11 we build rational homology $3$-spheres
$N(P,\lambda)$ with geometric structures modelled on $S^{3}$,
$S^{2}\times\mathbb{R}$, $\mathbb{R}^{3}$, $\mathbb{H}^{2}\times\mathbb{R}$,
and $\mathbb{H}^{3}$.

The paper is organized as follows.

In Section 1 we give main definitions and basic facts about real moment-angle
manifolds $\mathbb{R}\mathcal{Z}_{P}$ and their factor spaces $N(P,\Lambda)$.
In particular, in Proposition 1.13 and Corollary 1.14 we give the criterion
when the pseudomanifold $N(P,\Lambda)$ is closed and orientable.

In Section 2 we describe complexes $\mathcal{C}(P,c)$ corresponding to
colorings of facets of $P$ and their properties. In particular, in Proposition
2.6 and Corollary 2.7 we prove that all colorings of facets of the simplex
$\Delta^{n}$ in $r$ colors produce equivalent complexes.

In Sections 3 and 4 we describe the weakly equivariant classification of
spaces $N(P,\Lambda)$ defined by vector-colorings and $N(P,\lambda)$ defined
by affine colorings.

In Section 5 we give the criterion when $N(P,\Lambda)$ is a topological
manifold (Theorem 5.1) and give a Construction 5.8 of spheres $N(P,\Lambda)$.
In particular, in Example 5.9 for any face $G\subset P$ of codimension $k$ we
build a subgroup $H_{G}\subset\mathbb{Z}_{2}^{m}$ of codimension $k+1$ such
that $\mathbb{R}\mathcal{Z}_{P}/H_{G}\simeq S^{n}$. For a vertex of the
product $\Delta^{n_{1}}\times\dots\times\Delta^{n_{k}}$ this gives an action
of $\mathbb{Z}_{2}^{k-1}$ on $S^{n_{1}}\times\dots\times S^{n_{k}}$ with the
orbit space $S^{n_{1}+\dots+n_{k}}$ build by Dmitry Gugnin in [G19].

In Section 6 we give a sufficient condition for the space $\mathcal{Z}_{P}/H$
to be a closed topological manifold with a boundary (Proposition 6.1). This
condition is similar to Theorem 5.1 and can be extracted from the general
theory developed in [S09, AGo24]. Namely, if a subgroup $H$ is defined by an
integer vector-coloring
$\Lambda\colon\\{F_{1},\dots,F_{m}\\}\to\mathbb{Z}^{r}\setminus\\{\boldsymbol{0}\\}$
such that $\langle\Lambda_{1},\dots\Lambda_{m}\rangle=\mathbb{Z}^{r}$ and for
any vertex $v=F_{i_{1}}\cap\dots\cap F_{i_{n}}$ all the different vectors
among $\\{\Lambda_{i_{1}},\dots,\Lambda_{i_{n}}\\}$ form a part of a basis in
$\mathbb{Z}^{r}$, then $\mathcal{Z}_{P}/H$ is a closed topological
$(n+r)$-manifold. In Proposition 6.2 we give a sufficient condition for
$\mathcal{Z}_{P}/H$ to be homeomorphic to a sphere. As an application in
Example 6.4 we build an action of $\mathbb{T}^{k-1}$ on
$S^{n_{1}+1}\times\dots\times S^{n_{k}+1}$ with the orbit space
$S^{n_{1}+\dots+n_{k}+1}$ constructed in [AGu23].

In Section 7 we describe combinatorial properties of boolean simplicial prisms
important for a construction of hyperelliptic manifolds.

In Section 8 we give Construction 8.6 of special hyperelliptic manifolds
$N(P,\lambda)$ with a hyperelliptic involution
$\tau\in\mathbb{Z}_{2}^{m}/H(\lambda)$ such that
$\mathcal{C}(P,\lambda_{\tau})\simeq\mathcal{C}(n,r)$. In Theorem 8.14 for
these manifolds we classify all special hyperelliptic involutions
$\tau\in\mathbb{Z}_{2}^{m}/H(\lambda)$.

In Section 9 we give basic facts from the graph theory and theory of
$3$-polytopes and in Theorem 9.10 we prove that complexes $\mathcal{C}(P,c)$
corresponding to $3$-polytopes $P$ are exactly subdivisions of the $2$-sphere
arising from disjoint unions (perhaps empty) of simple curves and connected
$3$-valent graphs without bridges.

In Section 10 we prove that for an affine coloring $\lambda$ of rank $r$ of a
simple $3$-polytope $P$ the space $N(P,\lambda)$ is homeomorphic to $S^{3}$ if
and only if $\mathcal{C}(P,\lambda)$ is equivalent to $\mathcal{C}(3,r+1)$
(Theorem 10.1).

In Section 11 for affine colorings $\lambda$ of $3$-polytopes $P$ we classify
all hyperelliptic involutions in $\mathbb{Z}_{2}^{m}/H$ acting on
$N(P,\lambda)$ (Theorems 11.5 and 11.7).

In Section 12 we give a criterion when the space $N(P,\lambda)$ is a rational
homology $3$-sphere (Theorem 12.5) and consider examples of such spaces.

In Section 13 we gather known information on simple $3$-polytopes admitting
three consistent Hamiltonian cycles and build examples of such polytopes and
also of polytopes that do not have such a property.

##  1\. Real moment-angle manifolds and their factor spaces

For an introduction to the polytope theory we recommend the books [Z95] and
[Gb03]. In this paper by a polytope we call an $n$-dimensional combinatorial
convex polytope. Sometimes we implicitly use its geometric realization in
$\mathbb{R}^{n}$ and sometimes we use it explicitly. In the latter case we
call the polytope geometric. A polytope is simple, if any its vertex is
contained in exactly $n$ facets. Let $\\{F_{1},\dots,F_{m}\\}$ be the set of
all the facets, and $\mathbb{Z}_{2}={\mathbb{Z}}/2{\mathbb{Z}}$.

######  Definition 1.1.

For each geometric simple polytope $P$ one can associate an $n$-dimensional
real moment-angle manifold:

$\mathbb{R}\mathcal{Z}_{P}=P\times\mathbb{Z}_{2}^{m}/\sim,\text{ where
}(p,a)\sim(q,b)\text{ if and only if }p=q\text{ and
}a-b\in\langle\boldsymbol{e}_{i}\colon p\in F_{i}\rangle,$

where $\boldsymbol{e}_{1},\dots,\boldsymbol{e}_{m}$ is the standard basis in
$\mathbb{Z}_{2}^{m}$.

The space $\mathcal{Z}_{P}$ was introduced in [DJ91]. It can be showed that
$\mathcal{Z}_{P}$ has a structure of a smooth manifold such that the action of
$\mathbb{Z}_{2}^{m}$ is smooth (see [BP15]).

It is convenient to imagine $\mathbb{R}\mathcal{Z}_{P}$ as a space glued from
copies of the polytope $P$ along facets. If we fix an orientation on $P\times
0$, then define on the polytope $P\times a$ the same orientation, if $a$ has
an even number of unit coordinates, and the opposite orientation, in the other
case. A polytope $P\times a$ is glued to the polytope $P\times(a+e_{i})$ along
the facet $F_{i}$. At each vertex the polytopes are arranged as coordinate
orthants in $\mathbb{R}^{n}$, at each edge – as the orthants at a coordinate
axis, and at face of dimension $i$ – as the orthants at an $i$-dimensional
coordinate subspace. Therefore, $\mathbb{R}\mathcal{Z}_{P}$ has a natural
structure of an oriented piecewise-linear manifold. There is a natural action
of $\mathbb{Z}_{2}^{m}$ on $\mathbb{R}\mathcal{Z}_{P}$ induced from the action
on the second factor. The actions of basis vectors $e_{i}$ can be viewed as
reflections in facets of the polytope. In particular, it changes the
orientation.

######  Lemma 1.2.

The element $x=(x_{1},\dots,x_{m})\in\mathbb{Z}_{2}^{m}$ preserves the
orientation of $\mathbb{R}\mathcal{Z}_{P}$ if and only if it has an even
number of nonzero coordinates. In other words, if $x_{1}+\dots+x_{m}=0$.

######  Definition 1.3.

We will denote by $H_{0}$ the subgroup of $\mathbb{Z}_{2}^{m}$ consisting of
all the orientation preserving elements.

We consider manifolds obtained as orbit spaces of (not necessarily) free
actions of subgroups $H\subset\mathbb{Z}_{2}^{m}$ on
$\mathbb{R}\mathcal{Z}_{P}$. Each subgroup of $\mathbb{Z}_{2}^{m}$ is
isomorphic to $\mathbb{Z}_{2}^{m-r}$ for some $r$ and may be defined as a
kernel of a an epimorphism
$\Lambda\colon\mathbb{Z}_{2}^{m}\to\mathbb{Z}_{2}^{r}$. Such a mapping is
uniquely defined by the images $\Lambda_{i}\in\mathbb{Z}_{2}^{r}$ of all the
vectors $e_{i}\in\mathbb{Z}_{2}^{m}$ corresponding to facets $F_{i}$, $i=1$,…,
$m$. It can be shown that the action of the subgroup
$H(\Lambda)\subset\mathbb{Z}_{2}^{m}$ on $\mathbb{R}\mathcal{Z}_{P}$ is free
if and only if

$(*)\quad\text{for any face $F_{i_{1}}\cap\dots\cap F_{i_{k}}$ of $P$ the
vectors $\Lambda_{i_{1}}$, $\dots$, $\Lambda_{i_{k}}$ are linearly
independent.}$

Since any face of $P$ contains a vertex, it is sufficient to check this
condition only for vertices.

######  Definition 1.4.

We call a mapping $\Lambda\colon\\{F_{1},\dots,F_{m}\\}\to\mathbb{Z}_{2}^{r}$
such that the images $\Lambda_{j}$ of the facets $F_{j}$ span
$\mathbb{Z}_{2}^{r}$ a (general) vector-coloring of rank $r$. If,
additionally, the condition (*) holds we call such a vector-coloring linearly
independent.

######  Remark 1.5.

In [E22M] by definition any vector-coloring is assumed to be linearly
independent.

######  Remark 1.6.

Sometimes we call by a vector-coloring of rank $r$ a mapping
$\Lambda\colon\\{F_{1},\dots,F_{m}\\}\to\mathbb{Z}_{2}^{s}$ such that
$\dim\,\langle\Lambda_{1},\dots,\Lambda_{s}\rangle=r$.

Denote by $N(P,\Lambda)$ the orbit space
$\mathbb{R}\mathcal{Z}_{P}/H(\Lambda)$ of an action corresponding to a vector-
coloring of rank $r$. If we identify $\mathbb{Z}_{2}^{m}/{\rm Ker}\,\Lambda$
with $\mathbb{Z}_{2}^{r}$ via the mapping $\Lambda$, then

$N(P,\Lambda)=P\times\mathbb{Z}_{2}^{r}/\sim,\text{ where
}(p,a)\sim(q,b)\text{ if and only if }p=q\text{ and
}a-b\in\langle\Lambda_{i}\colon p\in F_{i}\rangle.$

In particular, the space $N(P,\Lambda)$ is glued from $2^{r}$ copies of $P$.
It has a canonical action of $\mathbb{Z}_{2}^{r}$ such that the orbit space is
$P$.

######  Definition 1.7.

We call $N(P,\Lambda)$ a space defined by a vector-coloring $\Lambda$.

######  Example 1.8.

For $r=m$ and the mapping $E(F_{i})=\boldsymbol{e}_{i}$, where
$\boldsymbol{e}_{1}$, $\dots$, $\boldsymbol{e}_{m}$ is the standard basis in
$\mathbb{Z}_{2}^{m}$, the space $N(P,E)$ is $\mathbb{R}\mathcal{Z}_{P}$.

For $r=n$ a linearly independent vector-coloring is called a characteristic
mapping, and the space $N(P,\Lambda)$ is called a small cover.

For $r=1$ and the constant mapping $\Lambda_{i}=1$ the corresponding subgroup
is the subgroup $H_{0}$ consisting of all the elements preserving the
orientation of $\mathbb{R}\mathcal{Z}_{P}$. The space $N(P,\Lambda)$ is glued
of two copies of $P$ along the common boundary. It is homeomorphic to $S^{n}$.

######  Proposition 1.9.

For vector-colorings $\Lambda_{1}$ and $\Lambda_{2}$ of ranks $r_{1}$ and
$r_{2}$ of a polytope $P$ we have $H(\Lambda_{1})\subset H(\Lambda_{2})$ if
and only if there is an epimorphism
$\Pi\colon\mathbb{Z}_{2}^{r_{1}}\to\mathbb{Z}_{2}^{r_{2}}$ such that
$\Pi\circ\Lambda_{1}=\Lambda_{2}$. In this case
$N(P,\Lambda_{2})=N(P,\Lambda_{1})/{\rm Ker\,}\Pi$, where ${\rm
Ker\,}\Pi\simeq H(\Lambda_{2})/H(\Lambda_{1})$. In particular, if the action
of ${\rm Ker\,}\Pi$ is free, then there is a covering $N(P,\Lambda_{1})\to
N(P,\Lambda_{2})$ with the fiber $H(\Lambda_{2})/H(\Lambda_{1})$.

###### Proof.

We have $H(\Lambda_{1})\subset H(\Lambda_{2})$ if and only if each row of the
matrix $\Lambda_{2}$ with columns $\Lambda_{2,i}$ is a linear combination of
rows of $\Lambda_{1}$. This is equivalent to the existence of a surjection
$\Pi\colon\mathbb{Z}_{2}^{r_{1}}\to\mathbb{Z}_{2}^{r_{2}}$ such that
$\Pi(\Lambda_{1,i})=\Lambda_{2,i}$ for all $i=1$, $\dots$, $m$. ∎

######  Corollary 1.10.

We have $H(\Lambda_{1})=H(\Lambda_{2})$ if and only if there is an isomorphism
$\Pi\colon\mathbb{Z}_{2}^{r_{2}}\to\mathbb{Z}_{2}^{r_{1}}$ such that
$\Lambda_{1}=\Pi\circ\Lambda_{2}$.

######  Corollary 1.11.

Let $\Lambda$ be a vector-coloring of rank $r$ of a simple polytope $P$. Then
there is a bijection between the subgroups
$H^{\prime}\subset\mathbb{Z}_{2}^{r}$ and the subgroups in
$\mathbb{Z}_{2}^{m}$ containing $H(\Lambda)$ given by the correspondance
$H^{\prime}={\rm Ker\,}\Pi\to{\rm Ker\,}\Pi\circ\Lambda$ (or by the
isomorphism $\mathbb{Z}_{2}^{r}\simeq\mathbb{Z}_{2}^{m}/{\rm Ker}\,\Lambda$).
Moreover, $N(P,\Lambda)/H^{\prime}\simeq N(P,\Pi\circ\Lambda)$.

######  Corollary 1.12.

We have $H(\Lambda_{1})\subset H(\Lambda_{2})$ if and only if there is a
change of coordinates in $\mathbb{R}^{r_{1}}$ such that $\mathbb{R}^{r_{2}}$
corresponds to the first $r_{2}$ coordinates, and
$\Lambda_{1,i}=(\Lambda_{2,i},\beta_{i})$ for each $i=1,\dots,m$ and some
$\beta_{i}\in\mathbb{R}^{r_{1}-r_{2}}$.

###### Proof.

Indeed, we can choose a basis $e_{1}$, $\dots$, $e_{r_{1}}$ in
$\mathbb{Z}_{2}^{r_{1}}$ such that $\Pi(e_{1})$, $\dots$, $\Pi(e_{r_{2}})$ is
the standard basis in $\mathbb{R}^{r_{2}}$, and $e_{r_{2}+1}$, $\dots$,
$e_{r_{1}}$ is a basis in ${\rm Ker}\,\Pi$. We have
$\mathbb{Z}_{2}^{r_{1}}=\langle e_{1},\dots,e_{r_{2}}\rangle\oplus\langle
e_{r_{2}+1},\dots,e_{r_{1}}\rangle$, and in this basis $\Pi(a,b)=a$. ∎

The space $N(P,\Lambda)$ is a pseudomanifold, perhaps with a boundary. It is
glued from $2^{r}$ copies of $P$, any facet of each copy belongs to at most
two copies of $P$, and for any two copies $P\times a$ and $P\times b$ there is
a sequence of polytopes $P\times a_{i}$, $i=0$, $\dots$, $l$, such that
$a_{i_{0}}=a$, $a_{l}=b$, and $P\times a_{i}\cap P\times a_{i+1}$ contains a
facet of both polytopes. After several barycentric subdivisions this condition
translates to a standard definition of a pseudomanifold as a simplicial
complex. In particular, the notion of an orientation of the space
$N(P,\Lambda)$ is well-defined. The boundary of $N(P,\Lambda)$ is glued of
copies of facets $F_{i}$ of $P$ with $\Lambda_{i}=0$. The following result is
a generalization of the result from [NN05], which considers only small covers
and the result from [E22M], which considers only free actions.

######  Proposition 1.13.

Let the vectors $\Lambda_{j_{1}},\dots,\Lambda_{j_{r}}$ form a basis in
$\mathbb{Z}_{2}^{r}$. Then the pseudomanifold $N(P,\Lambda)$ is orientable if
and only if any nonzero $\Lambda_{i}$ is a linear combination of an odd number
of these vectors. Moreover, if $N(P,\Lambda)$ is orientable, then the action
of an element $\boldsymbol{x}\in\mathbb{Z}_{2}^{r}$ preserves its orientation
if and only if $\boldsymbol{x}$ is a linear combination of an even number of
the vectors $\Lambda_{j_{1}},\dots,\Lambda_{j_{r}}$.

###### Proof.

For $N(P,\Lambda)=P\times\mathbb{Z}_{2}^{r}/\sim$ to be orientable it is
necessary and sufficient that for any facet $F_{i}$ of an oriented polytope
$P$ such that $\Lambda_{i}\neq 0$ the polytope
$P\times(\boldsymbol{a}+\Lambda_{i})$, which is glued to
$P\times\boldsymbol{a}$ along this facet, has an opposite orientation.
Starting from $P\times a$ and using only facets $F_{j_{1}}$, $\dots$,
$F_{j_{r}}$ we can come from $P\times\boldsymbol{a}$ to any
$P\times\boldsymbol{b}$, $\boldsymbol{b}\in\mathbb{Z}_{2}^{r}$, which defines
uniquely the orientation of any polytope $P\times\boldsymbol{b}$. For these
orientations to be consistent it is necessary and sufficient that for any
facet $F_{i}$ with $\Lambda_{i}\neq 0$ the polytope
$P\times(\boldsymbol{a}+\Lambda_{i})$ is achieved in an odd number of steps,
which is equivalent to the fact that $\Lambda_{i}$ is a linear combination of
an odd number of vectors $\Lambda_{j_{l}}$. The element
$\boldsymbol{x}\in\mathbb{Z}_{2}^{r}$ moves the polytope
$P\times\boldsymbol{a}$ to $P\times(\boldsymbol{a}+\boldsymbol{x})$, so it
preserves the orientation if and only if $\boldsymbol{x}$ is a linear
combination of an even number of the vectors
$\Lambda_{j_{1}},\dots,\Lambda_{j_{r}}$. ∎

This condition can be reformulated in a more invariant form.

######  Corollary 1.14.

The pseudomanifold $N(P,\Lambda)$ is orientable if and only if there is a
linear function $\boldsymbol{c}\in(\mathbb{Z}_{2}^{r})^{*}$ such that
$\boldsymbol{c}\Lambda_{i}=1$ for all $i$ with $\Lambda_{i}\neq 0$. Moreover,
if $N(P,\Lambda)$ is orientable, then the action of an element
$\boldsymbol{x}\in\mathbb{Z}_{2}^{r}$ preserves its orientation if and only if
$\boldsymbol{c}\boldsymbol{x}=0$.

###### Proof.

Indeed, if there is such a function
$\boldsymbol{c}\in(\mathbb{Z}_{2}^{r})^{*}$, then for a basis
$\Lambda_{j_{1}},\dots,\Lambda_{j_{r}}$ $\boldsymbol{c}\Lambda_{j_{s}}=1$ for
all $s$, hence if
$\Lambda_{j}=u_{1}\Lambda_{j_{1}}+\dots+u_{r}\Lambda_{j_{r}}$, then
$\boldsymbol{c}\Lambda_{j}=u_{1}+\dots+u_{r}=1$, and the number of nonzero
elements $u_{s}$ is odd. On the other hand, if any vector $\Lambda_{j}$ is a
linear combination of an odd number of basis vectors, then the sum of all the
coordinates is the desired linear function. ∎

######  Remark 1.15.

We can consider the function $\boldsymbol{c}\in(\mathbb{Z}_{2}^{r})^{*}$ from
Corollary 1.14 as the first coordinate in $\mathbb{Z}_{2}^{r}$. Then
$\Lambda_{i}=(1,\lambda_{i})$ if $\Lambda_{i}\neq 0$. More on this
correspondence see in Section 4.

######  Corollary 1.16.

The pseudomanifold $N(P,\Lambda)$ is closed and orientable if and only
$H(\Lambda)\subset H_{0}$, that is $H(\Lambda)$ consists of orientation
preserving involutions. Moreover, if $N(P,\Lambda)$ is closed and orientable,
then the subgroup of the orientation-preserving involutions
$H_{0}^{\prime}\subset\mathbb{Z}_{2}^{r}$ corresponds to the subgroup
$H_{0}/{\rm Ker}\,\Lambda$ under the isomorphism
$\mathbb{Z}_{2}^{r}\simeq\mathbb{Z}_{2}^{m}/{\rm Ker}\,\Lambda$.

###### Proof.

The subgroup $H_{0}$ corresponds to the mapping $\Lambda_{0}(F_{i})=1$ for all
$i$. Thus, this is the direct corollary of Proposition 1.9 and Corollary 1.11.
∎

######  Corollary 1.17.

The pseudomanifold $N(P,\Lambda)/H^{\prime}$, where
$H^{\prime}\subset\mathbb{Z}_{2}^{r}$, is closed and orientable if and only
$N(P,\Lambda)$ is closed and orientable and $H^{\prime}\subset
H_{0}^{\prime}$, that is $H^{\prime}$ consists of orientation-preserving
involutions.

###### Proof.

Let $H^{\prime}={\rm Ker}\,\Pi$ for a surjection
$\pi\colon\mathbb{Z}_{2}^{r}\to\mathbb{Z}_{2}^{k}$. Then
$N(P,\Lambda)/H^{\prime}=N(P,\Pi\circ\Lambda)$ is closed and orientable if and
only if ${\rm Ker}\,\Pi\circ\Lambda\subset H_{0}$. This holds if and only if
${\rm Ker}\,\Lambda\subset H_{0}$ and $H^{\prime}\subset H_{0}^{\prime}$. ∎

######  Remark 1.18.

Corollaries 1.16 and 1.17 can be explained in another way. The pseudomanifold
$N(P,\Lambda)=\mathbb{R}\mathcal{Z}_{P}/H(\Lambda)$ of dimension $n$ is closed
and orientable if and only if $H_{n}(N(P,\Lambda),\mathbb{Q})=\mathbb{Q}$.
There is the following result connected with the notion of a transfer.

######  Theorem 1.19.

(See [B72, Theorem 2.4]) Let $G$ be a finite group acting on a simplicial
complex $K$ by simplicial homeomorphisms. Then for any field $\mathbb{F}$ of
characteristic $0$ or prime to $|G|$ the mapping $\pi_{*}\colon
H_{*}(|K|,\mathbb{F})\to H_{*}(|K|/G,\mathbb{F})$ induces the isomorphism

$H_{*}(|K|,\mathbb{F})^{G}\simeq H_{*}(|K|/G,\mathbb{F}),$

where $H_{*}(|K|,\mathbb{F})^{G}$ consists of homology classes invariant under
the action of any $g_{*}$, $g\in G$.

The action of $\mathbb{Z}_{2}^{m}$ on $\mathbb{R}\mathcal{Z}_{P}$ as well as
$\mathbb{Z}_{2}^{r}$ on $N(P,\Lambda)$ is simplicial with respect to the
structure of a simplicial complex arising from the barycentric subdivision of
$P$, hence for $H_{n}(N(P,\Lambda)/H^{\prime},\mathbb{Q})$ to be isomorphic to
$\mathbb{Q}$ it is necessary and sufficient that
$H_{n}(N(P,\Lambda),\mathbb{Q})\simeq\mathbb{Q}$ (that is, $N(P,\Lambda)$ is
closed and orientable) and
$H_{n}(N(P,\Lambda),\mathbb{Q})^{G}=H_{n}(N(P,\Lambda),\mathbb{Q})$ (that is,
any element of $G$ preserves the orientation).

##  2\. A complex $\mathcal{C}(P,c)$ defined by a coloring $c$

######  Construction 2.1.

Let us call a surjective mapping
$c\colon\\{F_{1},\dots,F_{m}\\}\to[l]=\\{1,\dots,l\\}$ a coloring of the
polytope $P$ in $l$ colors. For any coloring $c$ define a complex
$\mathcal{C}(P,c)\subset\partial P$ as follows. Its “facets” are connected
components of unions of all the facets of $P$ of the same color, “$k$-faces”
are connected components of intersections of $(n-k)$ different facets. By
definition each $k$-face is a union of $k$-faces of $P$. Choose a linear order
of all the facets $G_{1}$, $\dots$, $G_{M}$.

By an equivalence of two complexes $\mathcal{C}(P,c)$ and
$\mathcal{C}(Q,c^{\prime})$ we mean a homeomorphism $P\to Q$ sending facets of
$\mathcal{C}(P,c)$ to facets of $\mathcal{C}(Q,c^{\prime})$. If there is such
an equivalence, we call $\mathcal{C}(P,c)$ and $\mathcal{C}(Q,c^{\prime})$
equivalent.

Denote
$\mathbb{R}^{k}_{\geqslant}=\\{(y_{1},\dots,y_{k})\in\mathbb{R}^{k}\colon
y_{i}\geqslant 0\text{ for all }i\\}$. For a subset $\omega\subset[m]$ denote
$P_{\omega}=\bigcup_{i\in\omega}F_{i}$.

######  Lemma 2.2.

Let a point $p\in\partial P$ belong to exactly $l\geqslant 0$ facets
$G_{i_{1}}$, $\dots$, $G_{i_{l}}$ of $\mathcal{C}(P,c)$. Then there is a
piecewise linear homeomorphism $\varphi$ of a neighbourhood $U\subset P$ of
$p$ to a neighbourhood
$V\subset\mathbb{R}^{l}_{\geqslant}\times\mathbb{R}^{n-l}$ such that
$\varphi(G_{j_{s}}\cap U)=V\cap\\{y_{s}=0\\}$, $s=1,\dots,l$.

###### Proof.

Take the face $G(p)=\bigcap_{F_{i}\ni p}F_{i}=F_{j_{1}}\cap\dots\cap
F_{j_{k}}$. Since the distance from $p$ to any facet $F_{j}$, $p\notin F_{j}$,
is positive, there is a neighbourhood $U(p)\subset\mathbb{R}^{n}$ such that
$U(p)\cap P=U(p)\cap S(p)$, where

$S(p)=\\{x\in\mathbb{R}^{n}\colon\boldsymbol{a}_{j_{1}}\boldsymbol{x}+b_{j_{1}}\geqslant
0,\dots,\boldsymbol{a}_{j_{k}}\boldsymbol{x}+b_{j_{k}}\geqslant 0\\},$

and $\boldsymbol{a}_{j}\boldsymbol{x}+b_{j}\geqslant 0$ is the halfspace
defined by a facet $F_{j}$.

For any vertex $v\in G(p)$ there is an affine change of coordinates
$y_{j}=\boldsymbol{a}_{j}\boldsymbol{x}+b_{j}$: $F_{j}\ni v$. In these
coordinates

$S(p)=\\{y_{j_{1}}\geqslant
0\\}\times\dots\times\mathbb{\\{}y_{j_{k}}\geqslant
0\\}\times\mathbb{R}^{n-k}=\mathbb{R}^{k}_{\geqslant}\times\mathbb{R}^{n-k},$

where for the point $p$ we have $y_{j_{1}}=\dots=y_{j_{k}}=0$ and $y_{j}>0$
for all the other $j$.

Let $G_{i_{s}}=P_{\omega_{i_{s}}}$. We have a decomposition
$\\{j_{1},\dots,j_{k}\\}=\omega_{i_{1}}(p)\sqcup\dots\sqcup\omega_{i_{l}}(p)$,
where $\omega_{i_{s}}(p)=\omega_{i_{s}}\cap\\{j_{1},\dots,j_{k}\\}$. Set
$p_{i_{s}}=|\omega_{i_{s}}(p)|$. Then

$S(p)=\mathbb{R}^{p_{i_{1}}}_{\geqslant}\times\dots\mathbb{R}^{p_{i_{l}}}_{\geqslant}\times\mathbb{R}^{n-k}.$

Each $\mathbb{R}^{p}_{\geqslant}$ is piecewise linearly homeomorphic to
$\mathbb{R}^{p-1}\times\mathbb{R}_{\geqslant}$. Namely

$\mathbb{R}^{p}_{\geqslant}={\rm
cone}\,(\boldsymbol{e}_{1},\dots,\boldsymbol{e}_{p})=\bigcup\limits_{j=1}^{p}{\rm
cone}\,(\boldsymbol{e}_{1},\dots,\boldsymbol{e}_{j-1},\boldsymbol{e}_{1}+\dots+\boldsymbol{e}_{p},\boldsymbol{e}_{j+1},\dots\boldsymbol{e}_{p}).$

Then the mapping

$\boldsymbol{e}_{1}\to\boldsymbol{e}_{1},\dots,\boldsymbol{e}_{p-1}\to\boldsymbol{e}_{p-1},\boldsymbol{e}_{p}\to-\boldsymbol{e}_{1}-\dots-\boldsymbol{e}_{p-1},\boldsymbol{e}_{1}+\dots+\boldsymbol{e}_{p}\to\boldsymbol{e}_{p}$

defines a linear homeomorphism of each cone to its image and a piecewise
linear homeomorphism
$\mathbb{R}^{p}_{\geqslant}\simeq\mathbb{R}^{p-1}\times\mathbb{R}_{\geqslant}$.
It maps
$\partial\mathbb{R}^{p}_{\geqslant}=\mathbb{R}^{p}_{\geqslant}\cap\bigcup_{i=1}^{p}\\{y_{i}=0\\}$
to $\mathbb{R}^{p-1}$. Then we have a homeomorphism

$S(p)=\mathbb{R}^{p_{i_{1}}}_{\geqslant}\times\dots\mathbb{R}^{p_{i_{l}}}_{\geqslant}\times\mathbb{R}^{n-k}\simeq(\mathbb{R}^{p_{1}-1}\times\mathbb{R}_{\geqslant})\times\dots\times(\mathbb{R}^{p_{l}-1}\times\mathbb{R}_{\geqslant})\times\mathbb{R}^{n-k}\simeq\mathbb{R}^{l}_{\geqslant}\times\mathbb{R}^{n-l},$

which sends each set $G_{i_{s}}\cap S(p)$ to the corresponding hyperplane
$\\{y_{s}=0\\}$. ∎

######  Corollary 2.3.

Any set $P_{\omega}$, $\omega\neq\varnothing,[m]$ is a topological
$n$-manifold with a boundary.

###### Proof.

To prove this it is sufficient to consider a coloring
$c(F_{i})=\begin{cases}1,&i\in\omega\\\ 2,&i\notin\omega\end{cases}$. ∎

######  Corollary 2.4.

Each $k$-face of $\mathcal{C}(P,c)$ is a topological $k$-manifold, perhaps
with a boundary.

The proof is similar.

######  Remark 2.5.

It follows from Lemma 2.2 that the polytope $P$ with the complex
$\mathcal{C}(P,c)$ on its boundary has the structure of a manifold with facets
in the sense of [BP15, Definition 7.1.2].

######  Proposition 2.6.

Let $c$ be a coloring of a simplex $\Delta^{n}$ in $r$ colors. Then there is a
homeomorphism of $\Delta^{n}$ to the set

$S^{n}_{r,\geqslant}=\\{(x_{1},\dots,x_{n+1})\in\mathbb{R}^{n+1}\colon
x_{1}\geqslant 0,\dots,x_{r}\geqslant
0,x_{1}^{2}+\dots+x_{n+1}^{2}=1\\}\subset S^{n}$

such that each facet $G_{i}$ of $\mathcal{C}(\Delta^{n},c)$ is mapped to
$S^{n}_{r,\geqslant}\cap\\{x_{i}=0\\}$, $i=1,\dots,r$.

###### Proof.

We can use the same argument as in the proof of Lemma 2.2. First let us
realize $\Delta^{n}$ as a regular simplex in $\mathbb{R}^{n+1}$:

$\displaystyle\Delta^{n}\simeq\\{(x_{1},\dots,x_{n+1})\in\mathbb{R}^{n+1}\colon
x_{1}\geqslant 0,\dots,x_{n+1}\geqslant 0,x_{1}+\dots+x_{n+1}=1\\}$
$\displaystyle\simeq(\mathbb{R}^{n+1}_{\geqslant}\setminus\\{0\\})/(\boldsymbol{x}\sim
t\boldsymbol{x},t>0)\simeq S^{n+1}_{n+1,\geqslant}.$

Without loss of generality we can assume that

$c(F_{i})=\begin{cases}1,&1\leqslant i\leqslant p_{1},\\\ 2,&p_{1}+1\leqslant
i\leqslant p_{1}+p_{2},\\\ \dots\\\ r,&p_{1}+\dots+p_{r-1}+1\leqslant
i\leqslant n+1\end{cases}$

As in the proof of Lemma 2.2 we have a piecewise linear homeomorphism

$\mathbb{R}^{n+1}_{\geqslant}\simeq\mathbb{R}^{p_{1}}_{\geqslant}\times\dots\times\mathbb{R}^{p_{r}}_{\geqslant}\to(\mathbb{R}_{\geqslant}\times\mathbb{R}^{p_{1}-1})\times\dots\times(\mathbb{R}_{\geqslant}\times\mathbb{R}^{p_{r}-1})\simeq\mathbb{R}^{r}_{\geqslant}\times\mathbb{R}^{n+1-r}$

which sends rays $t\boldsymbol{x}$, $t>0$, to rays $t\boldsymbol{y}$, and each
set $\mathbb{R}^{n+1}_{\geqslant}\cap\\{x_{i}=0\\}$ to
$(\mathbb{R}^{r}_{\geqslant}\times\mathbb{R}^{n+1-r})\cap\\{x_{c(i)}=0\\}$.
Then

$\Delta^{n}\simeq(\mathbb{R}^{n+1}_{\geqslant}\setminus\\{0\\})/(\boldsymbol{x}\sim
t\boldsymbol{x},t>0)\simeq(\mathbb{R}^{r}_{\geqslant}\times\mathbb{R}^{n+1-r})/(\boldsymbol{x}\sim
t\boldsymbol{x},t>0)\simeq S^{n+1}_{r,\geqslant},$

and each facet $F_{i}$ of $\Delta^{n}$ is mapped to
$S^{n+1}_{r,\geqslant}\cap\\{x_{c(i)}\geqslant 0\\}$. ∎

######  Corollary 2.7.

The complexes $\mathcal{C}(\Delta^{n},c)$ and
$\mathcal{C}(\Delta^{n},c^{\prime})$ are equivalent if and only if the
colorings $c$ and $c^{\prime}$ have equal numbers of colors.

######  Definition 2.8.

We will denote $\mathcal{C}(n,r)$ the equivalence class of complexes
$\mathcal{C}(\Delta^{n},c)$ corresponding to $r$ colors.

######  Example 2.9.

For any face $G=F_{i_{1}}\cap\dots\cap F_{i_{k}}$ of $P$ of codimension
$k\geqslant 1$ consider the coloring

$c_{G}(F_{j})=\begin{cases}s,&\text{ if }j=i_{s},\\\ k+1,&\text{
otherwise}.\end{cases}$

######  Proposition 2.10.

The complex $\mathcal{C}(P,c_{G})$ is equivalent to $\mathcal{C}(n,k+1)$.

###### Proof.

A central projection from a point $p\in{\rm relint}\,G$ induces a
homeomorphism between $P$ and the set

$B^{n}_{k,\geqslant}=\\{(x_{1},\dots,x_{n})\in\mathbb{R}^{n}\colon
x_{1}\geqslant 0,\dots,x_{k}\geqslant 0,x_{1}^{2}+\dots+x_{n}^{2}\leqslant
1\\}$

such that each facet $F_{i_{s}}$ is mapped to the set
$B^{n}_{k,\geqslant}\cap\\{x_{s}=0\\}$, $s=1,\dots,k$, and all the other
facets are mapped to
$B^{n}_{k,\geqslant}\cap\\{x_{1}^{2}+\dots+x_{n}^{2}=1\\}$. Hence, the
complexes $\mathcal{C}(P,c_{G})$ and $\mathcal{C}(Q,c_{G^{\prime}})$ are
equivalent, if $P$ and $Q$ are simple $n$-polytopes and $\dim G=\dim
G^{\prime}$. In particular, $\mathcal{C}(P,c_{G})$ is equivalent to
$\mathcal{C}(\Delta^{n},c_{\Delta^{n-k}})=\mathcal{C}(n,k+1)$. ∎

######  Corollary 2.11.

There is a homeomorphism

(1) $S^{n}_{r+1,\geqslant}\simeq B^{n}_{r,\geqslant},$

where one of the facets $\\{x_{i}=0\\}$ of $S^{n}_{r+1,\geqslant}$ is mapped
to the facet $\\{x_{1}^{2}+\dots+x_{n}^{2}=1\\}$ of $B^{n}_{r,\geqslant}$.

##  3\. A weakly equivariant classification of spaces $N(P,\Lambda)$

######  Definition 3.1.

Two spaces $X$ and $Y$ with actions of $\mathbb{Z}_{2}^{r}$ are called weakly
equivariantly homeomorphic if there is a homeomorphism $\varphi\colon X\to Y$
and an automorphism $\psi\colon\mathbb{Z}_{2}^{r}\to\mathbb{Z}_{2}^{r}$ such
that
$\varphi(\boldsymbol{a}\cdot\boldsymbol{x})=\psi(\boldsymbol{a})\cdot\varphi(\boldsymbol{x})$
for any $\boldsymbol{x}\in X$ and $\boldsymbol{a}\in\mathbb{Z}_{2}^{r}$.

######  Definition 3.2.

Let $\Lambda_{P}$ and $\Lambda_{Q}$ be vector-colorings of rank $r$ of simple
$n$-polytopes $P$ and $Q$. We call the pairs $(P,\Lambda_{P})$ and
$(Q,\Lambda_{Q})$ equivalent, if there is an equivalence $\sigma$ between
$\mathcal{C}(P,\Lambda_{P})$ and $\mathcal{C}(Q,\Lambda_{Q})$ and a linear
isomorphism $A\colon\mathbb{Z}_{2}^{r}\to\mathbb{Z}_{2}^{r}$ such that
$\Lambda_{Q}(\sigma(G_{i}))=A\Lambda_{P}(G_{i})$ for all $i=1,\dots,M$.

The following result generalizes the corresponding fact for linearly
independent vector-colorings (see Proposition 1.8 in [DJ91] and Proposition
7.3.8 in [BP15]).

######  Proposition 3.3.

The spaces $N(P,\Lambda_{P})$ and $N(Q,\Lambda_{Q})$ are weakly equivariantly
homeomorphic if and only if the pairs $(P,\Lambda_{P})$ and $(Q,\Lambda_{Q})$
are equivalent.

###### Proof.

Let the pairs $(P,\Lambda_{P})$ and $(Q,\Lambda_{Q})$ be equivalent. We will
denote by $G_{i}$ the facets of $\mathcal{C}(P,\Lambda_{P})$, by
$G_{j}^{\prime}$ the facets of $\mathcal{C}(Q,\Lambda_{Q})$, by $j=\sigma(i)$
the index such that $\sigma(G_{i})=G_{j}^{\prime}$. Also denote
$\Lambda_{i}=\Lambda_{P}(G_{i})$ and
$\Lambda_{j}^{\prime}=\Lambda_{Q}(G_{j}^{\prime})$.

Define a homeomorphism $P\times\mathbb{Z}_{2}^{r}\to
Q\times\mathbb{Z}_{2}^{r}$ as
$(\boldsymbol{p},\boldsymbol{a})\to(\sigma(\boldsymbol{p}),A\boldsymbol{a})$.

If
$\boldsymbol{a}_{1}-\boldsymbol{a}_{2}=\sum\limits_{i\colon\boldsymbol{p}\in
F_{i}}\Lambda_{i}x_{i}$, then

$A\boldsymbol{a}_{1}-A\boldsymbol{a}_{2}=\sum\limits_{i\colon\boldsymbol{p}\in
F_{i}}(A\Lambda_{i})x_{i}=\sum\limits_{i\colon\boldsymbol{p}\in
G_{i}}(A\Lambda_{i})\sum\limits_{k\colon\boldsymbol{p}\in F_{k}\subset
G_{i}}x_{k}=\sum\limits_{i\colon\boldsymbol{p}\in
G_{i}}(A\Lambda_{i})\widetilde{x}_{i}=\\\
\sum\limits_{i\colon\boldsymbol{p}\in
G_{i}}(\Lambda_{\sigma(i)}^{\prime})\widetilde{x}_{i}=\sum\limits_{j\colon\sigma(\boldsymbol{p})\in
G_{j}^{\prime}}\Lambda_{j}^{\prime}\widetilde{x}_{\sigma^{-1}(j)}=\sum\limits_{j\colon\sigma(\boldsymbol{p})\in
F_{k}^{\prime}}\Lambda_{k}^{\prime}x_{k}^{\prime}\text{ for some
}x_{k}^{\prime}\in\mathbb{Z}_{2}.$

Thus, the mapping preserves the equivalence classes, and we obtain the
homeomorphism $\varphi\colon N(P,\Lambda_{P})\to N(Q,\Lambda_{Q})$. Moreover,

$\displaystyle\varphi\left(\boldsymbol{a}\cdot\left[\boldsymbol{p},\boldsymbol{b}\right]\right)=\varphi\left[\boldsymbol{p},\boldsymbol{a}+\boldsymbol{b}\right]=\left[\sigma(\boldsymbol{p}),A\left(\boldsymbol{a}+\boldsymbol{b}\right)\right]=$
$\displaystyle\left[\sigma(\boldsymbol{p}),A\boldsymbol{a}+A\boldsymbol{b}\right]=(A\boldsymbol{a})\cdot\left[\sigma(\boldsymbol{p}),A\boldsymbol{b}\right]=(A\boldsymbol{a})\cdot\varphi\left[\boldsymbol{p},\boldsymbol{b}\right]$

Thus, $\varphi$ is a weakly equivariant homeomorphism.

Now assume that there is a weakly equivariant homeomorphism $\varphi\colon
N(P,\Lambda_{P})\to N(Q,\Lambda_{Q})$. Then there is $A\in
Gl_{r}(\mathbb{Z}_{2})$ such that
$\varphi(\boldsymbol{a}\cdot[\boldsymbol{p},\boldsymbol{b}])=(A\boldsymbol{a})\cdot\varphi[\boldsymbol{p},\boldsymbol{b}]$
for all $\boldsymbol{p}\in P$ and
$\boldsymbol{a},\boldsymbol{b}\in\mathbb{Z}_{2}^{r}$. Since $\varphi$ is
weakly equivariant, it induces a homeomorphism of orbit spaces
$\widehat{\varphi}\colon P\to Q$, where $\widehat{\varphi}(\partial
P)=\partial Q$. Moreover, the points in $N(P,\Lambda_{P})$ with a stabilizer
$H\subset\mathbb{Z}_{2}^{r}$ are mapped by $\varphi$ to the points in
$N(Q,\Lambda_{Q})$ with the stabilizer $A(H)$. For a facet $G_{i}$ of
$\mathcal{C}(P,\Lambda_{P})$ define its relative interior ${\rm
relint}\,G_{i}$ to be the interior of $G_{i}$ as a subset of $\partial P$.
Then the points over ${\rm relint}\,G_{i}$ have the stabilizer
$\langle\Lambda_{i}\rangle$ and are mapped to the points over relative
interiors of the facets $G_{j_{1}}^{\prime}$, $\dots$, $G_{j_{l}}^{\prime}$ of
$\mathcal{C}(Q,\Lambda_{Q})$ with the stabilizer $\langle
A\Lambda_{i}\rangle$. Since ${\rm relint}\,G_{i}$ is path-connected and each
${\rm relint}\,G_{j_{s}}^{\prime}$ is a connected component of
$\bigcup_{s}{\rm relint}\,G_{j_{s}}^{\prime}$ (because $G_{j_{s}}^{\prime}\cap
G_{j_{t}}^{\prime}=\varnothing$ for $s\neq t$), we have
$\widehat{\varphi}({\rm relint}\,G_{i})={\rm relint}\,G_{j_{s}}^{\prime}$ for
a single facet $G_{j_{s}}^{\prime}$. Also $\widehat{\varphi}(\partial
G_{i})=\partial G_{j_{s}}^{\prime}$, since $\widehat{\varphi}$ is continuous.
Thus, $\widehat{\varphi}$ is an equivalence between
$\mathcal{C}(P,\Lambda_{P})$ and $\mathcal{C}(Q,\Lambda_{Q})$ such that
$A\Lambda_{P}(G_{i})=\Lambda_{Q}(\sigma(G_{i}))$. The proof is finished. ∎

##  4\. A weakly equivariant classification of spaces defined by affine
colorings

Remark 1.15 leads to the following definition.

######  Definition 4.1.

We call a mapping $\lambda\colon\\{F_{1},\dots,F_{m}\\}\to\mathbb{Z}_{2}^{r}$
such that the images $\lambda_{j}$ of the facets $F_{j}$ affinely span
$\mathbb{Z}_{2}^{r}$ an affine coloring of rank $r$. If, additionally,

$(**)\quad\text{for any face $F_{i_{1}}\cap\dots\cap F_{i_{k}}$ of $P$ the
points $\lambda_{i_{1}}$, $\dots$, $\lambda_{i_{k}}$ are affinely
independent}$

we call $\lambda$ an affinely independent coloring.

######  Definition 4.2.

Let $\lambda$ be an affine coloring of a simple $n$-polytope $P$. Define
$\Lambda_{i}=(1,\lambda_{i})\in\mathbb{Z}_{2}^{r+1}$. We call the space
$N(P,\lambda)=N(P,\Lambda)$ a space defined by an affine coloring $\lambda$.
Set $H(\lambda)=H(\Lambda)$.

By definition $N(P,\lambda)$ is a closed orientable pseudomanifold and any
closed orientable pseudomanifold $N(P,\Lambda)$ has this form. There is a
canonical action of $\mathbb{Z}_{2}^{r+1}$ on $N(P,\lambda)$, and the subgroup
of orientation-preserving involutions is

$H_{0}^{\prime}=\mathbb{Z}_{2}^{r}=\\{(x_{0},\dots,x_{r})\in\mathbb{Z}_{2}^{r+1}\colon
x_{0}=0\\}.$

This subgroup can be considered as a vector space associated to the affine
space $\mathbb{Z}_{2}^{r}$ containing the points $\lambda_{1}$, $\dots$,
$\lambda_{m}$.

The following results follow from Proposition 1.9 and Corollary 1.17.

######  Corollary 4.3.

We have $H(\lambda_{1})\subset H(\lambda_{2})$ if and only if there is an
affine surjection $\widehat{\Pi}\colon\mathbb{Z}^{r_{1}}\to\mathbb{Z}^{r_{2}}$
such that $\lambda_{2}=\widehat{\Pi}\circ\lambda_{1}$. In this case
$N(P,\lambda_{2})=N(P,\lambda_{1})/H^{\prime}$, where $H^{\prime}\simeq
H(\lambda_{1})/H(\lambda_{2})$.

######  Corollary 4.4.

For a subgroup $H^{\prime}\subset\mathbb{Z}_{2}^{r+1}$ the space
$N(P,\lambda)/H^{\prime}$ is a closed orientable pseudomanifold if and only if
$H^{\prime}\subset\mathbb{Z}_{2}^{r}=H_{0}^{\prime}$. In this case
$N(P,\lambda)/H^{\prime}=N(P,\widehat{\Pi}\circ\lambda)$, where
$\widehat{\Pi}\colon\mathbb{Z}_{2}^{r}\to\mathbb{Z}_{2}^{r}/H^{\prime}$ is an
affine surjection.

######  Corollary 4.5.

For an affine coloring $\lambda$ of rank $r$ of a simple $n$-polytope $P$ the
subgroups $H\colon H(\lambda)\subset H\subset H_{0}\subset\mathbb{Z}_{2}^{m}$
are in bijection with

  * •

affine surjections
$\widehat{\Pi}\colon\mathbb{Z}_{2}^{r}\to\mathbb{Z}_{2}^{l}$ defined up to
affine changes of coordinates in $\mathbb{Z}_{2}^{l}$;

  * •

affine colorings $\lambda^{\prime}$ of rank $l$ of the form
$\lambda^{\prime}=\widehat{\Pi}\circ\lambda$ defined up to affine changes of
coordinates in $\mathbb{Z}_{2}^{l}$;

  * •

subgroups
$H^{\prime}\subset\mathbb{Z}_{2}^{r}=H_{0}^{\prime}\subset\mathbb{Z}_{2}^{r+1}$
of involutions preserving the orientation of $N(P,\lambda)$.

The correspondence between the projections and the subgroups is given as

$H^{\prime}\to\left[\mathbb{Z}_{2}^{r}\to\mathbb{Z}_{2}^{r}/H^{\prime}\simeq\mathbb{Z}_{2}^{l}\right],\quad\left[A\boldsymbol{x}+\boldsymbol{b}\colon\mathbb{Z}_{2}^{r}\to\mathbb{Z}_{2}^{l}\right]\to{\rm\mathrm{Ker}\,A}.$

######  Definition 4.6.

Let $\lambda_{P}$ and $\lambda_{Q}$ be affine colorings of rank $r$ of simple
$n$-polytopes $P$ and $Q$. We call the pairs $(P,\lambda_{P})$ and
$(Q,\lambda_{Q})$ equivalent, if there is an equivalence $\sigma$ between
$\mathcal{C}(P,\lambda_{P})$ and $\mathcal{C}(Q,\lambda_{Q})$ and an affine
isomorphism $\mathcal{A}\colon\mathbb{Z}_{2}^{r}\to\mathbb{Z}_{2}^{r}$ such
that $\lambda_{Q}(\sigma(G_{i}))=\mathcal{A}\lambda_{P}(G_{i})$ for all
$i=1,\dots,M$.

######  Corollary 4.7.

The spaces $N(P,\lambda_{P})$ and $N(Q,\lambda_{Q})$ are weakly equivariantly
homeomorphic if and only if the pairs $(P,\lambda_{P})$ and $(Q,\lambda_{Q})$
are equivalent.

###### Proof.

Indeed, linear isomorphism $\mathbb{Z}_{2}^{r+1}\to\mathbb{Z}_{2}^{r+1}$ such
that the vectors $(1,\lambda_{i})$ spanning $\mathbb{Z}_{2}^{r+1}$ are mapped
to vectors $(1,\lambda_{j}^{\prime})$ have the form
$(1,\boldsymbol{x})\to(1,C\boldsymbol{x}+\boldsymbol{b})$, where $\det C=1$,
that is they correspond to affine isomorphism
$\mathbb{Z}_{2}^{r}\to\mathbb{Z}_{2}^{r}$. ∎

##  5\. A criterion when $N(P,\Lambda)$ is a manifold

######  Theorem 5.1.

The space $N(P,\Lambda)$ defined by a vector-coloring $\Lambda$ of a rank $r$
of a simple $n$-polytope $P$ is a closed topological manifold if and only if
all the vectors $\Lambda_{i}$ are nonzero and for any vertex
$v=F_{i_{1}}\cap\dots\cap F_{i_{n}}$ all the different vectors among
$\\{\Lambda_{i_{1}},\dots,\Lambda_{i_{n}}\\}$ are linearly independent. It is
a topological manifold with a boundary if and only if $\Lambda_{j}=0$ for some
$j$, and for any vertex $v$ all the nonzero different vectors among
$\\{\Lambda_{i_{1}},\dots,\Lambda_{i_{n}}\\}$ are linearly independent.

######  Remark 5.2.

Theorem 5.1 can be extracted from general results by A.V. Mikhailova [M85] and
C. Lange [L19]. Nevertheless, we give a short self-sufficient proof here. For
$r=m-n+1$ Theorem 5.1 also follows from results of [G23].

######  Example 5.3.

In the case of $3$-polytopes the first condition means that at each vertex
$v=F_{i}\cap F_{j}\cap F_{k}$ either $\Lambda_{i}=\Lambda_{j}=\Lambda_{k}$, or
for a relabelling $\Lambda_{i}\neq\Lambda_{j}$ and
$\Lambda_{k}\in\\{\Lambda_{i},\Lambda_{j}\\}$, or the vectors $\Lambda_{i}$,
$\Lambda_{j}$, and $\Lambda_{k}$ are linearly independent.

######  Corollary 5.4.

The space $N(P,\Lambda)$ defined by a vector-coloring $\Lambda$ is a closed
topological manifold if and only if $\Lambda$ induces a linearly independent
coloring of the complex $\mathcal{C}(P,\Lambda)$.

######  Corollary 5.5.

The space $N(P,\lambda)$ defined by an affine coloring $\lambda$ is a closed
orientable topological manifold if and only if $\lambda$ induces an affinely
independent coloring of the complex $\mathcal{C}(P,\lambda)$.

###### Proof of Theorem 5.1.

Consider the complex $\mathcal{C}(P,\Lambda)$. By construction the mapping
$\Lambda$ induces the vector-coloring of its facets $G_{1},\dots,G_{M}$. We
have

(2) $N(P,\Lambda)=P\times\mathbb{Z}_{2}^{r}/\sim,\text{ where
}(p,a)\sim(q,b)\text{ if and only if }p=q\text{ and
}a-b\in\langle\Lambda_{i}\colon p\in G_{i}\rangle.$

If at each vertex $v=F_{i_{1}}\cap\dots\cap F_{i_{n}}$ all the different
vectors among $\\{\Lambda_{i_{1}},\dots,\Lambda_{i_{n}}\\}$ are linearly
independent, then for each point $p\in\partial P$, which belongs to exactly
$l$ facets $G_{i_{1}}$, $\dots$, $G_{i_{l}}$, the vectors
$\Lambda_{i_{1}},\dots,\Lambda_{i_{l}}$ are linearly independent. By Lemma 2.2
$p$ has a neighbourhood in $P$ homeomorphic to
$\mathbb{R}^{l}_{\geqslant}\times\mathbb{R}^{n-l}$. Then in $N(P,\Lambda)$ for
the point $p\times a$ these neighbourhoods are glued in the neighbourhood
homeomorphic to $\mathbb{R}^{l}\times\mathbb{R}^{n-l}$. Indeed, in $p\times a$
the copies
$P\times(a+\varepsilon_{1}\Lambda_{i_{1}}+\dots+\varepsilon_{l}\Lambda_{i_{l}})$,
$\varepsilon_{s}=\pm 1$, are glued locally as the sets
$\\{\varepsilon_{1}y_{1}\geqslant 0,\dots,\varepsilon_{l}y_{l}\geqslant 0\\}$,
where the addition of the vector $\Lambda_{i_{s}}$ corresponds to the
operation $y_{s}\to-y_{s}$. Hence, $N(P,\Lambda)$ is a closed topological
manifold.

On the other hand, if $\Lambda_{j}=0$ for some $j$ but at each vertex
$v=F_{i_{1}}\cap\dots\cap F_{i_{n}}$ all the nonzero different vectors among
$\\{\Lambda_{i_{1}},\dots,\Lambda_{i_{n}}\\}$ are linearly independent, then
for the the points $p$ lying in the facets $G_{j}$ with $\Lambda_{j}=0$ the
neighbourhoods of the form $\mathbb{R}^{l}_{\geqslant}\times\mathbb{R}^{n-l}$
are glued to $\mathbb{R}_{\geqslant}\times\mathbb{R}^{n-1}$, where the
coordinate $y_{s}\geqslant 0$ corresponds to the facet $G_{j}$. Thus,
$N(P,\Lambda)$ is topological manifold with a boundary glued from copies of
the facets $G_{j}$ with $\Lambda_{j}=0$.

Now assume that at some vertex $v=F_{i_{1}}\cap\dots\cap F_{i_{n}}$ we have
$\Lambda_{j_{k}}=\Lambda_{j_{1}}+\dots+\Lambda_{j_{k-1}}$ for
$\\{j_{1},\dots,j_{k}\\}\subset\\{i_{1},\dots,i_{n}\\}$ and all the vectors
$\Lambda_{j_{1}}$, $\dots$, $\Lambda_{j_{k}}$ are nonzero and different (in
particular, $k\geqslant 3$). Moreover, assume that $k$ is minimal. In
particular, the vectors $\Lambda_{j_{1}},\dots,\Lambda_{j_{k-1}}$ are linearly
independent. Consider a point $p$ such that $G_{j_{1}}$, $\dots$, $G_{j_{k}}$
are exactly the facets containing this point. Such a point exists by Lemma 2.2
applied to the point $v$. Also by this lemma some neighbourhood of $p$ in $P$
is homeomorphic to $\mathbb{R}^{k}_{\geqslant}\times\mathbb{R}^{n-k}$, and the
facets $G_{j_{s}}$ are mapped to the hyperplanes $y_{s}=0$. Then for the space
$N(P,\Lambda)$ in the point $p\times a$ the copies $(P\setminus
G_{j_{k}})\times(a+\varepsilon_{1}\Lambda_{j_{1}}+\dots+\varepsilon_{k-1}\Lambda_{j_{k-1}})$,
$\varepsilon_{s}=\pm 1$, are glued locally as the sets
$\\{\varepsilon_{1}y_{1}\geqslant 0,\dots,\varepsilon_{k-1}y_{k-1}\geqslant
0,y_{k}>0\\}$ and form
$\mathbb{R}^{k-1}\times\mathbb{R}_{>}\times\mathbb{R}^{n-k}$, where the
addition of the vector $\Lambda_{j_{s}}$ corresponds to the operation
$y_{s}\to-y_{s}$. The points in $G_{j_{k}}\subset P$ correspond to the points
in $\mathbb{R}^{k-1}\times\\{0\\}\times\mathbb{R}^{n-k}$. In $N(P,\Lambda)$
for these points we have the additional identification
$(x,a)\sim(x,a+\Lambda_{j_{k}})=(x,a+\Lambda_{j_{1}}+\dots+\Lambda_{j_{k-1}})$.
This means that the point $(y_{1},\dots,y_{k-1},0,y_{k+1},\dots,y_{n})$ is
identified with $(-y_{1},\dots,-y_{k-1},0,y_{k+1},\dots,y_{n})$. Equivalently,
the copies of $\mathbb{R}^{k}_{\geqslant}\times\mathbb{R}^{n-k}$ are glued to
the space $\mathbb{R}^{n}/\sim$, where
$(y_{1},\dots,y_{k},y_{k+1},\dots,y_{n})\sim(-y_{1},\dots,-y_{k},y_{k+1},\dots,y_{n})$,
and the point $p\times a$ corresponds to the equivalence class
$[\boldsymbol{y}_{0}]$ of some point
$\boldsymbol{y}_{0}=(0,\dots,0,y_{k+1}^{0},\dots,y_{n}^{0})$. In
$\mathbb{R}^{n}$ the point $\boldsymbol{y}_{0}$ has a ball neighbourhood $B$
of radius $\varepsilon$ with the boundary sphere $S^{n-1}$ homeomorphic to the
join

$S^{k-1}*S^{n-k-1}=S^{k-1}\times
S^{n-k-1}\times[0,1]/(a_{1},b,0)\sim(a_{2},b,0),\,(a,b_{1},1)\sim(a,b_{2},1)$

via the mapping $S^{k-1}*S^{n-k-1}\to
S^{n-1}\colon(a,b,t)\to(\sqrt{t}a,\sqrt{1-t}b)$. There is a homeomorphism
$B\simeq CS^{n-1}\simeq C(S^{k-1}*S^{n-k-1})$, where $CX$ is the cone over
$X$. In $\mathbb{R}^{n}/\sim$ this gives a neighbourhood homeomorphic to
$C(\mathbb{R}P^{k-1}*S^{n-k-1})=C\Sigma^{n-k}\mathbb{R}P^{k-1}$, where $\Sigma
X$ is a suspension over $X$. Then

$H_{i}(N(P,\Lambda),N(P,\Lambda)\setminus[p\times a])\simeq
H_{i}(C\Sigma^{n-k}\mathbb{R}P^{k-1},C\Sigma^{n-k}\mathbb{R}P^{k-1}\setminus\text{apex})\simeq\\\
H_{i}(C\Sigma^{n-k}\mathbb{R}P^{k-1},\Sigma^{n-k}\mathbb{R}P^{k-1})\simeq\widetilde{H}_{i-1}(\Sigma^{n-k}\mathbb{R}P^{k-1})\simeq\widetilde{H}_{i+k-n-1}(\mathbb{R}P^{k-1}).$

In particular, for $k\geqslant 3$ we have
$H_{n+2-k}(N(P,\Lambda),N(P,\Lambda)\setminus[p\times a])=\mathbb{Z}_{2}$, and
$N(P,\Lambda)$ is not a manifold. ∎

######  Corollary 5.6.

For any affine coloring of a simple $3$-polytope $P$ the space $N(P,\lambda)$
is a closed orientable manifold.

###### Proof.

This follows from the fact that any two or three different points in
$\mathbb{Z}_{2}^{r}$ are affinely independent. ∎

######  Corollary 5.7.

Let $e_{1},\dots,e_{r}$ be a basis in $\mathbb{Z}_{2}^{r}$. Then for any
mapping
$\Lambda\colon\\{F_{1},\dots,F_{m}\\}\to\\{e_{1},\dots,e_{r},e_{1}+\dots+e_{r}\\}$
the space $N(P,\Lambda)$ is a closed topological manifold. Moreover, for odd
$r$ it is orientable.

######  Construction 5.8.

Let $P$ be a simple $n$-polytope and $\lambda$ be its affine coloring of rank
$r$. If the complex $\mathcal{C}(P,\lambda)$ is equivalent to
$\mathcal{C}(n,r+1)$ then the induced coloring is affinely independent, the
polytope is homeomorphic to $S^{n}_{r+1,\geqslant}$, and the manifold
$N(P,\lambda)$ is homeomorphic to $S^{n}$ glued from $2^{r+1}$ copies of
$S^{n}_{r+1,\geqslant}$.

######  Example 5.9.

Examples for Construction 5.8 are provided by Example 2.9. Each face
$G=F_{i_{1}}\cap\dots\cap F_{i_{k}}$ corresponds to an affine coloring

$\lambda_{i}=\begin{cases}\boldsymbol{e}_{s},&\text{ if
}i=i_{s},s=1,\dots,k,\\\ \boldsymbol{0},&\text{ otherwise},\end{cases}$

where $\boldsymbol{e}_{1}=(1,0,\dots,0)$, $\dots$,
$\boldsymbol{e}_{k}=(0,\dots,0,1)\in\mathbb{Z}_{2}^{k}$. Then the subgroup
$H_{G}=H(\lambda)$ of rank $m-k-1$ is defined in $\mathbb{Z}_{2}^{m}$ by the
equations $x_{i_{1}}=0$, $\dots$, $x_{i_{k}}=0$, and $x_{1}+\dots+x_{m}=0$.
This is the intersection of the subgroup $H_{0}$ consisting of all the
orientation preserving involutions with the coordinate subgroup corresponding
to $G$. We have $\mathbb{R}\mathcal{Z}_{P}/H_{G}\simeq S^{n}$.

In particular, each vertex $v\in P$ corresponds to a subgroup $H_{v}$ of rank
$m-n-1$ such that $\mathbb{R}\mathcal{Z}_{P}/H_{v}\simeq S^{n}$. The
particular case of this construction was presented in [G19]. This corresponds
to the case when $P=\Delta^{n_{1}}\times\dots\times\Delta^{n_{k}}$ and $v$ is
any vertex. We obtain an action of $\mathbb{Z}_{2}^{k-1}$ on
$S^{n_{1}}\times\dots\times S^{n_{k}}$ with the orbit space
$S^{n_{1}+\dots+n_{k}}$.

######  Conjecture 5.10.

The space $N(P,\lambda)$ corresponding to an affine coloring $\lambda$ of rank
$r$ of a simple $n$-polytope $P$ is homeomorphic to $S^{n}$ if and only if
$\mathcal{C}(P,\lambda)\simeq\mathcal{C}(n,r+1)$.

######  Example 5.11.

In dimension $n=1$ we have $P=I^{1}=\Delta^{1}$ and the conjecture is valid.

In dimension $n=2$ the complex $\mathcal{C}(P_{m},\lambda)$ corresponding to
an $m$-gon is equivalent ether to $\mathcal{C}(2,1)$, or to
$\mathcal{C}(2,2)$, or to a complex $\mathcal{C}(P_{l},\lambda^{\prime})$
corresponding to an affinely independent coloring of an $l$-gon $P_{l}$,
$l\geqslant 3$. In the latter case
$N(P_{m},\lambda)=N(P_{l},\lambda^{\prime})$ is a sphere with $g$ handles,
where $\chi(N(P_{l},\lambda^{\prime}))=2-2g=2^{r-1}l-2^{r}l+2^{r+1}$.
Therefore, $g=1+2^{r-2}(l-4)$ and $N(P_{m},\lambda)\not\simeq S^{2}$ for
$l>3$. Thus, the conjecture is valid.

As we will see in Section 10 the conjecture is also valid in dimension $n=3$.

Now we will prove a fact about skeletons of the complexes
$\mathcal{C}(P,\Lambda)$ and $\mathcal{C}(P,\Pi\circ\Lambda)$ which we will
need below.

######  Proposition 5.12.

Let $\Lambda$ be a vector-coloring of rank $r$ of a simple $n$-polytope $P$
such that $N(P,\Lambda)$ is a manifold, and
$H^{\prime}\subset\mathbb{Z}_{2}^{r}$ be a subgroup of rank $k$ corresponding
to a vector-coloring $\Lambda^{\prime}=\Pi\circ\Lambda$, where
$\Pi\colon\mathbb{Z}_{2}^{r}\to\mathbb{Z}_{2}^{r}/H^{\prime}\simeq\mathbb{Z}_{2}^{r-k}$
is the canonical projection. Then any $q$-skeleton
$\mathcal{C}^{q}(P,\Lambda)$ belongs to the $(q+k)$-skeleton
$C^{q+k}(P,\Lambda^{\prime})$.

###### Proof.

Consider a point $\boldsymbol{x}\in\mathcal{C}^{q}(P,\Lambda)$. It lies in the
intersection of $(n-q)$ facets $G_{i_{1}}$, $\dots$, $G_{i_{n-q}}$. Let
$F_{j_{1}}\cap\dots\cap F_{j_{l}}$ be the minimal face of $P$ containing
$\boldsymbol{x}$. Then
$\\{\Lambda(F_{j_{1}}),\dots,\Lambda(F_{j_{l}})\\}=\\{\Lambda(G_{i_{1}}),\dots,\Lambda(G_{i_{n-q}})\\}$
and the latter set of vectors in linearly independent. If the set
$\\{\Lambda^{\prime}(G_{i_{1}}),\dots,\Lambda^{\prime}(G_{i_{n-q}})\\}$
consists of $n-s$ different vectors, then
$\boldsymbol{x}\in\mathcal{C}^{s}(P,\Lambda^{\prime})$. We have

$n-s\geqslant\dim\langle\Lambda^{\prime}(G_{i_{1}}),\dots,\Lambda^{\prime}(G_{i_{n-q}})\rangle=\\\
=\dim\langle\Lambda(G_{i_{1}}),\dots,\Lambda(G_{i_{n-q}})\rangle-\dim{\rm
Ker}\,\Pi\left.\right|_{\langle\Lambda(G_{i_{1}}),\dots,\Lambda(G_{i_{n-q}})\rangle}\geqslant\\\
\geqslant\dim\langle\Lambda(G_{i_{1}}),\dots,\Lambda(G_{i_{n-q}})\rangle-\dim{\rm
Ker}\,\Pi=n-q-k.$

Thus, $s\leqslant q+k$ and
$\boldsymbol{x}\in\mathcal{C}^{q+k}(P,\Lambda^{\prime})$. ∎

######  Corollary 5.13.

Let $\Lambda$ be a vector-coloring of rank $r$ of a simple $n$-polytope $P$
such that $N(P,\Lambda)$ is a manifold, and $\tau\subset\mathbb{Z}_{2}^{r}$ be
an involution. Then any vertex of $\mathcal{C}(P,\Lambda)$ is either a vertex
of $C(P,\Lambda_{\tau})$ or belongs to its $1$-face, where
$\Lambda_{\tau}=\Pi\circ\Lambda$, and
$\Pi\colon\mathbb{Z}_{2}^{r}\to\mathbb{Z}_{2}^{r}/\langle\tau\rangle\simeq\mathbb{Z}_{2}^{r-1}$
is the canonical projection.

##  6\. Manifolds with torus actions

Results obtained in Section 5 can be generalized to actions of compact torus
$\mathbb{T}^{m}=(S^{1})^{m}$ instead of $\mathbb{Z}_{2}^{m}$. Namely, let us
identify $S^{1}$ with $\mathbb{R}/\mathbb{Z}$ and $\mathbb{T}^{r}$ with
$\mathbb{R}^{r}/\mathbb{Z}^{r}$. Then for a mapping
$\Lambda\colon\\{F_{1},\dots,F_{m}\\}\to\mathbb{Z}^{r}$ such that
$\langle\Lambda_{1},\dots\Lambda_{m}\rangle=\mathbb{Z}^{r}$ one can define a
space

$M(P,\Lambda)=P\times\mathbb{T}^{r}/\sim,$

where
$(\boldsymbol{p}_{1},\boldsymbol{t}_{1})\sim(\boldsymbol{p}_{2},\boldsymbol{t}_{2})$
if and only if $\boldsymbol{p}_{1}=\boldsymbol{p}_{2}$ and
$\boldsymbol{t}_{1}-\boldsymbol{t}_{2}\in\left\\{\sum\limits_{i\colon\boldsymbol{p}_{1}\in
F_{i}}\Lambda_{i}\varphi_{i},\varphi_{i}\in\mathbb{R}/\mathbb{Z}\right\\}$.

We will call the mapping $\Lambda$ an integer vector-coloring of rank $r$.

The space $M(P,\Lambda)$ has a canonical action of $\mathbb{T}^{r}$ and
$M(P,\Lambda)/\mathbb{T}^{r}=P$.

When $\Lambda$ has an additional property

(3) $\\{\Lambda_{i_{1}},\dots,\Lambda_{i_{k}}\\}\text{ is a part of some basis
in }\mathbb{Z}^{r}\text{ if }F_{i_{1}}\cap\dots\cap F_{i_{k}}\neq\varnothing,$

then it is known that $M(P,\Lambda)$ is a topological (even smooth) manifold
obtained as an orbit space of a free action of the group

$H(\Lambda)=\\{(\varphi_{1},\dots,\varphi_{m})\in\mathbb{T}^{m}\colon\Lambda_{1}\varphi_{1}+\dots+\Lambda_{m}\varphi_{m}=\boldsymbol{0}\\}\simeq\mathbb{T}^{m-r}$

on the moment-angle manifold $\mathcal{Z}_{P}=M(P,E)$,
$E(F_{i})=\boldsymbol{e}_{i}$, where $\boldsymbol{e}_{1}$, $\dots$,
$\boldsymbol{e}_{m}$ is the standard basis in $\mathbb{Z}^{m}$ (see [DJ91,
BP15]). We have the following generalization.

######  Proposition 6.1.

Let $P$ be a simple $n$-polytope and
$\Lambda\colon\\{F_{1},\dots,F_{m}\\}\to\mathbb{Z}^{r}\setminus\\{\boldsymbol{0}\\}$
be an integer vector-coloring of rank $r$ such that for any vertex
$v=F_{i_{1}}\cap\dots\cap F_{i_{n}}$ all the different vectors among
$\\{\Lambda_{i_{1}},\dots,\Lambda_{i_{n}}\\}$ form a part of a basis in
$\mathbb{Z}^{r}$. Then $M(P,\Lambda)$ is a closed topological
$(n+r)$-manifold.

###### Proof.

Consider the complex $\mathcal{C}(P,\Lambda)$. There is an induced mapping
$\Lambda$ for the set of its facets $G_{1},\dots,G_{M}$. For each point
$p\in\partial P$, which belongs to exactly $l$ facets $G_{i_{1}}$, $\dots$,
$G_{i_{l}}$, the vectors $\Lambda_{i_{1}},\dots,\Lambda_{i_{l}}$ form a part
of a basin in $\mathbb{Z}^{r}$. By Lemma 2.2 $p$ has a neighbourhood in $P$
homeomorphic to $\mathbb{R}^{l}_{\geqslant}\times\mathbb{R}^{n-l}$. The open
set in $M(P,\Lambda)$ over this neighbourhood is homeomorphic to

$\mathbb{R}^{l}_{\geqslant}\times\mathbb{R}^{n-l}\times\mathbb{T}^{l}\times\mathbb{T}^{r-l}/\sim\simeq\mathbb{C}^{l}\times\mathbb{R}^{n-l}\times\mathbb{T}^{r-l}.$

Thus, $M(P,\Lambda)$ is a closed topological $(n+r)$-manifold. ∎

This result can be obtained as a corollary of results in [S09] and also of
[AGo24, Theorem 1.1].

######  Proposition 6.2.

Let $P$ be a simple $n$-polytope and
$\Lambda\colon\\{F_{1},\dots,F_{m}\\}\to\\{\boldsymbol{e}_{1},\dots,\boldsymbol{e}_{r}\\}$
be an epimorphism, where $\\{\boldsymbol{e}_{1},\dots,\boldsymbol{e}_{r}\\}$
is a basis in $\mathbb{Z}^{r}$. If the complex $\mathcal{C}(P,\Lambda)$ is
equivalent to $\mathcal{C}(n,r)$ then the polytope is homeomorphic to
$S^{n}_{r,\geqslant}$, and the manifold $M(P,\Lambda)$ is homeomorphic to
$S^{n+r}$.

###### Proof.

Indeed, $S^{n}_{r,\geqslant}\times\mathbb{T}^{r}/\sim\simeq S^{n+r}$, and the
homeomorphism is given as

$\left[(x_{1},\dots,x_{n+1}),(\varphi_{1},\dots,\varphi_{r})\right]\to(x_{1}\cos(2\pi\varphi_{1}),x_{1}\sin(2\pi\varphi_{1}),\dots,x_{r}\cos(2\pi\varphi_{r}),x_{r}\sin(2\pi\varphi_{r}),x_{r+1},\dots,x_{n+1}).$

∎

######  Example 6.3.

Examples for Proposition 6.2 are provided by Example 2.9. Each face
$G=F_{i_{1}}\cap\dots\cap F_{i_{k}}$ corresponds to a mapping

$\Lambda_{i}=\begin{cases}\boldsymbol{e}_{s},&\text{ if
}i=i_{s},s=1,\dots,k,\\\ \boldsymbol{e}_{k+1},&\text{ otherwise},\end{cases}$

where $\boldsymbol{e}_{1}=(1,0,\dots,0)$, $\dots$,
$\boldsymbol{e}_{k+1}=(0,\dots,0,1)\in\mathbb{Z}^{k+1}$. Then the subgroup
$H_{G}=H(\Lambda)\simeq\mathbb{T}^{m-k-1}$ is defined in $\mathbb{T}^{m}$ by
the equations $\varphi_{i_{1}}=0$, $\dots$, $\varphi_{i_{k}}=0$, and
$\varphi_{1}+\dots+\varphi_{m}=0$. We have $\mathcal{Z}_{P}/H_{G}\simeq
S^{n+k+1}$.

######  Example 6.4.

For each polytope $P$ the mapping $\Lambda_{i}=1\in\mathbb{Z}$ gives the
complex $\mathcal{C}(P,\Lambda)\simeq\mathcal{C}(n,1)$. The subgroup
$H_{0}=H(\Lambda)$ is defined by the equation
$\varphi_{1}+\dots+\varphi_{m}=0$. We have $\mathcal{Z}_{P}/H_{0}=S^{n+1}$.

For any vector-coloring $\Lambda$ such that there is a function
$\boldsymbol{c}=(c_{1},\dots,c_{r})\in(\mathbb{Z}^{r})^{*}$ with
$\boldsymbol{c}\Lambda_{i}=1$ for all $i$ we have $H(\Lambda)\subset H_{0}$
and on the space $M(P,\Lambda)$ there is an action of
$H_{0}^{\prime}=H_{0}/H(\Lambda)\simeq\mathbb{T}^{r-1}$ such that
$M(P,\Lambda)/H_{0}^{\prime}=\mathcal{Z}_{P}/H_{0}\simeq S^{n+1}$. The
subgroup $H_{0}^{\prime}$ is defined in $\mathbb{T}^{r}$ by the equation
$c_{1}\psi_{1}+\dots+c_{r}\psi_{r}=0$.

In particular, for the product of polytopes
$P^{n}=P_{1}^{n_{1}}\times\dots\times P_{k}^{n_{k}}$ each facet has the form
$P_{1}\times\dots F_{i,j}\times\dots\times P_{k}$, where $F_{i,j}$ is a facet
of $P_{i}$. We have a mapping $\Lambda(P_{1}\times\dots
F_{i,j}\times\dots\times P_{k})=\boldsymbol{e}_{i}$, where
$\boldsymbol{e}_{1}$, $\dots$, $\boldsymbol{e}_{k}$ is the standard basis in
$\mathbb{Z}^{k}$. For the function
$\boldsymbol{c}=(1,\dots,1)\in(\mathbb{Z}^{k})^{*}$ we have
$\boldsymbol{c}\boldsymbol{e}_{i}=1$ for all $i$. Then

$\mathcal{Z}_{P}=\mathcal{Z}_{P_{1}}\times\dots\times\mathcal{Z}_{P_{k}}\text{
and }M(P,\Lambda)=\mathcal{Z}_{P}/(H_{1,0}\times\dots\times
H_{k,0})=S^{n_{1}+1}\times\dots\times S^{n_{k}+1}.$

On this manifold there is an action of
$H_{0}^{\prime}=H_{0}/(H_{1,0}\times\dots\times
H_{k,0})\simeq\mathbb{T}^{k-1}$. This subgroup is defined in $\mathbb{T}^{k}$
by the equation $\psi_{1}+\dots+\psi_{k}=0$. Then
$S^{n_{1}+1}\times\dots\times
S^{n_{k}+1}/H_{0}^{\prime}\simeq\mathcal{Z}_{P}/H_{0}\simeq S^{n+1}$. This
torus analog of Dmitry Gugnin’s construction from [G19] was described in
[AGu23].

The latter example can be generalized as follows. Given integer vector-
colorings $\Lambda_{P_{i}}$ of ranks $r_{i}$ on polytopes $P_{i}$ such that
$\mathcal{C}(P_{i},\Lambda_{P_{i}})\simeq\mathcal{C}(n,r_{i})$ we have the
product coloring $\Lambda_{P}$ on $P_{1}\times\dots\times P_{k}$ such that
$M(P,\Lambda)\simeq S^{n_{1}+r_{1}}\times\dots\times S^{n_{k}+r_{k}}$ and an
action of $H_{0}^{\prime}\simeq\mathbb{T}^{r_{1}+\dots+r_{k}-1}$ such that
$S^{n_{1}+r_{1}}\times\dots\times
S^{n_{k}+r_{k}}/\mathbb{T}^{r_{1}+\dots+r_{k}-1}\simeq S^{n+1}$.

##  7\. Boolean simplices and simplicial prisms

On this section we will give definition and prove basic facts about the
notions we will need in subsequent sections.

######  Definition 7.1.

Let us call an affinely independent set of points
$\\{\boldsymbol{p}_{1},\dots,\boldsymbol{p}_{r+1}\\}\in\mathbb{Z}_{2}^{N}$ a
boolean $r$-simplex and denote it $\Delta_{2}^{r}$. By definition set
$\Delta^{-1}=\varnothing$. Let us call a set of points
$S\subset\mathbb{Z}_{2}^{N}$ affinely equivalent to the direct product
$\Delta^{r-1}_{2}\times\mathbb{Z}_{2}$ a boolean simplicial prism and denote
it $\Pi^{r}$. We have $\Pi^{1}=\mathbb{Z}_{2}=\Delta^{1}$ and
$\Pi^{2}=\mathbb{Z}_{2}^{2}$.

A boolean simplicial prism $\Pi^{r}$ consists of two disjoint boolean
$(r-1)$-simplices (“bases”) $\boldsymbol{a}_{1},\dots,\boldsymbol{a}_{r}$ and
$\boldsymbol{b}_{1},\dots,\boldsymbol{b}_{r}$ in $\mathbb{Z}_{2}^{r}$ such
that any two points $\boldsymbol{a}_{i},\boldsymbol{b}_{i}$ form a boolean
line parallel to the same vector $\boldsymbol{l}$ (“main direction”) that is
not parallel to bases. This means that
$\boldsymbol{l}=\boldsymbol{a}_{i}+\boldsymbol{b}_{i}$ for all $i$, and the
disjoint union of any base and a vertex of the other base is a simplex. It is
easy to see that for any $i$ there is a unique affine isomorphism exchanging
$\boldsymbol{a}_{i}$ and $\boldsymbol{b}_{i}$ and leaving all
$\boldsymbol{a}_{j}$ and $\boldsymbol{b}_{j}$ with $j\neq i$ fixed.

######  Lemma 7.2.

A subset of
$\Pi^{r}=\\{\boldsymbol{a}_{1},\boldsymbol{b}_{1},\dots,\boldsymbol{a}_{r},\boldsymbol{b}_{r}\\}$
is affinely independent if and only if it contains at most one pair
$\\{\boldsymbol{a}_{i},\boldsymbol{b}_{i}\\}$.

###### Proof.

The proof is straightforward using the equality
$\boldsymbol{a}_{i}+\boldsymbol{b}_{i}+\boldsymbol{a}_{j}+\boldsymbol{b}_{j}=0$.
∎

######  Corollary 7.3.

A subset $S\subset\Pi^{r}$ is an affine $2$-plane if and only if
$S=\\{\boldsymbol{a}_{i},\boldsymbol{b}_{i},\boldsymbol{a}_{j},\boldsymbol{b}_{j}\\}$
for $i\neq j$.

###### Proof.

Indeed, the points $\boldsymbol{a}_{i},\boldsymbol{b}_{i},\boldsymbol{a}_{j}$
are affinely independent and
$\boldsymbol{b}_{j}=\boldsymbol{a}_{i}+\boldsymbol{b}_{i}+\boldsymbol{a}_{j}$.
Hence,
$\\{\boldsymbol{a}_{i},\boldsymbol{b}_{i},\boldsymbol{a}_{j},\boldsymbol{b}_{j}\\}$
is an affine $2$-plane. On the other hand, if $S$ does not contain two pairs
$\\{\boldsymbol{a}_{i},\boldsymbol{b}_{i}\\}$, then $S$ is affinely
independent. ∎

######  Definition 7.4.

Consider two affine sets of points $S_{1}$, $S_{2}\subset\mathbb{Z}_{2}^{N}$.
If the planes ${\rm aff}(S_{1})$ and ${\rm aff}(S_{2})$ are skew, that is they
do not intersect and the intersection of the corresponding linear subspaces is
zero, then we call the set $S_{1}\sqcup S_{2}$ a join of $S_{1}$ and $S_{2}$
and denote it $S_{1}*S_{2}$. If $S_{1}$ and $S_{1}^{\prime}$ are affinely
equivalent as well as $S_{2}$ and $S_{2}^{\prime}$, then $S_{1}*S_{2}$ and
$S_{1}^{\prime}*S_{2}^{\prime}$ are also affinely equivalent. Therefore, up to
an affine equivalence we can define a join of any two sets $S_{1}$,
$S_{2}\subset\mathbb{Z}_{2}^{N}$, if we put them to skew planes. Then
$(S_{1}*S_{2})*S_{3}=S_{1}*(S_{2}*S_{3})$.

A join of a set $S$ and a point $\boldsymbol{p}$ is called a cone over $S$ and
is denoted $CS$. A cone $CS$ is a disjoint union of $S$ and a point
$\boldsymbol{p}\notin{\rm aff}(S)$. We have $C^{k}S=\Delta^{k-1}_{2}*S$. The
boolean simplex $\Delta^{r}_{2}$ is a join of its vertices and a cone over
$\Delta^{r-1}_{2}$.

######  Lemma 7.5.

Any full-dimensional subset $S$ of $\Pi^{r}$ of cardinality $r+k$ is affinely
isomorphic to $\Delta^{r-k-1}_{2}*\Pi^{k}=C^{r-k}\Pi^{k}$. In particular, for
$k=1$ it is $\Delta^{r}_{2}$, and for $k=2$ it is
$\Delta_{2}^{r-3}*\mathbb{Z}_{2}^{2}$.

###### Proof.

Indeed, $S$ is affinely isomorphic to
$\\{\boldsymbol{a}_{1},\dots,\boldsymbol{a}_{r},\boldsymbol{b}_{1},\dots,\boldsymbol{b}_{k}\\}$.
We have

${\rm
aff}(\boldsymbol{a}_{1},\dots,\boldsymbol{a}_{k},\boldsymbol{b}_{1},\dots,\boldsymbol{b}_{k})={\rm
aff}(\boldsymbol{a}_{1},\dots,\boldsymbol{a}_{k},\boldsymbol{b}_{1}).$

This plane is skew with ${\rm
aff}(\boldsymbol{a}_{k+1},\dots,\boldsymbol{a}_{r})$, since the points
$\\{\boldsymbol{a}_{1},\dots,\boldsymbol{a}_{r},\boldsymbol{b}_{1}\\}$ are
affinely independent. ∎

######  Lemma 7.6.

For $k\geqslant 2$ and $r\geqslant k+1$ the subsets $\Delta^{r-k-1}$ and
$\Pi^{k}$ are affine invariants of the join $\Delta^{r-k-1}*\Pi^{k}$.

###### Proof.

Indeed, $\Delta^{r-k-1}$ consists of points not lying in the affine hull of
the rest points. ∎

######  Lemma 7.7.

For $r\geqslant 3$, the main direction $\boldsymbol{l}$ is a unique direction
$\boldsymbol{d}$ such that $\Pi^{r}$ consists of $r$ lines of this direction.
In particular, $\boldsymbol{l}$ is an affine invariant of the boolean prism
$\Pi^{r}$. For $r=2$ we have $\Pi^{2}=\mathbb{Z}_{2}^{2}$ and any direction
can be chosen as main.

###### Proof.

Indeed, if $\boldsymbol{d}\neq\boldsymbol{l}$, then without loss of generality
we may assume that one line consists of $\boldsymbol{a}_{i}$ and
$\boldsymbol{a}_{j}$. Then
$\boldsymbol{d}=\boldsymbol{a}_{i}+\boldsymbol{a}_{j}=\boldsymbol{b}_{i}+\boldsymbol{b}_{j}$.
Since $r\geqslant 3$, there are at least three lines. Then either
$\boldsymbol{d}=\boldsymbol{a}_{k}+\boldsymbol{a}_{l}=\boldsymbol{b}_{k}+\boldsymbol{b}_{l}$
or
$\boldsymbol{d}=\boldsymbol{a}_{k}+\boldsymbol{b}_{l}=\boldsymbol{b}_{k}+\boldsymbol{a}_{l}$
for some $k\neq l$ such that $\\{k,l\\}\cap\\{i,j\\}=\varnothing$. We obtain a
contradiction to Lemma 7.2. ∎

######  Lemma 7.8.

For $r\geqslant k\geqslant 3$ the main direction $\boldsymbol{l}$ of $\Pi^{k}$
is a unique direction $\boldsymbol{d}$ such that the image of
$\Delta^{r-k-1}_{2}*\Pi^{k}$ under the projection
$\mathbb{Z}_{2}^{r}\to\mathbb{Z}_{2}^{r}/\langle\boldsymbol{d}\rangle$ is
$\Delta^{r-1}_{2}$.

For $k=2$ such directions are three main directions of $\Pi^{2}$.

For $k=1$ there are $\frac{r(r-1)}{2}$ such directions corresponding to the
pairs of vertices of $\Delta^{r-k-1}_{2}*\Pi^{k}=\Delta^{r}_{2}$.

###### Proof.

For $k=1$ the statement is trivial. Assume that $k>2$. The set
$\Delta^{r-k-1}_{2}*\Pi^{k}$ consisting of $r+k\geqslant r+2$ points lies on
$r$ lines of direction $\boldsymbol{d}$. Since at least two lines contain two
points and any point of $\Delta^{r-k-1}_{2}$ does not lie in the affine hull
of all the other points of $\Delta^{r-k-1}_{2}*\Pi^{k}$, each point of
$\Delta^{r-k-1}_{2}$ is a single point on the corresponding line. Hence,
$\Pi^{k}$ consists of $k$ lines of direction $\boldsymbol{d}$ and by Lemma 7.7
these lines have a main direction. Lemma 7.5 implies that a main direction
satisfies the desired condition. ∎

##  8\. Special hyperelliptic manifolds $N(P,\Lambda)$

######  Definition 8.1.

Following [VM99S1] we call a closed $n$-manifold $M$ hyperelliptic if it has
an involution $\tau$ such that the orbit space $M/\langle\tau\rangle$ is
homeomorphic to an $n$-sphere. The corresponding involution $\tau$ is called a
hyperelliptic involution.

In this section we consider hyperelliptic involutions $\tau$ in the group
$\mathbb{Z}_{2}^{r+1}$ canonically acting on the closed manifold
$N(P,\Lambda)$ corresponding to a vector-coloring of rank $r+1$ of a simple
$n$-polytope $P$. By Corollary 1.17 the manifold $N(P,\Lambda)$ should be
orientable. Hence, $N(P,\Lambda)=N(P,\lambda)$ for an affine coloring
$\lambda$ of rank $r$. Moreover, by Corollary 4.4 the involution $\tau$
preserves the orientation, that is $\tau\in\mathbb{Z}_{2}^{r}=H_{0}^{\prime}$.
Corollary 4.5 implies the following result.

######  Lemma 8.2.

Let $\lambda$ be an affine coloring of a simple $n$-polytope $P$. An
involution $\tau\in\mathbb{Z}_{2}^{r}$ is hyperelliptic if and only if
$N(P,\lambda)/\langle\tau\rangle=N(P,\lambda_{\tau})$ is homeomorphic to
$S^{3}$, where $\lambda_{\tau}$ is the composition $\widehat{\Pi}\circ\lambda$
of $\lambda$ and the affine surjection
$\widehat{\Pi}\colon\mathbb{Z}_{2}^{r}\to\mathbb{Z}_{2}^{r}/\langle\tau\rangle$.

######  Definition 8.3.

Let us call an involution $\tau\in\mathbb{Z}_{2}^{r}$ special, if the complex
$\mathcal{C}(P,\lambda_{\tau})$ is equivalent to $\mathcal{C}(n,r)$.

######  Proposition 8.4.

Any special involution is hyperelliptic.

###### Proof.

This follows from Construction 5.8. ∎

######  Definition 8.5.

Let us call a manifold $N(P,\lambda)$ equipped with a special hyperelliptic
involution $\tau$ a special hyperelliptic manifold of rank $r$.

It follows from the definition that any special hyperelliptic manifold is
obtained by the following construction.

######  Construction 8.6 (A special hyperelliptic manifold).

Let $P$ be a simple $n$-polytope and $c$ be its coloring in $r\geqslant 1$
colors such that the complex $\mathcal{C}(P,c)$ is equivalent to
$\mathcal{C}(n,r)$. Let $G_{1}$, $\dots$, $G_{r}$ be its facets. Choose any
coloring $\chi$ of $P$ in two colors $0$ and $1$ such that at least one
restriction $\chi\left.\right|_{G_{i}}$ is non-constant. Define a space
$N(P,c,\chi)=N(P,\lambda(c,\chi))$, where

$\lambda(c,\chi)(F_{i})=\begin{cases}\boldsymbol{a}_{c(i)},&\text{if
}\chi(F_{i})=1,\\\ \boldsymbol{b}_{c(i)},&\text{if }\chi(F_{i})=0,\end{cases}$

and
$\\{\boldsymbol{a}_{1},\dots,\boldsymbol{a}_{r},\boldsymbol{b}_{1},\dots,\boldsymbol{b}_{r}\\}\subset\mathbb{Z}_{2}^{r}$
is a boolean simplicial prism of dimension $r$. If we exchange the colors $0$
and $1$ at one facet $G_{i}$, then $\lambda(c,\chi)$ will be changed to an
affinely equivalent coloring, and the weakly equivariant type of $N(P,c,\chi)$
will remain the same. If $N(P,c,\chi)$ is a manifold, then by definition it is
a special hyperelliptic manifold of rank $r$ with the special hyperelliptic
involution
$\boldsymbol{l}=\boldsymbol{a}_{i}+\boldsymbol{b}_{i}\in\mathbb{Z}_{2}^{r}$.

######  Remark 8.7.

The image of the mapping
$\lambda(c,\chi)\colon\\{F_{1},\dots,F_{m}\\}\to\mathbb{Z}_{2}^{r}$ consists
of $r+k$ points if and only if $\chi\left.\right|_{G_{i}}$ is non-constant
exactly for $k$ facets $G_{i}$.

Lemma 7.2 and Corollary 7.3 imply the following criterion.

######  Corollary 8.8.

The space $N(P,c,\chi)$ is a manifold if and only if one of the following
equivalent conditions hold:

  1. (1)

$F_{i}\cap F_{j}\cap F_{k}\cap F_{l}=\varnothing$ whenever
$c(F_{i})=c(F_{j})\neq c(F_{k})=c(F_{l})$ and
$\chi(F_{i})=\chi(F_{k})\neq\chi(F_{j})=\chi(F_{l})$;

  2. (2)

$F_{i}\cap F_{j}\cap F_{k}\cap F_{l}=\varnothing$ whenever
$\lambda(c,\chi)(\\{F_{i},F_{j},F_{k},F_{l}\\})=\\{\boldsymbol{a}_{p},\boldsymbol{b}_{p},\boldsymbol{a}_{q},\boldsymbol{b}_{q}\\}$
for $p\neq q$.

  3. (3)

$F_{i}\cap F_{j}\cap F_{k}\cap F_{l}=\varnothing$ whenever
$\lambda(c,\chi)(\\{F_{i},F_{j},F_{k},F_{l}\\})$ is an affine $2$-plane.

######  Corollary 8.9.

In dimension $n=3$ in Construction 8.6 the space $N(P,c,\chi)$ is a special
hyperelliptic manifold for any $\chi$.

######  Remark 8.10.

Corollary 8.9 also follows Corollary 5.6.

######  Proposition 8.11.

Any special hyperelliptic manifold can be obtained by Construction 8.6.

###### Proof.

Indeed, if $\tau$ is a special involution on the manifold $N(P,\lambda)$, then
$\mathcal{C}(P,\lambda_{\tau})\simeq\mathcal{C}(n,r)$. Hence, we can choose
$c=\lambda_{\tau}$. The image of $c$ consists of affinely independent points
$\boldsymbol{p}_{1}$, $\dots$,
$\boldsymbol{p}_{r}\in\mathbb{Z}_{2}^{r}/\langle\tau\rangle$ corresponding to
facets $G_{1}$, $\dots$, $G_{r}$ of $\mathcal{C}(P,\lambda_{\tau})$. Let
$\widehat{\Pi}\colon\mathbb{Z}_{2}^{r}\to\mathbb{Z}_{2}^{r}/\langle\tau\rangle$
be the canonical projection. Choose for each $i$ some facet $F_{j_{i}}\subset
G_{i}$ and set $\boldsymbol{a}_{i}=\lambda_{j_{i}}$. Then the points
$\boldsymbol{a}_{1}$, $\dots$, $\boldsymbol{a}_{r}$ are affinely independent
and
$\widehat{\Pi}^{-1}(\boldsymbol{p}_{i})=\\{\boldsymbol{a}_{i},\boldsymbol{b}_{i}\\}$
for each $i$, where $\boldsymbol{b}_{i}=\boldsymbol{a}_{i}+\tau$. Thus,
setting $\boldsymbol{l}=\tau$ and $\chi(F_{i})=\begin{cases}1,&\text{if
}\lambda_{i}=\boldsymbol{a}_{i},\\\ 0,&\text{if
}\lambda_{i}=\boldsymbol{b}_{i}\end{cases}$ we finish the proof. ∎

Now let us enumerate all special involutions on a manifold $N(P,\lambda)$.

######  Proposition 8.12.

Let $N(P,c,\chi)$ be a special hyperelliptic manifold of rank $r$ and $G_{1}$,
$\dots$, $G_{r}$ be facets of $\mathcal{C}(P,c)\simeq\mathcal{C}(n,r)$.

  * •

If $\chi\left.\right|_{G_{i}}$ is non-constant exactly for one facet $G_{i}$
(that is, the image of
$\lambda(c,\chi)\colon\\{F_{1},\dots,F_{m}\\}\to\mathbb{Z}_{2}^{r}$ is a
boolean simplex), then $\tau\in\mathbb{Z}_{2}^{r}$ is a special involution if
and only if the vector $\tau$ connects two vertices of the simplex and
$\mathcal{C}(P,\lambda_{\tau})\simeq\mathcal{C}(n,r)$. There are at most
$\frac{r(r+1)}{2}$ such involutions.

  * •

If $\chi\left.\right|_{G_{i}}$ is non-constant exactly for two facets $G_{i}$
and $G_{j}$ (that is, the image of $\lambda(c,\chi)$ is
$\Delta^{r-3}*\Pi^{2}$), then $\tau\in\mathbb{Z}_{2}^{r}$ is a special
involution if and only if
$\tau\in\\{\boldsymbol{l},\boldsymbol{a}_{i}+\boldsymbol{a}_{j},\boldsymbol{a}_{i}+\boldsymbol{b}_{j}\\}$
(that is, tau is a main direction of $\Pi^{2}$) and
$\mathcal{C}(P,\lambda_{\tau})\simeq\mathcal{C}(n,r)$. There are at most three
such involutions.

  * •

If $\chi\left.\right|_{G_{i}}$ is non-constant for more than two facets
$G_{i}$ (that is, the image of $\lambda(c,\chi)$ is $\Delta^{r-k-1}*\Pi^{k}$
for $k\geqslant 3$), then $\tau\in\mathbb{Z}_{2}^{r}$ is a special involution
if and only if $\tau=\boldsymbol{l}$ (the main direction of $\Pi^{k}$). That
is, there is only one special involution.

###### Proof.

The Proposition follows from Lemma 7.8. ∎

We can summarise the above results as follows.

######  Definition 8.13.

For an affine coloring $\lambda$ of rank $r$ of a simple $n$-polytope $P$
denote
$I(\lambda)=\\{\lambda_{1},\dots,\lambda_{m}\\}\subset\mathbb{Z}_{2}^{r}$. For
a subset $S\subset\mathbb{Z}_{2}^{r}$ denote
$G(S)=\bigcup\limits_{q\colon\lambda_{q}\in S}F_{q}\subset\partial P$.

######  Theorem 8.14.

Let $\lambda$ be an affine coloring of rank $r$ of a simple $n$-polytope $P$.
The space $N(P,\lambda)$ is a special hyperelliptic manifold if and only if
$1\leqslant r\leqslant n+1$ and one of the following conditions hold:

  1. (1)

$I(\lambda)=\\{\boldsymbol{p}_{1},\dots,\boldsymbol{p}_{r+1}\\}$ is a boolean
$r$-simplex, and at least for one direction
$\tau=\boldsymbol{p}_{i}+\boldsymbol{p}_{j}$, $i\neq j$, the complex
$\mathcal{C}(P,\lambda_{\tau})$ is equivalent to $\mathcal{C}(n,r)$. Each
special hyperelliptic involution $\tau\in\mathbb{Z}_{2}^{r}$ has this form and
there are at most $\frac{r(r+1)}{2}$ such involutions.

  2. (2)

$I(\lambda)=\Delta^{r-3}*\Pi^{2}$,
$\bigcap_{\lambda_{j}\in\Pi_{2}}G(\lambda_{j})=\varnothing$ and at least for
one main direction $\tau$ of $\Pi^{2}$ the complex
$\mathcal{C}(P,\lambda_{\tau})$ is equivalent to $\mathcal{C}(n,r)$. Each
special hyperelliptic involution $\tau\in\mathbb{Z}_{2}^{r}$ has this form and
there are at most three such involutions.

  3. (3)

$I(\lambda)=\Delta^{r-k-1}*\Pi^{k}$, $k\geqslant 3$,
$\bigcap_{\lambda_{j}\in\Pi_{2}}G(\lambda_{j})=\varnothing$ for any $2$-plane
$\Pi^{2}\subset\Pi^{k}$, and for the main direction $\tau$ of $\Pi^{k}$ the
complex $\mathcal{C}(P,\lambda_{\tau})$ is equivalent to $\mathcal{C}(n,r)$.
The main direction $\tau$ is a unique special hyperelliptic involution in
$\mathbb{Z}_{2}^{r}$.

Moreover, in all these cases any vertex of $\mathcal{C}(P,\lambda)$ belongs to
the $1$-skeleton of $\mathcal{C}(P,\lambda_{\tau})\simeq\mathcal{C}(n,r)$.

###### Proof.

The proof follows from Corollary 8.8, Propositions 8.11 and 8.12, and
Corollary 5.13. ∎

We will specify this result for $3$-dimensional polytopes in Section 11.

######  Corollary 8.15.

If $N(P,\lambda)$ is a special hyperelliptic manifold of rank $r$, where
$1\leqslant r\leqslant n-2$, then the complex $C(P,\lambda)$ has not vertices.

###### Proof.

If follows from the fact that the $1$-skeleton of the complex
$\mathcal{C}(n,r)\simeq\mathcal{C}(P,\lambda_{\tau})$ is empty for $r\leqslant
n-2$, since the intersection of all its facets is $S^{n-r}$, $n-r\geqslant 2$.
∎

######  Example 8.16.

Example 5.9 produces the following special hyperelliptic manifolds. Each face
$G=F_{i_{1}}\cap\dots\cap F_{i_{k}}$ and an epimorphism
$\chi\colon\\{F_{1},\dots,F_{m}\\}\setminus\\{F_{i_{1}},\dots,F_{i_{k}}\\}\to\\{0,1\\}$
corresponds to an affine coloring of rank $k+1$

$\lambda_{i}=\begin{cases}\boldsymbol{e}_{s},&\text{ if
}i=i_{s},s=1,\dots,k;\\\ \boldsymbol{e}_{k+1},&\text{ if
}i\notin\\{i_{1},\dots,i_{k}\\}\text{ and }\chi(F_{i})=1;\\\
\boldsymbol{0},&\text{ if }i\notin\\{i_{1},\dots,i_{k}\\}\text{ and
}\chi(F_{i})=0,\end{cases}$

where $\boldsymbol{e}_{1}=(1,0,\dots,0)$, $\dots$,
$\boldsymbol{e}_{k+1}=(0,\dots,1)\in\mathbb{Z}_{2}^{k+1}$. Then the subgroup
$H_{G,\chi}=H(\lambda)$ of rank $m-k-2$ is defined in $\mathbb{Z}_{2}^{m}$ by
the equations $x_{i_{1}}=0$, $\dots$, $x_{i_{k}}=0$, $x_{1}+\dots+x_{m}=0$,
and $\sum_{i\colon\chi(F_{i})=1}x_{i}$. The space $N(P,\lambda)$ is a special
hyperelliptic manifold of rank $k+1$ with a special hyperelliptic involution
$\boldsymbol{e}_{k+1}\in\mathbb{Z}_{2}^{k+1}$.

######  Example 8.17.

If $\lambda$ is an affinely independent coloring of a simple $n$-polytope $P$
and $N(P,\lambda)$ is a special hyperelliptic manifold of rank $r$, then
$n-1\leqslant r\leqslant n+1$, and all the vertices of $P$ belong to the
$1$-skeleton of $\mathcal{C}(P,\lambda_{\tau})\simeq\mathcal{C}(n,r)$, which
is a subset of the graph of $P$. For $r=n-1$, this $1$-skeleton is a single
circle without vertices. We have a simple edge-cycle in the graph of $P$
containing all its vertices. Such cycles are called Hamiltonian. For $r=n$ the
$1$-skeleton of $\mathcal{C}(P,\lambda_{\tau})\simeq\mathcal{C}(n,r)$ is a
graph with two vertices and $n$ multiple edges. For $r=n+1$ it is a complete
graph $K_{n+1}$.

######  Example 8.18.

For $n=1$ the only small cover over $P=I^{1}=\Delta^{1}$ is
$N(P,\lambda)=\mathbb{R}P^{1}\simeq S^{1}$, and it is not a special
hyperelliptic manifold.

For $n=2$ any orientable small cover $N(P_{k},\lambda)$ over a $k$-gon $P_{k}$
is a special hyperelliptic manifold. In this case $k$ is even and $\lambda$
corresponds to a coloring of edges of $P_{k}$ in two colors such that adjacent
edges have different colors.

For $n=3$ special hyperelliptic small covers $N(P,\lambda)$ correspond to
Hamiltonian cycles on $P$. We will see such examples in Sections 12 and 13.
For example, there is a special hyperelliptic small cover over the
dodecahedron with three special involutions, see Fig. 9. It is a classical
fact that not any simple $3$-polytope admits a Hamiltonian cycle (see [T46,
G68]).

For $n=4$ if a polytope $P$ admits a special hyperelliptic small cover, then
$P$ has a Hamiltonian cycle $\gamma$ and all the facets of $P$ can be colored
in $3$ colors in such a way that any edge of $\gamma$ is an intersection of
facets of different colors. Moreover, the union of all the facets of each
color is a $3$-disk. Since $P$ has at least $5$ facets, there are two adjacent
facets $F_{i}$ and $F_{j}$ of the same color. Then no edge of the polygon
$F_{i}\cap F_{j}$ belongs to $\gamma$, and at each vertex of this polygon
$\gamma$ passes through two complementary edges. Then the colors of the facets
$F_{k}\neq F_{i},F_{j}$ containing the successive edges of $F_{i}\cap F_{j}$
alter. Thus, $F_{i}\cap F_{j}$ has an even number of edges.

######  Proposition 8.19.

If a simple $4$-polytope $P$ admits a special hyperelliptic small cover, then
$P$ has at least one $2$-face with an even number of edges.

######  Corollary 8.20.

The simplex $\Delta^{4}$ and the $120$-cell have no special hyperelliptic
small covers.

Moreover, it can be shown than the products $I\times\Delta^{3}$,
$I^{2}\times\Delta^{2}$, $\Delta^{2}\times\Delta^{2}$, and the cube $I^{4}$
admit no special hyperelliptic small covers.

######  Conjecture 8.21.

There are no special hyperelliptic small covers in dimensions $n\geqslant 4$.

##  9\. A structure of the complex $\mathcal{C}(P,c)$ for $3$-polytopes

###  9.1. Basic facts from the graph theory

######  Agreement 9.1.

In this article by a spherical graph we mean a graph realized on the sphere
$S^{2}$ piecewise linearly in some triangulation of $S^{2}$.

For additional details on the graph theory see [BE17I].

######  Definition 9.2.

A graph is simple if it has no loops and multiple edges.

Following [Z95] we call a connected graph $G$ with at least two edges
$2$-connected if it has no loops and a deletion of any vertex with all
incident edges leaves the graph connected.

A connected graph $G$ with at least four edges is called $3$-connected, if it
is simple and a deletion of any vertex or any two vertices with all incident
edges leaves the graph connected.

A face of a spherical graph $G\subset S^{2}$ is a connected component of the
complement $S^{2}\setminus G$. A vertex or an edge of $G$ is incident to a
face if it belongs to its closure. A vertex of an edge is incident to it.

Two spherical graphs are called combinatorially equivalent, if there is a
bijection between the sets of their vertices, edges and faces preserving the
incidence relation.

A bridge of a graph $G$ is an edge such that a deletion of this edge makes the
graph disconnected.

The proof of the following classical facts can be found in [BE17I, Lemmas
2.4.1 and 2.4.2] and [BE17S, Lemma 1.27].

######  Lemma 9.3.

A spherical graph $G$ with more than one vertex is connected if and only if
any its face is a disk (equivalently, has one connected component of the
boundary).

######  Lemma 9.4.

A simple spherical graph $G$ with more than one vertex is $3$-connected if and
only if any its face is bounded by a simple cycle and if the boundary cycles
of two faces intersect, then their intersection is a vertex or an edge.

To characterize the graphs of $3$-polytopes we will use the following result
(see [Z95]).

######  Theorem 9.5 (The Steinitz theorem).

A simple graph $G$ is a graph of some $3$-polytope if and only if it is planar
and $3$-connected.

Moreover, by H. Whitney’s theorem (see [Z95]) any two spherical realizations
of the graph of a $3$-polytope are combinatorially equivalent.

######  Corollary 9.6.

A connected simple spherical graph with more than one vertex is
combinatorially equivalent to a graph of a $3$-polytope if and only if any its
facet is bounded by a simple cycle and if the boundary cycles of two facets
intersect, then their intersection is a vertex or an edge.

######  Lemma 9.7.

For a connected $3$-valent spherical graph $G$ the following conditions are
equivalent:

  1. (1)

$G$ is $2$-connected (in particular, it has no loops);

  2. (2)

$G$ has no bridges;

  3. (3)

any face of $G$ is a disk bounded by a simple edge-cycle.

###### Proof.

If $G$ is $2$-connected, then it has no bridges, since the deletion of any
vertex of a bridge disconnects the graph. If $G$ has no bridges, then it has
no loops since the vertex of a loop necessarily belongs to a bridge. Also $G$
has at least $3$ edges, since it is $3$-valent. If a deletion of a vertex and
incident edges makes the graph disconnected, then at least one edge in this
vertex is a bridge. A contradiction. Thus, $G$ is $2$-connected and items (1)
and (2) are equivalent.

If each face of $G$ is a disk bounded by a simple edge-cycle, then $G$ has no
bridges since a bridge has the same face on both sides and the boundary cycle
of this face is not simple. Let $G$ have no bridges. Since $G$ is connected,
each its face is a disk. If a boundary cycle passes a vertex more than once,
then it passes an edge more than once since $G$ is $3$-valent. Then this edge
has the same face on both sides. Hence, it is a bridge, which is a
contradiction. Thus, items (2) and (3) are equivalent. ∎

######  Lemma 9.8.

Any $3$-valent graph $G$ has an even number of vertices.

###### Proof.

Indeed, if we cut each edge in two parts, then each vertex is incident to
three such parts, hence $3V=2E$, where $V$ and $E$ are numbers of vertices and
edges. In particular, $V$ is even. ∎

###  9.2. A characterization of complexes $\mathcal{C}(P,c)$ of $3$-polytopes

In dimension $n=3$ each facet of the complex $\mathcal{C}(P,c)$ with a non-
constant mapping $c$ is a sphere with holes. Its boundary consists of
$1$-faces and $0$-faces, which we call vertices. Each $1$-face belongs to two
different facets and each vertex – to three different facets and three
different $1$-faces. Each $1$-face is either the whole circle without
vertices, or a simple path connecting two different vertices.

######  Definition 9.9.

We call $1$-faces of $\mathcal{C}(P,c)$ containing no vertices circles, and
$1$-faces connecting two vertices edges.

Consider the $1$-skeleton $\mathcal{C}^{1}(P,c)$, which is the union of all
vertices and $1$-faces. Each its connected component is either a circle
without vertices or a connected $3$-valent spherical graph without loops and
bridges. Indeed, a bridge should have the same facet on both sides, hence it
can not be an intersection of two different facets.

######  Theorem 9.10.

Complexes $\mathcal{C}(P,c)$ corresponding to $3$-polytopes $P$ are exactly
subdivisions of the $2$-sphere arising from disjoint unions (perhaps empty) of
simple curves and connected $3$-valent graphs without bridges.

###### Proof.

We have already proof the theorem in one direction. Consider the other
direction. By Lemma 9.7 each connected $3$-valent spherical graph without
bridges has no loops. We will call by “facets” the connected components of the
complement in $S^{2}$ to a disjoint union of simple closed curves and
connected $3$-valent graphs without bridges, and by “circles” simple closed
curves from the union.

The empty union corresponds to a constant function $c$ on any polytope. Now
let us assume that the union is non-empty.

Consider a facet $C$ and a component $\gamma$ of $\partial C$ that is not a
circle. There is a vertex on $\gamma$. This vertex belongs to three different
edges and to closures of three different facets, for otherwise some of the
edges is a bridge. Two of these edges belong to $\gamma$ and the third edge
does not belong. Then $\gamma$ is a simple edge-cycle, since it passes each
vertex at most once, and $C$ is a sphere with holes bounded by such simple
edge-cycles and circles from the union. Each edge or circle belongs to the
closures of exactly two different facets, and each vertex – to the closures of
three different facets.

Now we will add edges to this data to obtain a $1$-skeleton of some simple
$3$-polytope. Each edge will have two new different $3$-valent vertices and
will divide a facet into two new different facets. If a facet $C$ is not a
disk, we can first add edges connecting points on the same boundary component
to subdivide $C$ into rings, and then for each ring add three edges connecting
different boundary components to subdivide it into three “quadrangles” (see
Fig. 1a).

Figure 1. a) A subdivision of a sphere with holes; b) Cutting off the common
edges

After this procedure we obtain a new subdivision of a $S^{2}$ with $3$-valent
vertices and each facet being a disk bounded by a simple edge-cycle or a
circle without vertices. In the latter case $\mathcal{C}(P,c)$ consists of two
disks glued along the common boundary circle. We can add two edges to these
disks to obtain the boundary complex of a simplex. Thus, we can assume that
each facet has at least one vertex on the boundary. Then there are at least
two vertices, for otherwise the adjacent facet is not bounded by a simple
curve. If there are exactly two vertices, we add an edge separating the
$2$-gon into two triangles. Repeating this step for all $2$-gons, we obtain a
$3$-valent partition of $S^{2}$ into polygons with at list $3$-edges. The
graph defining this partition is simple. Indeed, there are no loops by
construction. If two edges have the same vertices, then they form a simple
closed curve dividing the sphere into two disks. The third edges at both
vertices should lie in the same disk, for otherwise there arise two equal
facets in both vertices. Thus, two multiple edges bound a $2$-gon. A
contradiction.

At the end of this step we obtain a simple spherical graph with each facet
bounded by a simple cycle with at least $3$ edges. Now we will add edges to
this partition to obtain another $3$-valent partition such that each facet is
bounded by a simple edge-cycle with at least $3$ edges and the closures of two
different facets have at most one edge in common. The last condition is
equivalent to the condition that all the edges of any facet belong to
different facets surrounding it. The graph of the new partition is
$3$-connected and by the Steinitz theorem it corresponds to a boundary of a
simple $3$-polytope. Then the original complex is obtained from $P$ by a
sequence of operations of a deletion of an edge and has the form
$\mathcal{C}(P,c)$, where $c(F_{i})=c(F_{j})$ if and only if the facets of $P$
belong to the same facet in the initial partition.

Now let us describe the last step. If a facet has with another facet more than
one common edge, then their intersection consists of a disjoint set of edges
lying on the boundary of each facet. We can “cut off” all but one these edges.
Namely, for each edge we add inside the first facet an edge with vertices on
its boundary close to the vertices of the chosen edge outside it. As a result
the edge is substituted by a quadrangle adjacent to $4$ different facets (see
Fig. 1b). Repeating this procedure we will obtain a new partition of the
sphere such that all the edges of the chosen facet belong to different facets
and all the arising quadrangles also satisfy this condition. Applying this
argument to all the facets one by one we see that at each step there arise no
new bad facets, and their total number is decreasing by one. ∎

##  10\. A criterion when $N(P,\Lambda)$ is a sphere for $3$-polytopes

In this section we will give a criterion when a manifold $N(P,\Lambda)$
corresponding to a vector-coloring $\Lambda$ of rank $r+1$ of a simple
$3$-polytope is homeomorphic to a sphere $S^{3}$. Since $N(P,\Lambda)$ should
be closed and orientable, it has the form $N(P,\lambda)$ for an affine
coloring $\lambda$ of rank $r$. Thus we will consider only affine colorings.

Following [VM99S1] we call a $3$-valent graph consisting of $2$ vertices and
three edges connecting them a theta-graph. By $K_{n}$ we denote a complete
graph on $n$-vertices.

######  Theorem 10.1.

Let $\lambda$ be an affine coloring of rank $r$ of a simple $3$-polytope $P$.
The space $N(P,\lambda)$ is homeomorphic to $S^{3}$ if and only if
$\mathcal{C}(P,\lambda)$ is equivalent to $\mathcal{C}(3,r+1)$. In other
words, if and only if one of the following conditions holds:

  1. (1)

$r=0$ and $\mathcal{C}^{1}(P,\lambda)$ is empty;

  2. (2)

$r=1$ and $\mathcal{C}^{1}(P,\lambda)$ is a circle;

  3. (3)

$r=2$ and $\mathcal{C}^{1}(P,\lambda)$ is a theta-graph;

  4. (4)

$r=3$ and $\mathcal{C}^{1}(P,\lambda)$ is the complete graph $K_{4}$.

In all these cases the image of $\lambda$ is a boolean $(r+1)$-simplex.

######  Remark 10.2.

The spheres in the theorem arise in Construction 5.8 and can be imagined as
follows. In the first case $S^{3}$ is glued of two copies of a polytope along
the boundaries. In the second case – of $4$ copies of the ball with the
boundary sphere subdivided into two hemispheres. If we glue two copies along
the hemispheres we obtain a ball with the boundary subdivided into two
hemispheres again. Then we glue two copies of this ball along boundaries. In
the third case $N(P,\lambda)$ is glued of $8$ copies of the ball with the
boundary sphere subdivided into three $2$-gons by the theta-graph. We can
imagine this sphere as follows. Let the vertices of the theta-graph be the
north and the south poles and edges be three meridians. The sphere and the
ball are subdivided by the equatorial plane into two balls combinatorially
equivalent to a $3$-simplex $\Delta^{3}$. Then $8$ copies of this simplex are
glued at one vertex to an octahedron as the coordinate octants in
$\mathbb{R}^{3}$. The resulting sphere is glued of two copies of this
octahedron along the boundaries. In the case of $K_{4}$ the space
$N(P,\lambda)$ is homeomorphic to $\mathbb{R}\mathcal{Z}_{\Delta^{3}}\simeq
S^{3}$. All these $4$ cases arise if we subdivide the standard $3$-sphere in
$\mathbb{R}^{4}$ into $3$-disks by $1$, $2$, $3$, or $4$ coordinate
hyperplanes.

###### Proof of Theorem 10.1.

The “if” direction follows from Construction 5.8.

Now let us prove the theorem in the “only if” direction. By Corollary 5.6
$N(P,\lambda)$ is a closed orientable $3$-manifold for any affine coloring
$\lambda$ of a simple $3$-polytope $P$.

If a facet $G_{j}$ of $\mathcal{C}(P,\lambda)$ is a sphere with at least two
holes, then there is a simple closed curve $\gamma$ inside $G_{j}$ separating
its two boundary components. Then $\mathcal{C}(P,\lambda)$ can be represented
as a connected sum of spherical complexes $\mathcal{C}(P,\lambda^{\prime})$
and $\mathcal{C}(P,\lambda^{\prime\prime})$ arising if we change the points at
all the facets of $P$ inside one of the connected component of $\partial
P\setminus\gamma$ to $\lambda_{j}$. Denote $r^{\prime}={\rm
rk}\,\lambda^{\prime}$ and $r^{\prime\prime}={\rm
rk}\,\lambda^{\prime\prime}$. Both spaces $N(P,\lambda^{\prime})$ and
$N(P,\lambda^{\prime\prime})$ are closed orientable manifolds by Corollary
5.6.

######  Lemma 10.3.

There is a homeomorphism

(4) $N(P,\lambda)\simeq
N(P,\lambda^{\prime})^{\\#2^{r-r^{\prime}}}\\#N(P,\lambda^{\prime\prime})^{\\#2^{r-r^{\prime\prime}}}\\#(S^{2}\times
S^{1})^{\\#\left[2^{r}-2^{r-r^{\prime}}-2^{r-r^{\prime\prime}}+1\right]}$

######  Corollary 10.4.

If $\mathcal{C}(P,\lambda)$ has a facet, which is a sphere with at least two
holes, then in the Knezer-Milnor prime decomposition of $N(P,\lambda)$ there
is a summand $S^{1}\times S^{2}$. In particular, $N(P,\lambda)$ is not
homeomorphic to a sphere and it is not a homology sphere for any coefficient
group.

###### Proof.

Indeed, in the Knezer-Milnor decomposition of $N(P,\lambda)$ there is a
summand $\\#(S^{2}\times
S^{1})^{\\#\left[2^{r}-2^{r-r^{\prime}}-2^{r-r^{\prime\prime}}+1\right]}$. But
$1\leqslant r^{\prime},r^{\prime\prime}\leqslant r$, since on both sides of
the curve $\gamma$ there is a facet with $\lambda_{i}\neq\lambda_{j}$, where
$G_{j}$ is a chosen facet, which is a sphere with at least two holes. Also
$r^{\prime}+r^{\prime\prime}=r+\dim{\rm aff}(\lambda^{\prime})\cap{\rm
aff}(\lambda^{\prime\prime})\geqslant r$, since $\lambda_{j}\in{\rm
aff}(\lambda^{\prime})\cap{\rm aff}(\lambda^{\prime\prime})$. Hence,

$2^{r}-2^{r-r^{\prime}}-2^{r-r^{\prime\prime}}+1=2^{r-r^{\prime}}(2^{r^{\prime}}-1)-(2^{r-r^{\prime\prime}}-1)\geqslant\\\
\geqslant
2^{r-r^{\prime}}(2^{r^{\prime}}-1)-(2^{r^{\prime}}-1)=(2^{r-r^{\prime}}-1)(2^{r^{\prime}}-1)\geqslant
0$

Moreover, if the left part is equal to zero, then
$r^{\prime}=r-r^{\prime\prime}$ and either $r^{\prime}=0$ or $r=r^{\prime}$
(then $r^{\prime\prime}=0$). A contradiction. ∎

If a facet of $\mathcal{C}(P,\lambda)$ is the whole sphere, then
$\mathcal{C}^{1}(P,\lambda)=\varnothing$. Thus, we can assume that each facet
of $\mathcal{C}(P,\lambda)$ is a disk. If the intersection of two facets
$G_{i}$ and $G_{j}$ is a boundary circle of both facets, then
$\mathcal{C}^{1}(P,\lambda)$ is a single circle. Thus, we can assume that a
nonempty intersection of each two disks $G_{i}$ and $G_{j}$ consists of a
disjoint union of edges. If there are more then one edge, consider a simple
closed curve $\gamma$ consisting of two simple paths connecting the points
inside two common edges – one path in $G_{i}$ and the other in $G_{j}$.

Then $\mathcal{C}(P,\lambda)$ can be represented as a connected sum of
spherical complexes $\mathcal{C}(P,\lambda^{\prime})$ and
$\mathcal{C}(P,\lambda^{\prime\prime})$ arising if we change the points at all
the facets of $P\setminus G_{i}$ inside one of the connected component of
$\partial P\setminus\gamma$ to $\lambda_{j}$. Denote $r^{\prime}={\rm
rk}\,\lambda^{\prime}$ and $r^{\prime\prime}={\rm
rk}\,\lambda^{\prime\prime}$. Both spaces $N(P,\lambda^{\prime})$ and
$N(P,\lambda^{\prime\prime})$ are closed orientable manifolds by by Corollary
5.6.

######  Lemma 10.5.

There is a homeomorphism

(5) $N(P,\lambda)\simeq
N(P,\lambda^{\prime})^{\\#2^{r-r^{\prime}}}\\#N(P,\lambda^{\prime\prime})^{\\#2^{r-r^{\prime\prime}}}\\#(S^{2}\times
S^{1})^{\\#\left[2^{r-1}-2^{r-r^{\prime}}-2^{r-r^{\prime\prime}}+1\right]}$

######  Corollary 10.6.

Let each facet of $\mathcal{C}(P,\lambda)$ be a disk and an intersection of
two different facets be a disjoint set of at least two edges. Then in the
Knezer-Milnor prime decomposition of the manifold $N(P,\lambda)$ there is a
summand $S^{1}\times S^{2}$. In particular, $N(P,\lambda)$ is not homeomorphic
to a sphere and it is not a homology sphere for any coefficient group.

###### Proof.

Indeed, if the Knezer-Milnor prime decomposition of $N(P,\lambda)$ there is a
summand $\\#(S^{2}\times
S^{1})^{\\#\left[2^{r-1}-2^{r-r^{\prime}}-2^{r-r^{\prime\prime}}+1\right]}$.
But $2\leqslant r^{\prime},r^{\prime\prime}\leqslant r$, since on both sides
of the curve $\gamma$ there is a vertex of a common edge, and therefore a
facet with $\lambda_{k}\notin\\{\lambda_{i},\lambda_{j}\\}$, where $G_{i}$ and
$G_{j}$ are the facets under consideration. Also
$r^{\prime}+r^{\prime\prime}=r+\dim{\rm aff}(\lambda^{\prime})\cap{\rm
aff}(\lambda^{\prime\prime})\geqslant r+1$, since
$\lambda_{i},\lambda_{j}\in{\rm aff}(\lambda^{\prime})\cap{\rm
aff}(\lambda^{\prime\prime})$. Hence,

$2^{r-1}-2^{r-r^{\prime}}-2^{r-r^{\prime\prime}}+1=2^{r-r^{\prime}}(2^{r^{\prime}-1}-1)-(2^{r-r^{\prime\prime}}-1)\geqslant\\\
\geqslant
2^{r-r^{\prime}}(2^{r^{\prime}-1}-1)-(2^{r^{\prime}-1}-1)=(2^{r-r^{\prime}}-1)(2^{r^{\prime}-1}-1)\geqslant
0$

Moreover, if the left part is equal to zero, then
$r^{\prime}-1=r-r^{\prime\prime}$ and either $r^{\prime}=1$ or $r=r^{\prime}$
(then $r^{\prime\prime}=1$). A contradiction. ∎

Thus, we can assume that any facet of $\mathcal{C}(P,\lambda)$ is a disk
bounded by a simple edge-cycle and any nonempty intersection of two facets is
an edge. We know, that the boundary cycle of a facet can not contain only one
vertex. If there are only two vertices $v$ and $w$ on the boundary of a facet
$G_{i}$, then the vertex $v$ belongs to some other facets $G_{j}$ and $G_{k}$.
Moreover, each facet $G_{j}$ and $G_{k}$ has a common edge with $G_{i}$, and
this edge contains $w$. Then $G_{j}\cap G_{k}$ is an edge connecting $v$ and
$w$, and $\mathcal{C}^{1}(P,\lambda)$ is a theta-graph.

Now assume that each facet has at least $3$ vertices on its boundary. Then
$\mathcal{C}^{1}(P,\lambda)$ has no multiple edges, for otherwise a $2$-gonal
facet arises. Then $\mathcal{C}^{1}(P,\lambda)$ is a simple planar
$3$-connected graph with at least $4$ edges, and by the Steinitz theorem it
corresponds to a boundary of some simple $3$-polytope $Q$. This polytope has
an induced affinely independent coloring $\lambda$ and
$N(P,\lambda)=N(Q,\lambda)$, where $N(Q,\lambda)$ is a quotient space of a
free action of a subgroup $K$ of $\mathbb{Z}_{2}^{l}$ on
$\mathbb{R}\mathcal{Z}_{Q}$. In particular, it is covered by
$\mathbb{R}\mathcal{Z}_{Q}$. Hence, if $N(P,\lambda)$ is a sphere, then
$N(P,\lambda)=\mathbb{R}\mathcal{Z}_{Q}$.

Assume that $P\neq\Delta^{3}$. If $P$ has a $3$-belt, that is a triple of
facets $G_{i}$, $G_{j}$ and $G_{k}$ with an empty intersection such that any
two of them are adjacent, then $Q$ is a connected sum of two polytopes $Q_{1}$
and $Q_{2}$ along vertices (see details in [E22M]). It is proved in [E22M,
Corollary 3.8] that there is a homeomorphism

$\mathbb{R}\mathcal{Z}_{P}\simeq\mathbb{R}\mathcal{Z}_{P_{1}}^{\\#2^{l-l_{1}}}\\#\mathbb{R}\mathcal{Z}_{P_{2}}^{\\#2^{l-l_{2}}}\\#(S^{2}\times
S^{1})^{\\#\left[(2^{l-l_{1}}-1)\cdot(2^{l-l_{2}}-1)\right]},$

where $l$, $l_{1}$ and $l_{2}$ are the numbers of facets of $Q$, $Q_{1}$ and
$Q_{2}$ respectively. Also $l_{1},l_{2}\leqslant l-1$. Hence, if $Q$ contains
a $3$-belt, then $\mathbb{R}\mathcal{Z}_{Q}$ contains a summand $S^{2}\times
S^{1}$ in its Knezer-Milnor decomposition. If $Q\neq\Delta^{3}$ has no
$3$-belts, then $Q$ is a flag polytope and $\mathbb{R}\mathcal{Z}_{Q}$ is
aspherical (that is $\pi_{i}(\mathbb{R}\mathcal{Z}_{Q})=0$ for $i\geqslant 2$,
see [DJS98, Theorem 2.2.5] or [D08, Proposition 1.2.3]). Thus, if
$N(P,\lambda)\simeq S^{3}$, then $Q=\Delta^{3}$ and the theorem is proved. ∎

######  Corollary 10.7.

Let $\lambda$ be an affine coloring of rank $r$ of a simple $3$-polytope $P$.
Then any hyperelliptic involution $\tau\in\mathbb{Z}_{2}^{r}$ is special, that
is $\mathcal{C}(P,\lambda_{\tau})\simeq\mathcal{C}(3,r)$.

######  Definition 10.8.

Let us call by a theta-subgraph and a $K_{4}$-subgraph of $P$ the image of an
embedding of the theta-graph or the compete graph $K_{4}$ to the $1$-skeleton
of $P$ such that each vertex of the embedded graph is mapped to a vertex of
$P$ and each edge – to a simple edge-path.

######  Corollary 10.9.

Let $P$ be a simple $3$-polytope. The subgroups $H\neq H_{0}$ of
$\mathbb{Z}_{2}^{m}$ such that $N(P,H)\simeq S^{3}$ are in one-to-one
correspondence with simple edge-cycles, theta-subgraphs and $K_{4}$-subgraphs
of $P$. The subgroup corresponding to a subgraph is defined by the linear
equations $\sum\limits_{F_{i}\subset G}x_{i}=0$ corresponding to its facets
$G$.

######  Example 10.10.

Any facet $F_{i}$ is bounded by a simple edge-cycle. This fits Example 2.9 for
$G=F_{i}$.

######  Example 10.11.

It is known that for any two different vertices of $P$ there is a theta-
subgraph with these vertices. This is one of the equivalent definitions of the
$3$-connectivity of the graph (see [Gb03, Section 11.3]). Each edge $F_{i}\cap
F_{j}$ of $P$ corresponds to a theta-subgraph according to Example 2.9. Its
two additional edges are formed by edges of the facets $F_{i}$ and $F_{j}$
complementary to $F_{i}\cap F_{j}$.

######  Example 10.12.

Each vertex $F_{i}\cap F_{j}\cap F_{k}$ of $P$ corresponds to a
$K_{4}$-subgraph according to Example 2.9. Its edges are $F_{i}\cap F_{j}$,
$F_{j}\cap F_{k}$, $F_{k}\cap F_{i}$, and three additional edges formed by
edges of the facets $F_{i}$, $F_{j}$ and $F_{k}$ complementary to the first
three edges.

######  Example 10.13.

It is known that any simple $3$-polytope can be combinatorially obtained from
$\Delta^{3}$ by a sequence of operations of cutting off a vertex or a set of
successive edges of some facet by a single plane (V. Eberhard (1891), M.
Brückner (1900), see [Gb03]). Each operation corresponds to a subdivision of a
facet of a graph into two facets by a new edge. Each sequence of such
operations connecting $\Delta^{3}$ and $P$ corresponds a $K_{4}$-subgraph of
$P$.

There is the following characterisation of complexes $\mathcal{C}(3,k)$.

######  Lemma 10.14.

Let $c$ be a coloring of a simple $3$-polytope $P$. Then

  1. (1)

$\mathcal{C}^{1}(P,c)$ is empty (equivalently,
$\mathcal{C}(P,c)\simeq\mathcal{C}(3,1)$) if and only if the complex
$\mathcal{C}(P,c)$ has exactly one facet;

  2. (2)

$\mathcal{C}^{1}(P,c)$ is a circle (equivalently,
$\mathcal{C}(P,c)\simeq\mathcal{C}(3,2)$) if and only if the complex
$\mathcal{C}(P,c)$ has exactly two facets;

  3. (3)

$\mathcal{C}^{1}(P,c)$ is a theta-graph (equivalently,
$\mathcal{C}(P,c)\simeq\mathcal{C}(3,3)$) if and only if $\mathcal{C}(P,c)$
has exactly three facets and all of them are disks;

  4. (4)

$\mathcal{C}^{1}(P,c)$ is a $K_{4}$-graph (equivalently,
$\mathcal{C}(P,c)\simeq\mathcal{C}(3,4)$) if and only if $\mathcal{C}(P,c)$
has exactly four facets, all of them are disks and any two of them intersect.

###### Proof.

The “only if” part follows from the definition. If $\mathcal{C}(P,c)$ has
exactly two facets, then both of them are disks and they intersect at the
common boundary circle $\mathcal{C}^{1}(P,c)$. If $\mathcal{C}(P,c)$ has
exactly three facets and all of them are disks, consider two of them. Their
intersection should be an edge, and the complement to their union is the
interior of the third disk. Thus, $\mathcal{C}^{1}(P,c)$ is a theta-graph. If
$\mathcal{C}(P,c)$ has exactly four facets, all of them are disks and any two
of them intersect, consider two disks. Their intersection can be either an
edge, or a pair of edges, for otherwise there are more than $4$ facets. If the
intersection is a pair of edges, then the complementary two facets do not
intersect, which is a contradiction. Thus, the intersection of any two facets
is an edge and any edge belongs to two facets. Then any facet is a triangle
and $\mathcal{C}^{1}(P,c)$ is a $K_{4}$-graph. ∎

##  11\. Hyperelliptic manifolds $N(P,\lambda)$ over $3$-polytopes

######  Definition 11.1.

A Hamiltonian cycle of a polytope $P$ is a simple edge-cycle in the graph of
$P$ containing all the vertices of $P$. Let us call a theta-subgraph or a
$K_{4}$-subgraph of $P$ Hamiltonian if it contains all the vertices of $P$.
More generally, for a coloring $\kappa$ of a simple polytope $P$ we call an
empty set $\varnothing$, a simple cycle, a theta-subgraph or a
$K_{4}$-subgraph of $\mathcal{C}^{1}(P,\kappa)$ Hamiltonian, if it contains
all the vertices of $\mathcal{C}(P,\kappa)$. Here by a simple cycle we mean
either a circle (that is a $1$-face without vertices) or a simple edge-cycle
in $\mathcal{C}^{1}(P,\kappa)$. In particular, if an empty set or a circle is
Hamiltonian, then $\mathcal{C}^{1}(P,\kappa)$ has no vertices, and it is a
disjoint union of circles.

In the papers [M90, VM99M, VM99S2] the authors constructed examples of
hyperelliptic $3$-manifolds in five Thurston’s geometries $\mathbb{R}^{3}$,
$\mathbb{H}^{3}$, $\mathbb{S}^{3}$, $\mathbb{H}^{2}\times\mathbb{R}$, and
$S^{2}\times\mathbb{R}$. In each case $M$ is obtained as $X/\pi$, where $X$ is
a geometry and $\pi$ is a discrete group of isometries acting freely on $X$.
These examples were build using a right-angled $3$-polytope $P$ equipped with
a Hamiltonian cycle, a Hamiltonian theta-subgraph, or a Hamiltonian
$K_{4}$-subgraph.

In this section we will enumerate all hyperelliptic $3$-manifolds
$N(P,\lambda)$ corresponding to affine colorings of rank $r$ such that the
hyperelliptic involution belongs to the group
$\mathbb{Z}_{2}^{r}=H_{0}^{\prime}$ canonically acting on $N(P,\lambda)$. In
turns out that in the case of a right-angled polytope $P$ and an affinely
independent coloring $\lambda$ these are exactly manifolds built by A.D.
Mednykh and A.Yu. Vesnin. In general case these manifolds correspond to proper
Hamiltonian cycles, theta- and $K_{4}$-subgraphs in the complexes
$\mathcal{C}(P,\kappa)$ defined by colorings $\kappa$ of simple $3$-polytopes.

######  Construction 11.2 (An affine coloring induced by a Hamiltonian
subgraph).

Let $\kappa$ be a coloring of a simple polytope $P$. Given a proper
Hamiltonian cycle, theta-, or $K_{4}$-subgraph
$\Gamma\subset\mathcal{C}^{1}(P,\kappa)$ one can define an affine coloring
$\Lambda_{\Gamma}$ induced by $\Gamma$ and a special hyperelliptic manifold
$N(P,\kappa,\Gamma)=N(P,\lambda_{\Gamma})$ as follows.

Consider a facet $D$ of $\Gamma$ such that $D$ is a union of more than one
facets of $\mathcal{C}(P,\kappa)$. Such a facet exists if
$\Gamma\neq\mathcal{C}^{1}(P,\kappa)$. The facet $D$ is a disk bounded by a
simple cycle of $\mathcal{C}^{1}(P,\kappa)$ and containing no vertices of
$\mathcal{C}^{1}(P,\kappa)$ in its interior. Consider the adjacency graph
$G_{D}$ of the facets of $\mathcal{C}(P,\kappa)$ lying in $D$. Its vertices
are facets and its edges correspond to $1$-faces of $\mathcal{C}(P,\kappa)$
lying in two facets. The graph $G_{D}$ is connected. If its edge $E$
corresponds to an edge $e$ of $\mathcal{C}(P,\kappa)$, then $e$ has vertices
on $\partial D$ and $E$ is a bridge. If $E$ corresponds to the circle of
$\mathcal{C}(P,\kappa)$, then $E$ is also a bridge. Thus, $G_{D}$ is a tree
and its vertices can be colored in two colors such that adjacent vertices have
different colors. Hence, the facets of $\Gamma$ define a coloring $c$ of $P$
constant on them, and the tree corresponding to each facet defines the
$0/1$-coloring $\chi$ in Construction 8.6. We obtain an affine coloring
$\lambda_{\Gamma}=\lambda(c,\chi)$ and a special hyperelliptic manifold
$N(P,\kappa,\Gamma)=N(P,\lambda_{\Gamma})$ of rank $r$, where $r=2$ for a
Hamiltonian cycle, $r=3$ for a Hamiltonian theta-subgraph, and $r=4$ for a
Hamiltonian $K_{4}$-subgraph. Moreover,
$\mathcal{C}^{1}(P,(\lambda_{\Gamma})_{\tau})=\Gamma$.

Similarly, a proper Hamiltonian empty set $\Gamma=\varnothing$ induces an
affine coloring $\lambda_{\Gamma}$ and defines a special hyperelliptic
manifold $N(P,\kappa,\Gamma)=N(P,\lambda_{\Gamma})$ of rank $r=1$. Namely, if
the complex $\mathcal{C}(P,\kappa)$ has no vertices, then
$\mathcal{C}^{1}(P,\kappa)$ is a disjoint union of circles and each circle
divides the sphere $\partial P$ into two disks. Then the adjacency graph of
facets of $\mathcal{C}(P,\kappa)$ is a tree and we can define the
$0/1$-coloring $\chi$ and the constant coloring $c$ in Construction 8.6.

######  Remark 11.3.

It is not true that if the manifolds $N(P,\kappa,\Gamma)$ and
$N(Q,\kappa^{\prime},\Gamma^{\prime})$ are weakly equivariantly homeomorphic,
then there is an equivalence
$\mathcal{C}(P,\kappa)\to\mathcal{C}(Q,\kappa^{\prime})$ such that
$\Gamma\to\Gamma^{\prime}$. Two combinatorially different Hamiltonian
subgraphs in $\mathcal{C}(P,\kappa)$ may induce the same affine coloring
$\lambda(c,\chi)$. In Fig. 17 there is a polytope $P$ with three Hamiltonian
cycles inducing the same affine coloring of rank $2$ in four colors. Two of
these cycles can be moved to each other by a combinatorial equivalence of $P$,
but the third can not.

######  Definition 11.4.

A matching of a graph $G$ is a disjoint set of edges. A matching is perfect,
if it contains all the vertices of $G$. Perfect matching is also called a
$1$-factor. A $1$-factorization is a partition of the set of edges of $G$ into
disjoint $1$-factors. A perfect pair from a $1$-factorization is a pair of
$1$-factors whose union induces a Hamiltonian cycle. A perfect
$1$-factorization of a graph is a 1-factorization having the property that
every pair of $1$-factors is a perfect pair.

Any Hamiltonian cycle $\Gamma$ in a $3$-valent graph $G$ defines the following
$1$-factorisation of $G$. Each edge of $G$ not lying in $\Gamma$ connects two
different vertices of $G$ and any vertex belongs to a unique edge of this
type. We obtain a $1$-factor. Then there are even number of vertices and edges
in $\Gamma$ and it is partitioned into two additional $1$-factors.

We will call a Hamiltonian cycle in a $3$-valent graph $k$-Hamiltonian, if the
corresponding $1$-factorization has exactly $k$ perfect pairs.

######  Theorem 11.5.

Let $\lambda$ be an affine coloring of rank $r$ of a simple $3$-polytope $P$.
Then $N(P,\lambda)$ is a hyperelliptic manifold with a hyperelliptic
involution lying in the group $\mathbb{Z}_{2}^{r}=H_{0}^{\prime}$ of
orientation preserving involutions canonically acting on $N(P,\lambda)$ if and
only if $1\leqslant r\leqslant 4$ and $\lambda$ is induced by

  1. (1)

a Hamiltonian empty set in $\mathcal{C}^{1}(P,\lambda)$ for $r=1$;

  2. (2)

a Hamiltonian cycle in $\mathcal{C}^{1}(P,\lambda)$ for $r=2$;

  3. (3)

a Hamiltonian theta-subgraph in $\mathcal{C}^{1}(P,\lambda)$ for $r=3$;

  4. (4)

a Hamiltonian $K_{4}$-subgraph in $\mathcal{C}^{1}(P,\lambda)$ for $r=4$.

Hyperelliptic involutions in $\mathbb{Z}_{2}^{r}$ bijectively correspond to
the Hamiltonian subgraphs of the above type inducing the coloring $\lambda$.
Moreover,

  1. (1)

for $r=1$ there is a unique hyperelliptic involution;

  2. (2)

for $r=2$ there can be $1$, $2$ or $3$ such involutions. If the Hamiltonian
cycle is a circle, then there is a unique hyperelliptic involution. For the
Hamiltonian edge-cycle each involution corresponds to a perfect pair of
$1$-factors. In particular, there are $k\geqslant 2$ hyperelliptic involutions
if and only if $\mathcal{C}^{1}(P,\lambda)$ is a connected $3$-valent graph
and $\lambda$ is induced by a $k$-Hamiltonian cycle.

  3. (3)

for $r=3$ and

    1. (a)

$I(\lambda)=4$ there can be $1$, $2$, $3$, $4$ or $6$ hyperelliptic
involutions;

    2. (b)

$I(\lambda)=5$ there can be $1$, $2$ or $3$ such involutions;

    3. (c)

$I(\lambda)=6$ there is a unique hyperelliptic involution;

  4. (4)

for $r=4$ and

    1. (a)

$I(\lambda)=5$ there can be $1$, $2$ or $6$ hyperelliptic involutions;

    2. (b)

$I(\lambda)=6$ there can be $1$ or $2$ such involutions;

    3. (c)

$I(\lambda)\in\\{7,8\\}$ there is a unique hyperelliptic involution;

We will obtain this result as a corollary of the following lemma and a more
technical theorem.

######  Lemma 11.6.

Let $\lambda$ be an affine coloring of rank $r$ of a simple $3$-polytope $P$
and $\tau\in\mathbb{Z}_{2}^{r}$. Then
$\mathcal{C}(P,\lambda_{\tau})\simeq\mathcal{C}(3,r)$ (that is, $\tau$ is a
hyperelliptic involution) if and only if one of the following conditions hold:

  1. (1)

$r=1$ and $\mathcal{C}^{1}(P,\lambda_{\tau})$ is a Hamiltonian empty set in
$\mathcal{C}^{1}(P,\lambda)$;

  2. (2)

$r=2$ and $\mathcal{C}^{1}(P,\lambda_{\tau})$ is a Hamiltonian cycle in
$\mathcal{C}^{1}(P,\lambda)$;

  3. (3)

$r=3$ and $\mathcal{C}^{1}(P,\lambda_{\tau})$ is a Hamiltonian theta-subgraph
in $\mathcal{C}^{1}(P,\lambda)$;

  4. (4)

$r=4$ and $\mathcal{C}^{1}(P,\lambda_{\tau})$ is a Hamiltonian
$K_{4}$-subgraph in $\mathcal{C}^{1}(P,\lambda)$.

In all these cases $\lambda$ is induced by the corresponding Hamiltonian
subgraph.

###### Proof.

The lemma follows from Theorem 10.1 and Corollary 5.13. ∎

######  Theorem 11.7.

Let $\lambda$ be an affine coloring of rank $r$ of a simple $3$-polytope $P$.
Then $N(P,\lambda)$ is a hyperelliptic manifold with a hyperelliptic
involution lying in the group $\mathbb{Z}_{2}^{r}=H_{0}^{\prime}$ of
orientation preserving involutions canonically acting on $N(P,\lambda)$ if and
only if $1\leqslant r\leqslant 4$ and one of the following conditions holds:

  1. (1)

$I(\lambda)=\\{\boldsymbol{p}_{1},\dots,\boldsymbol{p}_{r+1}\\}$ is a boolean
$r$-simplex, $1\leqslant r\leqslant 4$, and at least for one vector
$\tau=\boldsymbol{p}_{i}+\boldsymbol{p}_{j}$, $i\neq j$, the complex
$\mathcal{C}(P,\lambda_{\tau})$ is equivalent to $\mathcal{C}(3,r)$. Each
hyperelliptic involution $\tau\in\mathbb{Z}_{2}^{r}$ has this form and there
are at most $\frac{r(r+1)}{2}$ such involutions. More precisely, an involution
$\tau\in\mathbb{Z}_{2}^{r}$ is hyperelliptic if and only if
$\tau=\boldsymbol{p}_{i}+\boldsymbol{p}_{j}$, $i\neq j$, and for

     * •

$r=1$ it is equal to $1\in\mathbb{Z}_{2}$. This is always a unique
hyperelliptic involution.

     * •

$r=2$ the set $G(\boldsymbol{p}_{k})$, $\\{i,j,k\\}=\\{1,2,3\\}$ is a disk.
There can be $0$, $1$, $2$, or $3$ such involutions.

     * •

$r=3$ each set $G(\boldsymbol{p}_{i},\boldsymbol{p}_{j})$,
$G(\boldsymbol{p}_{k})$, $G(\boldsymbol{p}_{l})$,
$\\{i,j,k,l\\}=\\{1,2,3,4\\}$ is a disk. There can be $0$, $1$, $2$, $3$, $4$
or $6$ such involutions.

     * •

$r=4$ each set $G(\boldsymbol{p}_{i},\boldsymbol{p}_{j})$,
$G(\boldsymbol{p}_{k})$, $G(\boldsymbol{p}_{l})$, $G(\boldsymbol{p}_{s})$,
$\\{i,j,k,l,s\\}=\\{1,2,3,4,5\\}$ is a disk and any two of these disks
intersect. There can be $0$, $1$, $2$ or $6$ hyperelliptic involutions.

The classification of complexes with more than one hyperelliptic involution
and the corresponding manifolds $N(P,\lambda)$ is presented in Fig. 2.

Figure 2. All complexes with more than one hyperelliptic involution for the
case when $I(\lambda)$ is a boolean simplex. On the top we write the
homeomorphism type of $N(P,\lambda)$, where $k\geqslant 0$ is the number of
dashed circles

  2. (2)

$I(\lambda)=\Pi^{2}*\Delta^{r-3}$, $2\leqslant r\leqslant 4$, where
$\Pi^{2}=\\{\boldsymbol{q}_{1},\boldsymbol{q}_{2},\boldsymbol{q}_{3},\boldsymbol{q}_{4}\\}\simeq\mathbb{Z}^{2}$
is a boolean $2$-plane and
$\Delta^{r-3}=\\{\boldsymbol{p}_{1},\dots,\boldsymbol{p}_{r-2}\\}$ is a
boolean simplex, and at least for one vector
$\tau=\boldsymbol{q}_{i}+\boldsymbol{q}_{j}$, $i\neq j$, the complex
$\mathcal{C}(P,\lambda_{\tau})$ is equivalent to $\mathcal{C}(3,r)$. Each
hyperelliptic involution $\tau\in\mathbb{Z}_{2}^{r}$ has this form and there
are at most three such involutions. More precisely, an involution
$\tau\in\mathbb{Z}_{2}^{r}$ is hyperelliptic if and only if
$\tau=\boldsymbol{q}_{i}+\boldsymbol{q}_{j}=\boldsymbol{q}_{k}+\boldsymbol{q}_{l}$
for some partition $\\{1,2,3,4\\}=\\{i,j\\}\sqcup\\{k,l\\}$, and one of the
following conditions hold

     * •

$r=2$ and $G(\boldsymbol{q}_{i},\boldsymbol{q}_{j})$ is a disk (then
$G(\boldsymbol{q}_{k},\boldsymbol{q}_{l})$ is also a disk bounded by the same
Hamiltonian cycle $\Gamma$ in $\mathcal{C}^{1}(P,\lambda)$). There can be $0$,
$1$, $2$, or $3$ such involutions. Moreover, there are $k\geqslant 2$
hyperelliptic involutions if and only if $\mathcal{C}^{1}(P,\lambda)$ is a
connected $3$-valent graph and $\Gamma$ is a $k$-Hamiltonian cycle in it. For
$k=3$ additionally $\mathcal{C}(P,\lambda)\simeq\partial Q$ for a simple
$3$-polytope $Q$.

     * •

$r=3$ and each set $G(\boldsymbol{q}_{i},\boldsymbol{q}_{j})$,
$G(\boldsymbol{q}_{k},\boldsymbol{q}_{l})$ and $G(\boldsymbol{p}_{1})$ is a
disk. There can be $0$, $1$, $2$ or $3$ such involutions. Moreover, if there
are $2$ hyperelliptic involutions, then $G(\boldsymbol{p}_{1})$ is a
quadrangle, a triangle, or a bigon, and the complex $\mathcal{C}(P,\lambda)$
can be reduced to a complex $\mathcal{C}(P,\lambda^{\prime})$ for an affine
coloring $\lambda^{\prime}$ of rank $2$ either

       * –

with $2$ or $3$ hyperelliptic involutions by reductions (a)-(d), or (f) in
Fig. 3, or

       * –

with $2$ hyperelliptic involutions by reduction (e).

If there are $3$ hyperelliptic involutions, then $G(\boldsymbol{p}_{1})$ is a
triangle and $\mathcal{C}(P,\lambda)$ can be reduced to
$\mathcal{C}(P,\lambda^{\prime})$ of rank $2$ with $3$ hyperelliptic
involutions by reduction (e).

     * •

$r=4$ each set $G(\boldsymbol{q}_{i},\boldsymbol{q}_{j})$,
$G(\boldsymbol{q}_{k},\boldsymbol{q}_{l})$, $G(\boldsymbol{p}_{1})$ and
$G(\boldsymbol{p}_{2})$ is a disk and any two of these disks intersect. There
can be $0$, $1$ or $2$ hyperelliptic involutions. Moreover, if there are $2$
hyperelliptic involutions, then $G(\boldsymbol{p}_{1},\boldsymbol{p}_{2})$ is
a quadrangle, a triangle, or a $2$-gon, and the complex
$\mathcal{C}(P,\lambda)$ can be reduced to a complex
$\mathcal{C}(P,\lambda^{\prime})$ for an affine coloring $\lambda^{\prime}$ of
rank $2$ with $2$ or $3$ hyperelliptic involutions by reductions (a)-(f) in
Fig. 3.