# Four-manifolds defined by vector-colorings of simple polytopes

Nikolai Erokhovets  Steklov Mathematical Institute of Russian Academy of
Sciences, Moscow, Russia &Department of Mechanics and Mathematics, Lomonosov
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
$\mathbb{R}\mathcal{Z}_{P}$ over a simple $n$-polytope $P$. The orbit space
$N(P,H)=\mathbb{R}\mathcal{Z}_{P}/H$ has an action of $\mathbb{Z}_{2}^{m}/H$.
For general $n$ we introduce the notion of a Hamiltonian
$\mathcal{C}(n,k)$-subcomplex generalizing the three-dimensional notions of a
Hamiltonian cycle, theta- and $K_{4}$-subgraphs. Each
$\mathcal{C}(n,k)$-subcomplex $C\subset\partial P$ corresponds to a subgroup
$H_{C}$ such that $N(P,H_{C})\simeq S^{n}$. We prove that in dimensions
$n\leqslant 4$ this correspondence is a bijection. Any subgroup
$H\subset\mathbb{Z}_{2}^{m}$ defines a complex
$\mathcal{C}(P,H)\subset\partial P$. We prove that each Hamiltonian
$\mathcal{C}(n,k)$-subcomplex $C\subset\mathcal{C}(P,H)$ inducing $H$
corresponds to a hyperelliptic involution $\tau_{C}\in\mathbb{Z}_{2}^{m}/H$ on
the manifold $N(P,H)$ (that is, an involution with the orbit space
homeomorphic to $S^{n}$) and in dimensions $n\leqslant 4$ this correspondence
is a bijection. We prove that for the geometries $\mathbb{X}=\mathbb{S}^{4}$,
$\mathbb{S}^{3}\times\mathbb{R}$, $\mathbb{S}^{2}\times\mathbb{S}^{2}$,
$\mathbb{S}^{2}\times\mathbb{R}^{2}$, $\mathbb{S}^{2}\times\mathbb{L}^{2}$,
and $\mathbb{L}^{2}\times\mathbb{L}^{2}$ there exists a compact right-angled
$4$-polytope $P$ with a free action of $H$ such that the geometric manifold
$N(P,H)$ has a hyperelliptic involution in $\mathbb{Z}_{2}^{m}/H$, and for
$\mathbb{X}=\mathbb{R}^{4}$, $\mathbb{L}^{4}$,
$\mathbb{L}^{3}\times\mathbb{R}$ and $\mathbb{L}^{2}\times\mathbb{R}^{2}$
there are no such polytopes.

###### Key words and phrases:

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

###### 2010 Mathematics Subject Classification:

57S12, 57S17, 57S25, 52B05, 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 Introduction
  2. 2 Rational homology $4$-spheres among $N(P,\Lambda)$
  3. 3 A criterion when $N(P,\Lambda)\simeq S^{4}$
  4. 4 Hyperelliptic manifolds $N(P,\Lambda)$
    1. 4.1 Hamiltonian subcomplexes
    2. 4.2 Bipartite Hamiltonian subcomplexes and orbit spaces of involutions
    3. 4.3 Bipartite Hamiltonian subcomplexes and the four color theorem
    4. 4.4 Subspaces of $N(P,\Lambda)$ corresponding to faces of $\mathcal{C}(P,\Lambda)$
    5. 4.5 Bipartite Hamiltonian subcomplexes and two-sheeted branched coverings
    6. 4.6 $\mathcal{C}(n,k)$-subcomplexes and hyperelliptic involutions
    7. 4.7 Geometric hyperelliptic manifolds $N(P,\Lambda)$
  5. 5 Acknowledgements

##  1\. Introduction

The paper is motivated by questions arisen during discussions of results of
[E24] with A.D. Mednykh.

For an introduction to the polytope theory we refer to [Gb03, Z95]. We will
use definitions and notations from [E24], but for convenience try to write
them explicitly. In this paper for topological spaces the notation $X\simeq Y$
means that $X$ and $Y$ are homeomorphic, and for complexes $C_{1}\simeq C_{2}$
means that $C_{1}$ and $C_{2}$ are equivalent.

Toric topology (see [BP15, DJ91]) assigns to each simple $n$-polytope with $m$
facets $F_{1},\dots,F_{m}$ the real moment-angle manifold

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

and $\boldsymbol{e}_{1},\dots,\boldsymbol{e}_{m}$ is the standard basis in
$\mathbb{Z}_{2}^{m}$. $\mathbb{R}\mathcal{Z}_{P}$ is a smooth manifold with a
smooth action of $\mathbb{Z}_{2}^{m}$ such that
$\mathbb{R}\mathcal{Z}_{P}/\mathbb{Z}_{2}^{m}=P$.

We consider (non-necessarily free) actions of subgroups
$H\subset\mathbb{Z}_{2}^{m}$ on $\mathbb{R}\mathcal{Z}_{P}$. Each subgroup $H$
of rank $m-r$ can be described as a kernel of an epimorphism
$\mathbb{Z}_{2}^{m}\to\mathbb{Z}_{2}^{r}$ mapping the basis vector
$\boldsymbol{e}_{i}$ to $\Lambda_{i}\in\mathbb{Z}_{2}^{r}$.

######  Definition 1.1.

We call a mapping $\Lambda\colon\\{F_{1},\dots,F_{m}\\}\to\mathbb{Z}_{2}^{r}$,
$F_{i}\to\Lambda_{i}$, such that
$\langle\Lambda_{1},\dots,\Lambda_{m}\rangle=\mathbb{Z}_{2}^{r}$ a vector-
coloring of $P$ of rank $r$. The subgroup $H$ is uniquely defined by
$\Lambda$, while $\Lambda$ is defined up to a linear automorphism of
$\mathbb{Z}_{2}^{r}$. A vector-coloring is linearly independent if for each
face $F_{i_{1}}\cap\dots\cap F_{i_{k}}\neq\varnothing$ the vectors
$\Lambda_{i_{1}}$, $\dots$, $\Lambda_{i_{k}}$ are linearly independent. Each
vector-coloring $\Lambda$ of rank $r$ defines the orbit space
$N(P,\Lambda)=\mathbb{R}\mathcal{Z}_{P}/H$ with an action of
$\mathbb{Z}_{2}^{r}\simeq\mathbb{Z}_{2}^{m}/H$.

It can be shown that the action of $H$ is free if and only if $\Lambda$ is
linearly independent. In this case $N(P,\Lambda)$ is a smooth manifold. In
particular, linearly independent vector-colorings of rank $n$ correspond to
small covers $N(P,\Lambda)$ introduced in [DJ91].

Linearly independent-vector colorings also arise in the following construction
of geometric manifolds developed in the papers by A.Yu. Vesnin and A.D.
Mednykh in [MV86, V87, M90, VM99M, V17].

######  Construction 1.2.

Let $P$ be a compact right-angled polytope in some geometry $\mathbb{X}$ (in
this paper we will use geometries of the form
$\mathbb{X}=\mathbb{X}_{1}\times\dots\times\mathbb{X}_{k}$, where
$\mathbb{X}_{i}\in\\{\mathbb{S}^{n_{i}},\mathbb{R}^{n_{i}},\mathbb{L}^{n_{i}}\\}$
and by a right-angled polytope we mean the product of the corresponding right-
angled polytopes).

The polytope $P$ corresponds to a right-angled coxeter group

$\langle\rho_{1},\dots,\rho_{m}\rangle/(\rho_{1}^{2},\dots,\rho_{m}^{2},\rho_{i}\rho_{j}=\rho_{j}\rho_{i},\text{
if }F_{i}\cap F_{j}\neq\varnothing).$

This group is isomorphic to a subgroup $G(P)$ of isometries of $\mathbb{X}$
generated by reflections in facets of $P$, where $\rho_{i}$ corresponds to the
reflection in $F_{i}$. The group $G(P)$ acts on $\mathbb{X}$ discretely, $P$
is a fundamental domain and the orbit space, and for any point of $P$ its
stabilizer is generated by reflections in facets containing this point (see
[VS88, Theorem 1.2 in Chapter 5]).

A linearly independent vector coloring $\Lambda$ of rank $r$ defines the
epimorphism $\varphi_{\Lambda}\colon G(P)\to\mathbb{Z}_{2}^{r}$ by the rule
$\varphi_{\Lambda}(\rho_{i})=\Lambda(F_{i})$. The subgroup ${\rm
Ker\,}\varphi_{\Lambda}$ acts freely on $\mathbb{X}$, and the quotient space
$\mathbb{X}/{\rm Ker\,}\varphi_{\Lambda}$ is a closed manifold with the
geometric structure modelled on $\mathbb{X}$. It is easy to see that
$N(P,\Lambda)\simeq\mathbb{X}/{\rm Ker\,}\varphi_{\Lambda}$ (see more details
in [E22M, Construction 4.11]).

For a general (not necessarily linearly independent) vector-coloring $\Lambda$
is was proved in [E24, Theorem 5.1] (jointly with D.V. Gugnin) that
$N(P,\Lambda)$ is a closed topological manifold if and only if for each face
$F_{i_{1}}\cap\dots\cap F_{i_{k}}\neq\varnothing$ different vectors among
$\\{\Lambda_{i_{1}},\dots,\Lambda_{i_{k}}\\}$ are linearly independent. (This
result also can be extracted from the general results by M.A. Mikhailova and
C. Lange [M85, LM16, L19].) This gives rise to the following definition.

######  Definition 1.3.

[E24, Definition 2.1] A coloring $c$ of a simple polytope $P$ in $l$ colors is
a surjective mapping from $\\{F_{1},\dots,F_{m}\\}$ to a finite set of $l$
elements. For convenience we identify the set with $[l]=\\{1,\dots,l\\}$, but
in this paper often this is a subset in $\mathbb{Z}_{2}^{r}$. 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 $P$.
We choose a linear order of all the facets $G_{1}$, $\dots$, $G_{M}$. Two
complexes $\mathcal{C}(P,c)$ and $\mathcal{C}(Q,c^{\prime})$ are equivalent
(we write $\mathcal{C}(P,c)\simeq\mathcal{C}(Q,c^{\prime})$) if there is a
homeomorphism $P\to Q$ sending facets of $\mathcal{C}(P,c)$ to facets of
$\mathcal{C}(Q,c^{\prime})$.

Each $k$-face of $\mathcal{C}(P,c)$ is a connected orientable $k$-manifold,
perhaps with a boundary (see Lemma 2.9). The closed manifold $N(P,\Lambda)$ is
orientable if and only if for some change of coordinates in
$\mathbb{Z}_{2}^{r}$ we have $\Lambda_{i}=(1,\lambda_{i})$ (see [E24,
Corollary 1.15]). We call the mapping $\lambda\colon F_{i}\to\lambda_{i}$ an
affine coloring of $P$ of rank $(r-1)$.

It can be shown (see [E24, Proposition 2.6]) that for any coloring $c$ of the
simplex $\Delta^{n}$ in $k$ colors the complex $\mathcal{C}(\Delta^{n},c)$ is
equivalent to the complex given on

$S^{n}_{k,\geqslant 0}=\\{x_{1}^{2}+\dots+x_{n+1}^{2}=1,x_{1}\geqslant
0,\dots,x_{k}\geqslant 0\\}$

by the facets $S^{n}_{k,\geqslant}\cap\\{x_{i}=0\\}$, $i=1,\dots,k$, and
([E24, Corollary]) to the complex given on

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

by the facets $B^{n}_{k-1,\geqslant}\cap\\{x_{s}=0\\}$, $s=1,\dots,k-1$, and
$B^{n}_{k-1,\geqslant}\cap\\{x_{1}^{2}+\dots+x_{n}^{2}=1\\}$. We denote the
equivalence class of these complexes $\mathcal{C}(n,k)$. For $n=3$ the complex
$\mathcal{C}(3,1)$ has a single facet $\partial P$, the complex
$\mathcal{C}(3,1)$ corresponds a simple edge-cycle in $\partial P$,
$\mathcal{C}(3,2)$ – to a theta-subgraph in $\partial P$ consisting of two
disjoint vertices of $P$ and three disjoint (outside these vertices) simple
edge-paths connecting them, and $\mathcal{C}(3,4)$ corresponds to a
$K_{4}$-subgraph in $\partial P$ consisting of $4$ disjoint vertices of $P$
pairwise connected by a set of disjoint simple edge-paths.

It is easy to see that if for a vector coloring of rank $r$ the complex
$\mathcal{C}(P,\Lambda)$ is equivalent to $\mathcal{C}(n,r)$, then
$N(P,\Lambda)\simeq S^{n}$ (see [E24, Construction 5.8]). The converse
trivially holds for $n=2$. It was proved in [E24, Theorem 10.1] that the
converse also holds for $n=3$. The first main result (Theorem 3.14) of this
paper states that this holds for $n=4$. Namely, for a vector-coloring
$\Lambda$ of rank $r$ of a simple polytope of dimension $n\leqslant 4$ we have
$N(P,\Lambda)\simeq S^{n}$ if and only if
$\mathcal{C}(P,\Lambda)\simeq\mathcal{C}(n,r)$. For general $n$ we prove
(Lemma 3.10 and Corollary 3.11) that if $N(P,\Lambda)\simeq S^{n}$, then for
each $\omega=\omega_{1}\sqcup\dots\sqcup\omega_{k}\subset[M]$ the set
$\left(\bigcup\limits_{i_{1}\in\omega_{1}}G_{i_{1}}\right)\cap\dots\cap\left(\bigcup\limits_{i_{k}\in\omega_{k}}G_{i_{k}}\right)$
is a rational homology $(n-k)$-disk if $\omega\neq[M]$, and a rational
homology $(n-k)$-sphere if $\omega=[M]$. In particular, $M\leqslant n+1$ and
each $k$-face of the complex $\mathcal{C}(P,\Lambda)$ is either a $k$-$RHD$ or
a $k$-$RHS$ (by definition we assume that for $k<0$ a $k$-$RHD$ is a point,
and a $k$-$RHS$ is empty).

On the base of these results we study hyperelliptic involutions in the group
$\mathbb{Z}_{2}^{r}$ acting on the manifold $N(P,\Lambda)$.

######  Definition 1.4.

Following [VM99S1] we call a closed $n$-manifold $M$ hyperelliptic if it has
an involution $\tau$ such that $M/\langle\tau\rangle\simeq S^{n}$. The
corresponding involution $\tau$ is called a hyperelliptic involution.

In dimension $n=3$ in the papers [M90, VM99M, VM99S2] A.D. Mednykh and
A.Yu.Vesnin constructed examples of hyperelliptic $3$-manifolds in five of
eight Thurston’s geometries: $\mathbb{S}^{3}$, $\mathbb{R}^{3}$,
$\mathbb{L}^{3}$, $\mathbb{S}^{2}\times\mathbb{R}$, and
$\mathbb{L}^{2}\times\mathbb{R}$. Each manifold was constructed using a
Hamiltonian cycle, a Hamiltonian theta-subgraph or a Hamiltonian
$K_{4}$-subgraph in a compact right-angled $3$-polytope, where a subgraph is
Hamiltonian, if it contains all the vertices of $P$. In [E24, Theorem 11.5] it
was proved that for a $3$-manifold $N(P,\Lambda)$ over a $3$-polytope
hyperelliptic involutions in $\mathbb{Z}_{2}^{r}$ are in bijection with a
Hamiltonian empty set, cycles, theta-subgraphs and $K_{4}$-subgraphs in
$\mathcal{C}(P,\Lambda)$ inducing $\Lambda$, where for a Hamiltonian subgraph
$\Gamma$ of the above types in the $1$-skeleton of $\mathcal{C}(P,c)$ there is
a canonical way to construct a vector-coloring $\widetilde{\Lambda}_{\Gamma}$
of $P$ induced by $\Gamma$ such that
$\mathcal{C}(P,c)=\mathcal{C}(P,\widetilde{\Lambda}_{\Gamma})$. Also in [E24,
Theorem 11.7] the classification of vector-colorings with more than one
hyperelliptic involutions was presented.

In this paper we generalize [E24, Theorem 11.5] to dimension $n=4$. For this
purpose we introduce the notion of a Hamiltonian $\mathcal{C}(n,k)$-subcomplex
$C\subset\mathcal{C}(P,c)$ (Definition 4.32) and explain (Construction 4.38)
how to build a canonical induced vector-coloring $\widetilde{\Lambda}_{C}$ of
rank $k+1$ such that $\mathcal{C}(P,c)=\mathcal{C}(P,\widetilde{\Lambda}_{C})$
and $N(P,\widetilde{\Lambda}_{C})$ has a canonical hyperelliptic involution in
$\mathbb{Z}_{2}^{k+1}$. Then for a vector-coloring $\Lambda$ of rank $r$ any
Hamiltonian $\mathcal{C}(n,r-1)$-subcomplex $C\subset\mathcal{C}(P,\Lambda)$
inducing $\Lambda$ corresponds to a hyperelliptic involution
$\tau_{C}\in\mathbb{Z}_{2}^{r}$. Our second main result (Theorem 4.39) is that
in dimensions $n\leqslant 4$ this correspondence is a bijection.

The third main result (Theorem 4.76 and Remark 4.47) of this paper is the
answer to the following question in dimension $n=4$.

######  Question 1.

For which geometries of the form
$\mathbb{X}=\mathbb{X}_{1}\times\dots\times\mathbb{X}_{k}$, where
$\mathbb{X}_{i}\in\\{\mathbb{S}^{n_{i}},\mathbb{R}^{n_{i}},\mathbb{L}^{n_{i}}\\}$
theres exists a right-angled polytope $P$ and a linearly independent vector-
coloring $\Lambda$ of rank $r$ such that the geometric manifold $N(P,\Lambda)$
has a hyperelliptic involution in the group $\mathbb{Z}_{2}^{r}$ canonically
acting on it.

In dimension $n=4$ there are $10$ geometries of this form. We prove that for
the geometries $\mathbb{S}^{4}$, $\mathbb{S}^{3}\times\mathbb{R}$,
$\mathbb{S}^{2}\times\mathbb{S}^{2}$, $\mathbb{S}^{2}\times\mathbb{R}^{2}$,
$\mathbb{S}^{2}\times\mathbb{L}^{2}$, and $\mathbb{L}^{2}\times\mathbb{L}^{2}$
there is such a pair $(P,\Lambda)$, and for the geometries $\mathbb{R}^{4}$,
$\mathbb{L}^{4}$, $\mathbb{L}^{3}\times\mathbb{R}$,
$\mathbb{L}^{2}\times\mathbb{R}^{2}$ there are no such pairs.

The paper is organized as follows.

In Section 2 on the base of results from [CP17, CP20] we give a criterion
(Lemma 2.4) when $N(P,\Lambda)$ is a rational homology $n$-sphere and specify
it (Theorem 2.7) for dimensions $n\leqslant 4$.

In Section 3 we prove our first main result (Theorem 3.14) – a criterion when
$N(P,\Lambda)\simeq S^{n}$ for $n\leqslant 4$. For the proof we use the
Armstrong theorem and results from Section 2. For general $n$ we give a
necessary condition (Lemma 3.10 and Corollary 3.11).

In Section 4 we study hyperelliptic involutions in $\mathbb{Z}_{2}^{r}$ acting
on manifolds $N(P,\Lambda)$.

In Subsection 4.1 we introduce the notions of a subcomplex
$\mathcal{C}(P,c_{2})\subset\mathcal{C}(P,c_{1})$ (Construction 4.2) and a
Hamiltonian subcomplex (Definition 4.3). We give a criterion (Lemma 4.4) when
the subcomplex $\mathcal{C}(P,c_{2})\subset\mathcal{C}(P,c_{1})$ is
Hamiltonian. We introduce (Definition 4.5) the notion of defining
$(n-2)$-faces of a subcomplex and reformulate (Proposition 4.6) the criterion
in terms of the defining faces. In Proposition 4.7 we give a criterion when a
set of disjoint $(n-2)$-faces is a set of defining faces of a Hamiltonian
subcomplex.

In Subsection 4.2 we introduce (Definition 4.8) the notion of the adjacency
graph of a Hamiltonian subcomplex
$\mathcal{C}(P,c_{2})\subset\mathcal{C}(P,c_{1})$. If this graph is bipartite
we call the Hamiltonian subcomplex bipartite (Definition 4.9).

  * •

In Construction 4.11 we show how to induce the vector-coloring from a
bipartite Hamiltonian subcomplex and in Example 4.13 we explain how a
bipartite Hamiltonian subcomplex $C$ defines a canonical vector-coloring
induced by $C$.

  * •

In Proposition 4.14 we show that for a closed manifold $N(P,\Lambda)$ there is
a bijection between the involutions in $\mathbb{Z}_{2}^{r}\setminus\\{0\\}$
and proper bipartite Hamiltonian subcomplexes $C\subset\mathcal{C}(P,\Lambda)$
such that $\Lambda$ is induced from some vector-coloring of $C$.

  * •

In Proposition 4.15 we give a criterion when for a closed manifold
$N(P,\Lambda)$ and an involution $\tau\in\mathbb{Z}_{2}^{r}$ the orbit space
$N(P,\Lambda)/\langle\tau\rangle$ is also a closed manifold.

  * •

In Proposition 4.16 we prove that if the vector-coloring of a bipartite
Hamiltonian subcomplex defines a closed manifold, then so does the induced
vector-coloring.

In Subsection 4.3 using the notion of a bipartite Hamiltonian subcomplex we
reformulate (Corollary 4.20) the four color theorem for a simple $3$-polytope
$P$ in terms of a disjoint union of circles containing all the vertices of
$P$.

In Subsection 4.4 we study the structure of faces of the complex
$\mathcal{C}(P,c)$ (Lemmas 4.22 and 4.23, Corollary 4.25).

  * •

In Proposition 4.27 for a closed manifold $N(P,\Lambda)$ and a face $G$ of
$\mathcal{C}(P,\Lambda)$ we describe the submanifold in $N(P,\Lambda)$ arising
as a preimage of $G$ under the projection $N(P,\Lambda)\to P$.

  * •

In Proposition 4.28 we describe the face structure of defining faces $M_{q}$
of a Hamiltonian subcomplex $C\subset\mathcal{C}(P,c)$ in terms of the
intersections of $M_{q}$ with faces of $C$.

  * •

In Proposition 4.29 and Corollary 4.30 we describe $k$-faces of a Hamiltoninan
subcomplex $C\subset\mathcal{C}(P,c)$ as unions of $k$-faces of
$\mathcal{C}(P,c)$ separated by $(k-1)$-faces of defining faces.

In Subsection 4.5 for a closed manifold $N(P,\Lambda)$ and an involution
$\tau\in\mathbb{Z}_{2}^{r}$ such that $N(P,\Lambda)/\langle\tau\rangle$ is
also a closed manifold we describe the mapping $N(P,\Lambda)\to
N(P,\Lambda)/\langle\tau\rangle$ as a two-sheeted branched covering.

In Subsection 4.6 we introduce (Definition 4.32) the notions of a
$\mathcal{C}(n,k)$-subcomplex and a Hamiltonian $\mathcal{C}(n,k)$-subcomplex.

  * •

In Corollary 4.33 we describe eplicitly the subgroup
$H(C)\subset\mathbb{Z}_{2}^{m}$ corresponding to a
$\mathcal{C}(n,k)$-subcomplex $C\subset\partial P$.

  * •

In Corollary 4.34 we reformulate our first main result in terms of
$\mathcal{C}(n,k)$-subcomplexes $C\subset\partial P$ of a polytope $P$ of
dimension $n\leqslant 4$.

  * •

In Proposition 4.35 we prove that for any facet $\widetilde{G}$ of a proper
Hamiltonian $\mathcal{C}(n,k)$-subcomplex $C\subset\mathcal{C}(P,c)$ the
adjacency graph of facets of $\mathcal{C}(P,c)$ lying in $\widetilde{G}$ is a
tree. In particular, any proper Hamiltonian $\mathcal{C}(n,k)$-subcomplex is
bipartite.

  * •

In Construction 4.38 we show that any proper Hamiltonian
$\mathcal{C}(n,k)$-subcomplex $C\subset\mathcal{C}(P,c)$ induces a canonical
vector-coloring $\widetilde{\Lambda}_{C}$ of rank $(k+1)$ such that
$\mathcal{C}(P,c)=\mathcal{C}(P,\widetilde{\Lambda}_{C})$ and
$N(P,\widetilde{\Lambda}_{C})$ is a hyperelliptic manifold with a canonical
hyperelliptic involution in $\mathbb{Z}_{2}^{k+1}$ corresponding to $C$.

  * •

In Theorem 4.39 we formulate our second main result, namely that for
$n\leqslant 4$ and a vector-coloring of rank $r$ of a simple $n$-polytope $P$
such that $N(P,\Lambda)$ is a closed manifold hyperelliptic involutions are in
bijection with proper Hamiltonian $\mathcal{C}(n,r-1)$-subcomplexes
$C\subset\mathcal{C}(P,\Lambda)$ inducing $\Lambda$.

  * •

In Proposition 4.41 we prove that for a proper Hamiltonian
$\mathcal{C}(n,r)$-subcomplex $C\subset\mathcal{C}(P,c)$ if a $k$-face
$\widetilde{G}$ of $C$ is not a circle, then the adjacency graph
$\Gamma(\widetilde{G})$ of $k$-faces $G$ of $\mathcal{C}(P,c)$ lying in
$\widetilde{G}$ is a tree.

  * •

Using this fact in Proposition 4.43 we obtain the formula expressing the
number of $k$-faces of a complex $\mathcal{C}(P,c)$ in terms of the numbers of
$k$-faces and $(k-1)$-faces of defining faces of a proper Hamiltonian
$\mathcal{C}(n,r)$-subcomplex.

  * •

In Proposition 4.44 we prove that for any defining $(n-2)$-face $M_{q}$ of a
proper Hamiltonian $\mathcal{C}(n,r)$-subcomplex $C\subset\mathcal{C}(P,c)$
its facets can be colored in $(r-1)$ colors in such a way that adjacent facets
have different colors.

In Subsection 4.7 we study geometric structures on hyperelliptic manifolds
$N(P,\Lambda)$. Our main tool is Construction 4.45 of a geometric
hyperelliptic manifold using a proper Hamiltonian
$\mathcal{C}(n,k)$-subcomplex $C$ in the boundary of a compact right-angled
polytope $P$ in some geometry $\mathbb{X}$. This construction is a direct
generalization of the construction from [M90, VM99M, VM99S2] based on a
Hamiltonian cycle, theta- or $K_{4}$-subgraph in the boundary of a right-
angled $3$-polytope.

  * •

In Corollary 4.49 we show that $\mathbb{I}^{n}$ does not admit Hamiltonian
$\mathcal{C}(n,k)$-subcomplexes for $n\leqslant 4$. In particular, the
geometry $\mathbb{R}^{n}$ does not admit Construction 4.45 for $n\geqslant 4$.

  * •

In Corollaries 4.51 and 4.53 and Remark 4.52 we show that if $P$ admits a
Hamiltonian $\mathcal{C}(n,n-1)$-subcomplex, then each defining face is a
polytope with only even-gonal $2$-faces.

  * •

In Proposition 4.56 we show that if a $4$-polytope $P$ admits a proper
Hamiltonian $\mathcal{C}(4,r)$-subcomplex, then $P$ has at least one $2$-face
with three or four edges. In particular, the geometry $\mathbb{L}^{4}$ does
not admit Construction 4.45, as well as $\mathbb{L}^{n}$ for $n\geqslant 5$
since in the latter geometries there are no compact right-angled polytopes due
to results by V.V. Nikulin [N82] (see [V17]).

  * •

In Proposition 4.57 we show that if a $4$-polytope $P=Q\times I$ admits a
proper Hamiltonian $\mathcal{C}(4,r)$-subcomplex, then one of the defining
faces is a triangle. In particular, the geometries
$\mathbb{L}^{3}\times\mathbb{R}$ and $\mathbb{L}^{2}\times\mathbb{R}^{2}$ do
not admit Construction 4.45.

  * •

In Proposition 4.58 we show that $\Delta^{n}$ admits a Hamiltonian
$\mathcal{C}(n,n)$-subcomplex. In particular, the geometry $\mathbb{S}^{n}$
admits Construction 4.45 for $n\geqslant 2$.

  * •

In Proposition 4.61 we show that the prism $\Delta^{n-1}\times I$ admits
Hamiltonian $\mathcal{C}(n,n)$ and $\mathcal{C}(n,n+1)$-subcomplexes for
$n\geqslant 4$. In particular, the geometry $\mathbb{S}^{n-1}\times\mathbb{R}$
admits Construction 4.45 for $n\geqslant 3$.

  * •

In Construction 4.62 we describe the operation of cutting off a face and show
that the initial polytope defines a proper Hamiltonian subcomplex in the
resulting polytope. As a result in Corollary 4.68 we show that the polytope
$P^{n}=\Delta^{p}\times\Delta^{q}$ has a Hamiltonian
$\mathcal{C}(n,n+1)$-subcomplex for $p,q\geqslant 1$. In particular, the
geometry $\mathbb{S}^{p}\times\mathbb{S}^{q}$, $p,q\geqslant 2$, admits
Construction 4.45.

  * •

In Proposition 4.70 we show that if $P^{n}$ has a Hamiltonian
$\mathcal{C}(n,n+1)$-subcomplex such that its defining faces
$\\{M_{1},\dots,M_{s}\\}$ do not intersect a facet $F_{i}\simeq\Delta^{n-1}$,
then for any $k\geqslant 1$ the polytope $\Delta^{k}\times P$ has a
Hamiltonian $\mathcal{C}(n+k,n+k+1)$-subcomplex, and in Corollary 4.71 we show
that for any $p,q\geqslant 1$ the polytope
$P^{n}=\Delta^{p}\times\Delta^{q}\times I$ has a Hamiltonian
$C(n,n+1)$-subcomplex. In particular, the geometries
$\mathbb{S}^{p}\times\mathbb{S}^{q}\times\mathbb{R}$, $p,q\geqslant 2$, and
$\mathbb{S}^{p}\times\mathbb{R}^{2}$, $p\geqslant 2$, admit Construction 4.45.

  * •

In Proposition 4.72 we show that if an $n$-polytope $P$ has a Hamiltonian
$\mathcal{C}(n,n+1)$-subcomplex $C$ such that its defining faces
$\\{M_{1},\dots,M_{s}\\}$ do not intersect an $(n-2)$-face $G\subset P$, then
$G\simeq\Delta^{n-2}$, and $P$ has a Hamiltonian
$\mathcal{C}(n,n)$-subcomplex. Using this fact in Corollary 4.73 we show that
$\Delta^{2}\times\Delta^{n-2}$ has a Hamiltonian $\mathcal{C}(n,n)$-subcomplex
for $n\geqslant 3$, and in Corollary 4.74 we show that for $p\geqslant 1$ the
polytope $P^{n}=\Delta^{p}\times P_{k}$, where $P_{k}$ is a $k$-gon,
$k\geqslant 3$, admits Hamiltonian $\mathcal{C}(n,n)$\- and
$\mathcal{C}(n,n+1)$-subcomplexes. In particular, the geometries
$\mathbb{S}^{p}\times\mathbb{S}^{2}$, $\mathbb{S}^{p}\times\mathbb{R}^{2}$ and
$\mathbb{S}^{p}\times\mathbb{L}^{2}$, $p\geqslant 2$, admit Construction 4.45.

  * •

In Example 4.75 we show that the polytope $P_{5}\times P_{5}$ admits a
Hamiltonian $\mathcal{C}(4,5)$-subcomplex. In particular, the geometry
$\mathbb{L}^{2}\times\mathbb{L}^{2}$ admits Construction 4.45.

  * •

In Lemma 4.60 we show that if for a defining $(n-2)$-face $F$ of a Hamiltonian
$\mathcal{C}(n,k)$-subcomplex $C\subset\partial P$ different facets of $F$
correspond to different facets of $C$, then in the branch set of the covering
$N(P,\widetilde{\Lambda}_{C})\to S^{n}$ the preimage of $F$ is a disjoint
union of $2^{k-1-m_{F}}$ copies of $\mathbb{R}\mathcal{Z}_{F}$, where $m_{F}$
is the number of facets of $F$.

  * •

Using Lemma 4.60 and Proposition 4.31 for each Hamiltonian
$\mathcal{C}(n,k)$-subcomplex $C\subset\partial P$ arising in this subsection
we describe the branch set of the corresponding $2$-sheeted branched covering
of $S^{n}$.

We conclude Subsection 4.7 with our third main result – Theorem 4.76.

##  2\. Rational homology $4$-spheres among $N(P,\Lambda)$

######  Definition 2.1.

We call a topological space $X$ a rational homology $n$-sphere ($n$-$RHS$), if
$X$ is a compact closed topological $n$-manifold and
$H_{k}(X,\mathbb{Q})=H_{k}(S^{n},\mathbb{Q})$ for all $k$. By definition $X$
is an $n$-$RHS$ for $n<0$, if $X=\varnothing$.

We call $X$ a rational homology $n$-disk ($n$-$RHD$), if $X$ is a compact
orientable topological $n$-manifold with a boundary such that
$H_{k}(X,\mathbb{Q})=H_{k}(D^{n},\mathbb{Q})$ for all $k$, where
$D^{n}=\\{(x_{1},\dots,x_{n})\in\mathbb{R}^{n}\colon
x_{1}^{2}+\dots+x_{n}^{2}\leqslant 1\\}$ is an $n$-disk. By definition $X$ is
an $n$-$RHD$ for $n<0$, if $X={\rm pt}$.

######  Lemma 2.2.

If $X$ is an $n$-$RHD$, then $\partial X$ is an $(n-1)$-$RHS$ for $n\geqslant
1$.

###### Proof.

This follows from the long exact sequence in homology

$\dots\to\widetilde{H}_{k}(\partial X)\to\widetilde{H}_{k}(X)\to
H_{k}(X,\partial X)\to\widetilde{H}_{k-1}(\partial X)\to\dots$

the Poincare-Lefschets duality $H_{k}(X,\partial X)\simeq H^{n-k}(X)$ and the
universal coefficients formula. ∎

We will use the following result, which was first proved for small covers and
$\mathbb{Q}$ coefficients in [ST12, T12]. Let us identify the subsets
$\omega\subset[m]=\\{1,\dots,m\\}$ with vectors
$\boldsymbol{x}\in\mathbb{Z}_{2}^{m}$ by the rule $\omega=\\{i\colon
x_{i}=1\\}$. For a vector-coloring $\Lambda$ of rank $(r+1)$ denote by ${\rm
row}\,\Lambda$ the subspace in $\mathbb{Z}_{2}^{m}$ generated by the row
vectors of the matrix $\Lambda$. Equivalently,

${\rm
row}\,\Lambda=\\{(x_{1},\dots,x_{m})\in\mathbb{Z}_{2}^{m}\colon\exists\boldsymbol{c}\in(\mathbb{Z}_{2}^{r+1})^{*}\colon
x_{i}=\boldsymbol{c}\Lambda_{i},i=1,\dots,m\\}.$

Remind that $P_{\omega}=\bigcup_{i\in\omega}F_{i}$.

######  Theorem 2.3.

[CP17, Theorem 4.5](see also [CP20, Theorem 1.1]) Let $\Lambda$ be a vector-
coloring of rank $(r+1)$ of a simple $n$-polytope $P$ and $R$ be a commutative
ring in which $2$ is a unit. Then there is an $R$-linear isomorphism

$H^{k}(N(P,\Lambda),R)\simeq\bigoplus_{\omega\in{\rm
row}\,\Lambda}\widetilde{H}^{k-1}(P_{\omega},R)$

Let is remind that a closed orientable manifold $N(P,\Lambda)$ is defined by a
an affine coloring $\lambda$ of rank $r$, where for some change of coordinates
in $\mathbb{Z}_{2}^{r+1}$ we have $\Lambda_{i}=(1,\lambda_{i})$. Theorem 2.3
and the universal coefficients formula imply

######  Proposition 2.4.

Let $\lambda$ be an affine coloring of rank $r$ of a simple $n$-polytope $P$.
The space $N(P,\lambda)$ is a rational homology $n$-sphere if and only if one
of the following equivalent conditions holds:

  1. (1)

$\bigcup\limits_{i\colon\lambda_{i}\in\pi}F_{i}$ is an $(n-1)$-$RHD$ for any
affine hyperplane $\pi\subset\mathbb{Z}_{2}^{r}$;

  2. (2)

$\bigcup\limits_{i\colon\lambda_{i}\in\pi}F_{i}$ is an $(n-1)$-$RHD$ for any
affine hyperplane $\pi\subset\mathbb{Z}_{2}^{r}$ passing through some pint
$\boldsymbol{p}\in\mathbb{Z}_{2}^{r}$.

We will use the following Generalized Shoenflies theorem.

######  Theorem 2.5.

[B60, Theorem 5] Let $h$ be a homeomorphic embedding of $S^{n-1}\times I$ into
$S^{n}$. Then the closure of either complementary domain of
$h(S^{n-1}\times\frac{1}{2})$ is homeomorphic to $D^{n}$.

######  Lemma 2.6.

Let $n\leqslant 4$. If $P_{\omega}$ is an $(n-1)$-$RHD$, then it is
homeomorphic to an $(n-1)$-disk $D^{n-1}$.

###### Proof.

For $n\leqslant 2$ this is trivial. Let $n\in\\{3,4\\}$. If $P_{\omega}$ is an
$(n-1)$-$RHD$, then by Lemma 2.2 $\partial P_{\omega}$ is an $(n-2)$-$RHS$.
For $n\in\\{3,4\\}$ this means that $\partial P_{\omega}$ is homeomorphic to
$S^{n-2}$. Since $P_{\omega}$ and $P_{[m]\setminus\omega}$ are compact
piecewise linear manifolds (see Lemma 2.10), their common boundary $\partial
P_{\omega}$ has a collar neighbourhood in both spaces and the embedding
$S^{n-2}\simeq\partial P_{\omega}$ to $\partial P\simeq S^{n-1}$ can be
extended to an embedding of $S^{n-2}\times I$. Then by Theorem 2.5
$P_{\omega}$ and $P_{[m]\setminus\omega}$ are topological $(n-1)$-disks. ∎

This leads to the following result.

######  Theorem 2.7.

Let $\lambda$ be an affine coloring of rank $r$ of a simple $n$-polytope $P$,
where $2\leqslant n\leqslant 4$. The space $N(P,\lambda)$ is a rational
homology $n$-sphere if and only if one of the following equivalent conditions
holds:

  1. (1)

$\bigcup\limits_{i\colon\lambda_{i}\in\pi}F_{i}$ is an $(n-1)$-disk for any
affine hyperplane $\pi\subset\mathbb{Z}_{2}^{r}$;

  2. (2)

$\bigcup\limits_{i\colon\lambda_{i}\in\pi}F_{i}$ is an $(n-1)$-disk for any
affine hyperplane $\pi\subset\mathbb{Z}_{2}^{r}$ passing through some pint
$\boldsymbol{p}\in\mathbb{Z}_{2}^{r}$.

######  Definition 2.8.

We call a subset $A\subset B$ proper, if $\varnothing\neq A\neq B$. For a
collection of pairwise disjoint subsets $\omega_{1}$, $\dots$,
$\omega_{k}\subset[m]$ define
$P_{\omega_{1},\dots,\omega_{k}}=P_{\omega_{1}}\cap
P_{\omega_{2}}\cap\dots\cap P_{\omega_{k}}$.

######  Lemma 2.9.

For any simple $n$-polytope $P$ and any collection of pairwise disjoint proper
subsets $\omega_{1}$, $\dots$, $\omega_{k}\subset[m]$, if
$P_{\omega_{1},\dots,\omega_{k}}\neq\varnothing$, then it is an orientable
$(n-k)$-manifold, perhaps with a boundary.

###### Proof.

As it is mentioned in [E24, Corollary 2.4], $P_{\omega_{1},\dots,\omega_{k}}$
is an $(n-k)$-manifold, perhaps with a boundary. This is based on the
following fact which we will use later.

######  Lemma 2.10.

[E24, 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
$p$ such that $U\cap G_{j}=\varnothing$ for $j\notin\\{i_{1},\dots,i_{l}\\}$
to a neighbourhood $V\subset\mathbb{R}^{l}_{\geqslant}\times\mathbb{R}^{n-l}$
such that $\varphi(G_{j_{s}}\cap U)=V\cap\\{y_{s}=0\\}$, $s=1,\dots,l$.

######  Remark 2.11.

It is not stated explicitly in [E24, Lemma 2.2] that $U\cap G_{j}=\varnothing$
for $j\notin\\{i_{1},\dots,i_{l}\\}$ but this condition can be be achieved
using a smaller neighbourhood.

Let us proof that the manifold $P_{\omega_{1},\dots,\omega_{k}}$ is
orientable. Indeed for $k=1$ the set $P_{\omega}$ is a manifold with a
boundary, and $P_{\omega}\setminus\partial P_{\omega}$ is in open subset in
$\partial P\simeq S^{n-1}$. Hence, $P_{\omega}$ is orientable. Assume that our
claim is valid for $1,\dots,k-1$. Then
$P_{\omega_{1},\dots,\omega_{k}}=P_{\omega_{1},\dots,\omega_{k-1}}\cap
P_{\omega_{k}}$ and $P_{\omega_{1},\dots,\omega_{k}}\setminus\partial
P_{\omega_{1},\dots,\omega_{k}}$ is an open subset in $\partial
P_{\omega_{1},\dots,\omega_{k-1}}$, which is an orientable manifold, since it
is a boundary of an orientable manifold. Thus,
$P_{\omega_{1},\dots,\omega_{k}}$ is orientable. ∎

######  Lemma 2.12.

Let $P$ ne an $n$-polytope, and let $\omega_{1},\omega_{2}\subset[m]$ be
subsets such that $\omega_{1}\cap\omega_{2}=\varnothing$, and
$P_{\omega_{1}}$, $P_{\omega_{2}}$ are $(n-1)$-$RHD$.

  * •

If $P_{\omega_{1}\sqcup\omega_{2}}$ is an $(n-1)$-$RHD$, then
$P_{\omega_{1}}\cap P_{\omega_{2}}$ is an $(n-2)$-$RHD$.

  * •

If $P_{\omega_{1}\sqcup\omega_{2}}$ is an $(n-1)$-$RHS$, then
$\omega_{1}\sqcup\omega_{2}=[m]$ and $P_{\omega_{1}}\cap P_{\omega_{2}}$ is an
$(n-2)$-$RHS$.

###### Proof.

The first item follows from Lemma 2.9 and the Mayer-Vietoris sequence in
reduced homology

$\dots\to\widetilde{H}_{k}(P_{\omega_{1}}\cap
P_{\omega_{2}})\to\widetilde{H}_{k}(P_{\omega_{1}})\oplus\widetilde{H}_{k}(P_{\omega_{2}})\to\widetilde{H}_{k}(P_{\omega_{1}\sqcup\omega_{2}})\to\widetilde{H}_{k-1}(P_{\omega_{1}}\cap
P_{\omega_{2}})\to\dots$

The second item follows from these facts and the Alexander duality
$\widetilde{H}^{k}(P_{\omega})=\widetilde{H}_{n-2-k}(P_{[m]\setminus\omega})$
in the sphere $\partial P\simeq S^{n-1}$, since $P_{[m]\setminus\omega}$ is
homotopy equivalent to $\partial P\setminus P_{\omega}$. ∎

######  Corollary 2.13.

Let $P$ ne an $n$-polytope, $n\leqslant 4$, and let
$\omega_{1},\omega_{2}\subset[m]$ be subsets such that
$\omega_{1}\cap\omega_{2}=\varnothing$, and $P_{\omega_{1}}$, $P_{\omega_{2}}$
are $(n-1)$-disks.

  * •

If $P_{\omega_{1}\sqcup\omega_{2}}$ is an $(n-1)$-disk, then
$P_{\omega_{1}}\cap P_{\omega_{2}}$ is an $(n-2)$-disk.

  * •

If $P_{\omega_{1}\sqcup\omega_{2}}$ is an $(n-1)$-sphere, then
$\omega_{1}\sqcup\omega_{2}=[m]$, and $P_{\omega_{1}}\cap P_{\omega_{2}}$ is
an $(n-2)$-sphere.

##  3\. A criterion when $N(P,\Lambda)\simeq S^{4}$

We will use the following result.

######  Theorem 3.1.

(Armstrong, [A65, Theorem 3]). Let $X$ be a connected simply connected
simplicial complex and let a finite group $G$ acts on $X$ by simplicial
homeomorphisms. Let $H$ be a subgroup in $G$ generated by all the elements
$h\in G$ such that the set of fixed points $X^{h}$ is nonempty. Then
$\pi_{1}(X/G)$ is isomorphic to $G/H$.

For any, not necessarily right-angled, simple polytope $P$ define the right-
angled Coxeter group

$\mathcal{G}(P)=\langle\rho_{1},\dots,\rho_{m}\rangle/(\rho_{1}^{2}=\dots=\rho_{m}^{2}=e,\rho_{i}\rho_{j}=\rho_{j}\rho_{i},\text{
if }F_{i}\cap F_{j}\neq\varnothing).$

######  Definition 3.2 (see [V71, D83, D08]).

Define a space

$\displaystyle W(P)=P\times\mathcal{G}(P)/\sim,$ $\displaystyle\text{ where
}(p,g_{1})\sim(q,g_{2})\text{ if and only if }p=q\text{ and
}g_{1}^{-1}g_{2}\in\langle\rho_{i}\colon p\in F_{i}\rangle$

######  Proposition 3.3.

[D83, Theorems 10.1 and 13.5] We have $\pi_{1}(W(P))=0$.

The proof of the following fact is straightforward.

######  Proposition 3.4.

The space $W(P)$ is a connected topological $n$-manifold. Moreover, it has a
structure of a simplicial complex such that $\mathcal{G}(P)$ acts on it
simplicially (the action comes from the action of $\mathcal{G}(P)$ on itself
$g(x)=gx$). Moreover, $W(P)/\mathcal{G}(P)\simeq P$.

Any vector-coloring $\Lambda$ defines an epimorphism
$\varphi_{\Lambda}\colon\mathcal{G}(P)\to\mathbb{Z}_{2}^{r}$.

######  Proposition 3.5.

We have $N(P,\Lambda)=W(P)/{\mathrm{Ker}\,\varphi}$.

######  Corollary 3.6.

We have $\pi_{1}(N(P,\Lambda))\simeq{\mathrm{Ker}\,\varphi}/K$, where the
subgroup $K$ is generated by kernels of the mappings $\langle
g\rho_{i_{1}}g^{-1},\dots
g\rho_{i_{n}}g^{-1}\rangle\simeq\mathbb{Z}_{2}^{n}\to\mathbb{Z}_{2}^{r}$,
$g\rho_{i_{s}}g^{-1}\to\Lambda_{i_{s}}$, for all elements $g\in G$ and all
vertices $F_{i_{1}}\cap\dots\cap F_{i_{n}}$ of $P$.

###### Proof.

We have $St_{\mathcal{G}(P)}[p,g]=\langle g\rho_{i}g^{-1}\colon p\in
F_{i}\rangle$. Therefore,
$St_{\mathrm{Ker}\,\varphi}[p,g]={\mathrm{Ker}\,\varphi}\cap\langle
g\rho_{i}g^{-1}\colon p\in F_{i}\rangle$. Any such a subgroup lies in a
subgroup corresponding to some vertex of $P$. ∎

######  Lemma 3.7.

Let $\pi_{1}(N(P,\Lambda))=0$. Then different vectors among
$\Lambda_{1},\dots,\Lambda_{m}$ are linearly independent and form a basis in
$\mathbb{Z}_{2}^{r}$.

###### Proof.

If $\pi_{1}(N(P,\Lambda))=0$, then ${\mathrm{Ker}\,\varphi}$ is generated by
kernels of the mappings $\langle g\rho_{i_{1}}g^{-1},\dots
g\rho_{i_{n}}g^{-1}\rangle\simeq\mathbb{Z}_{2}^{n}\to\mathbb{Z}_{2}^{r}$,
$g\rho_{i_{s}}g^{-1}\to\Lambda_{i_{s}}$, for all elements $g\in\mathcal{G}(P)$
and all vertices $F_{i_{1}}\cap\dots\cap F_{i_{n}}$ of $P$.

Consider the composition
$\mathcal{G}(P)\to\mathbb{Z}_{2}^{m}\to\mathbb{Z}_{2}^{r}$, where the first
mapping $\pi\colon\rho_{i}\to\boldsymbol{e}_{i}$ is the abelenization
homomorphism, and the second mapping $\Lambda$ is a linear mapping defined by
the condition $\boldsymbol{e}_{i}\to\Lambda_{i}$. We have $\pi^{-1}({\rm
Ker}\,\Lambda)={\rm Ker}\,\varphi$ and

$\pi\langle g\rho_{i_{1}}g^{-1},\dots
g\rho_{i_{n}}g^{-1}\rangle=\pi\langle\rho_{i_{1}},\dots\rho_{i_{n}}\rangle=\langle\boldsymbol{e}_{i_{1}}\dots\boldsymbol{e}_{i_{n}}\rangle.$

In particular, the restrictions of $\pi$ to $\langle g\rho_{i_{1}}g^{-1},\dots
g\rho_{i_{n}}g^{-1}\rangle$ is injective for any vertex $v$ and any
$g\in\mathcal{G}(P)$. Also

$\pi^{-1}({\rm
Ker}\,\Lambda\left.\right|_{\langle\boldsymbol{e}_{i_{1}}\dots\boldsymbol{e}_{i_{n}}\rangle})={\rm
Ker}\,\varphi\left.\right|_{\langle g\rho_{i_{1}}g^{-1},\dots
g\rho_{i_{n}}g^{-1}\rangle}.$

Since ${\mathrm{Ker}\,\varphi}$ is generated by kernels ${\rm
Ker}\,\varphi\left.\right|_{\langle g\rho_{i_{1}}g^{-1},\dots
g\rho_{i_{n}}g^{-1}\rangle}$, the group $\pi({\mathrm{Ker}\,\varphi})={\rm
Ker}\,\Lambda$ is generated by the subgroups $\pi({\rm
Ker}\,\varphi\left.\right|_{\langle g\rho_{i_{1}}g^{-1},\dots
g\rho_{i_{n}}g^{-1}\rangle})={\rm
Ker}\,\Lambda\left.\right|_{\langle\boldsymbol{e}_{i_{1}}\dots\boldsymbol{e}_{i_{n}}\rangle}$
for all vertices $F_{i_{1}}\cap\dots\cap F_{i_{n}}$ of $P$, that is

(1) ${\rm Ker}\,\Lambda=\sum\limits_{v\in P}{\rm
Ker}\,\Lambda\left.\right|_{\langle\boldsymbol{e}_{i_{1}}\dots\boldsymbol{e}_{i_{n}}\rangle}$

If $N(P,\Lambda)$ is a closed topological manifold, then by [E24, Theorem 5.1]
at each vertex $F_{i_{1}}\cap\dots\cap F_{i_{n}}$ different vectors among
$\\{\Lambda_{i_{1}},\dots,\Lambda_{i_{n}}\\}$ are linearly independent. In
particular, ${\rm
Ker}\,\Lambda\left.\right|_{\langle\boldsymbol{e}_{i_{1}}\dots\boldsymbol{e}_{i_{n}}\rangle}$
is generated by the vectors $\boldsymbol{e}_{i_{a}}+\boldsymbol{e}_{i_{b}}$
with $\Lambda_{i_{a}}=\Lambda_{i_{b}}$. We have
$[m]=\\{1,\dots,m\\}=S_{1}\sqcup\dots\sqcup S_{t}$, where
$\Lambda_{i}=\Lambda_{j}$ (equivalently,
$\boldsymbol{e}_{i}+\boldsymbol{e}_{j}\in{\rm Ker}\,\Lambda$) if and only if
$i$ and $j$ lie in the same set $S_{k}$. Assume that there is a linear
dependence $\Lambda_{j_{1}}+\dots+\Lambda_{j_{l}}=0$, where all the vectors
$\\{\Lambda_{j_{1}},\dots,\Lambda_{j_{l}}\\}$ are pairwise different, that is
$e_{j_{p}}\in S_{i_{p}}$, where $i_{p}\neq i_{q}$ if $j_{p}\neq j_{q}$. Then
$\boldsymbol{e}_{j_{1}}+\dots+\boldsymbol{e}_{j_{l}}\in{\rm Ker}\,\Lambda$,
and by (1)

$\boldsymbol{e}_{j_{1}}+\dots+\boldsymbol{e}_{j_{l}}=\sum\limits_{i=1}^{t}\sum\limits_{a,b\in
S_{i},a<b}\lambda_{a,b}(\boldsymbol{e}_{a}+\boldsymbol{e}_{b}).$

Since there is a direct sum
$\mathbb{Z}_{2}^{m}=\bigoplus\limits_{i=1}^{t}\langle\boldsymbol{e}_{j}\colon
j\in S_{i}\rangle$, we have $\boldsymbol{e}_{j_{p}}=\sum\limits_{a,b\in
S_{i_{p}},a<b}\lambda_{a,b}(\boldsymbol{e}_{a}+\boldsymbol{e}_{b})$. But on
the left side the sum $x_{1}+\dots+x_{m}$ of all the coordinates of the vector
is $1$, and on the right side it is $0$. A contradiction. Thus, all different
vectors among $\\{\Lambda_{1},\dots,\Lambda_{m}\\}$ are linearly independent.
In particular, they form a basis in $\mathbb{Z}_{2}^{r}$. ∎

######  Definition 3.8.

For a subset $\omega\subset[M]=\\{1,\dots,M\\}$ define
$G_{\omega}=\bigcup\limits_{i\in\omega}G_{i}$. For a collection of pairwise
disjoint subsets $\omega_{1}$, $\dots$, $\omega_{k}\subset[M]$ define
$G_{\omega_{1},\dots,\omega_{k}}=G_{\omega_{1}}\cap\dots\cap G_{\omega_{k}}$.

######  Corollary 3.9.

Let $N(P,\Lambda)\simeq S^{n}$. Then for each proper subset $\omega\subset[M]$
the set $G_{\omega}$ is an $(n-1)$-$RHD$. For $\omega=[M]$ we have
$G_{\omega}=\partial P\simeq S^{n-1}$.

###### Proof.

This follows directly from Theorem 2.3 and Proposition 2.4. ∎

######  Lemma 3.10.

Let $N(P,\Lambda)\simeq S^{n}$. Then for each collection of pairwise disjoint
proper subsets $\omega_{1}$, $\dots$, $\omega_{k}\subset[M]$ the set
$G_{\omega_{1},\dots,\omega_{k}}$ is

  * •

an $(n-k)$-$RHD$, if $\omega_{1}\sqcup\dots\sqcup\omega_{k}\neq[M]$;

  * •

an $(n-k)$-$RHS$, if $\omega_{1}\sqcup\dots\sqcup\omega_{k}=[M]$;

###### Proof.

We will prove this by induction on $k$. For $k\leqslant 2$ this follows from
Lemma 2.12 and Corollary 3.9. Let the lemma hold for $1$, $\dots$, $k-1$. We
have
$G_{\omega_{1},\dots,\omega_{k}}=G_{\omega_{1},\dots,\omega_{k-2},\omega_{k-1}}\cap
G_{\omega_{1},\dots,\omega_{k-2},\omega_{k}}$, where
$G_{\omega_{1},\dots,\omega_{k-2},\omega_{k-1}}\cup
G_{\omega_{1},\dots,\omega_{k-2},\omega_{k}}=G_{\omega_{1},\dots,\omega_{k-2},\omega_{k-1}\sqcup\omega_{k}}$.
Now the proof follows from the Mayer-Vietoris sequence and Lemma 2.9. ∎

######  Corollary 3.11.

Let $N(P,\Lambda)\simeq S^{n}$. Then $M\leqslant n+1$ and each $k$-face of the
complex $C(P,\Lambda)$ is either a $k$-$RHD$ or a $k$-$RHS$.

###### Proof.

Indeed, $G_{1}\cap\dots\cap G_{M-1}$ is an $(n-M+1)$-$RHD$. In particular, it
is nonempty. But the intersection of any $n+1$ facets of $P$ is empty, hence
$M-1\leqslant n$. Each $k$-face of $C(P,\Lambda)$ is a connected component of
$G_{i_{1}}\cap\dots\cap G_{i_{n-k}}$, hence by Lemma 3.10 it is a $k$-$RHD$ if
$n-k<M$, and a $k$-$RHD$ if $n-k=M$. ∎

######  Corollary 3.12.

Let $n\leqslant 4$. If $N(P,\Lambda)\simeq S^{n}$, then $M\leqslant n+1$ and
for each collection of pairwise disjoint proper subsets $\omega_{1}$, $\dots$,
$\omega_{k}\subset[M]$ the set $G_{\omega_{1},\dots,\omega_{k}}$ is

  * •

an $(n-k)$-disk, if $\omega_{1}\sqcup\dots\sqcup\omega_{k}\neq[M]$;

  * •

an $(n-k)$-sphere, if $\omega_{1}\sqcup\dots\sqcup\omega_{k}=[M]$.

###### Proof.

This follows from Lemmas 3.10 and 2.6 and the fact that an $n$-$RHS$ is a
topological $n$-sphere for $n\leqslant 2$ and an $n$-$RHD$ is a topological
$2$-disk for $n\leqslant 2$. ∎

######  Lemma 3.13.

Let $n\leqslant 4$. If for a complex $\mathcal{C}(P,\Lambda)$ we have
$M\leqslant n+1$ and for each collection of pairwise disjoint proper subsets
$\omega_{1}$, $\dots$, $\omega_{k}\subset[M]$ the set
$G_{\omega_{1},\dots,\omega_{k}}$ is

  * •

an $(n-k)$-disk, if $\omega_{1}\sqcup\dots\sqcup\omega_{k}\neq[M]$;

  * •

an $(n-k)$-sphere, if $\omega_{1}\sqcup\dots\sqcup\omega_{k}=[M]$,

then $\mathcal{C}(P,\Lambda)$ is equivalent to $\mathcal{C}(n,M)$.

###### Proof.

For $n=2$ the statement is trivial.

Let $n=3$. If $M=1$, then $\mathcal{C}(P,\Lambda)\simeq\mathcal{C}(n,1)$. If
$M=2$, then $G_{1}$ and $G_{2}$ are $2$-disks such that $\partial
G_{1}=\partial G_{2}=G_{1}\cap G_{2}$ is a circle. Hence,
$\mathcal{C}(P,\Lambda)\simeq\mathcal{C}(n,2)$. If $M=3$, then $G_{1}$,
$G_{2}$ and $G_{3}$ are $2$-disks such that $G_{i}\cap G_{j}$ is a simple path
for any $i\neq j$, and all these three paths have common ends, which form the
$0$-sphere $G_{1}\cap G_{2}\cap G_{3}$. Thus, the $1$-skeleton of
$\mathcal{C}(P,\Lambda)$ is a theta-graph, and
$\mathcal{C}(P,\Lambda)\simeq\mathcal{C}(n,3)$. If $M=4$, then each $G_{i}$ is
a disk, the intersection $G_{i}\cap G_{j}$ of each two such disks is a simple
path, the intersection of any three such disks is a point, and the
intersection of all the $4$-disks is empty. Then the $1$-skeleton of
$\mathcal{C}(P,\Lambda)$ is a $K_{4}$-graph, and
$\mathcal{C}(P,\Lambda)\simeq\mathcal{C}(n,3)$.

Let $n=4$. If $M=1$, then $\mathcal{C}(P,\Lambda)\simeq\mathcal{C}(n,1)$. If
$M=2$, then $G_{1}$ and $G_{2}$ are $3$-disks such that $\partial
G_{1}=\partial G_{2}=G_{1}\cap G_{2}$ is a $2$-sphere. Hence,
$\mathcal{C}(P,\Lambda)\simeq\mathcal{C}(n,2)$. If $M=3$, then $G_{1}$,
$G_{2}$ and $G_{3}$ are $3$-disks, such that $G_{i}\cap G_{j}$ is a $2$-disk
for $i<j$, and $G_{1}\cap G_{2}\cap G_{3}$ is a circle. We have $G_{1}\cup
G_{2}$ is a $3$-disk with a circle $G_{1}\cap G_{2}\cap
G_{3}=\partial(G_{1}\cap G_{2})$ on its boundary, and the complement to this
disk is the interior of $G_{3}$. Hence,
$\mathcal{C}(P,\Lambda)\simeq\mathcal{C}(n,3)$. If $M=4$, then $G_{1}$,
$G_{2}$, $G_{3}$, $G_{4}$ are $4$-disks, $G_{i}\cap G_{j}$ is a $2$-disk for
any $i<j$, $G_{i}\cap G_{j}\cap G_{k}$ is a simple path for any $i<j<k$, and
all these four paths have the common ends, which form the $0$-sphere
$G_{1}\cap G_{2}\cap G_{3}\cap G_{4}$. Then $G_{1}\cup G_{2}\cup G_{3}$ is a
$3$-disk with the theta-graph on the boundary formed by paths $G_{i}\cap
G_{j}\cap G_{4}$, $i<j<4$. There is the path $G_{1}\cap G_{2}\cap G_{3}$
inside this disk connecting the vertices of the theta-graph and each
$G_{i}\cap G_{j}$ is a $2$-disk bounded by the paths $G_{1}\cap G_{2}\cap
G_{3}$ and $G_{i}\cap G_{j}\cap G_{4}$. The complement to $G_{1}\cup G_{2}\cup
G_{3}$ in $\partial P$ is the interior of $G_{4}$. Thus,
$\mathcal{C}(P,\Lambda)\simeq\mathcal{C}(n,4)$. If $M=5$, then each $G_{i}$ is
a $3$-disk, each intersection $G_{i}\cap G_{j}$, $i<j$, is a $2$-disk, each
intersection $G_{i}\cap G_{j}\cap G_{k}$, $i<j<k$, is a simple path, each
intersection $G_{i}\cap G_{j}\cap G_{k}\cap G_{l}$, $i<j<k<l$, is a point, and
$G_{1}\cap G_{2}\cap G_{3}\cap G_{4}\cap G_{5}=\varnothing$. Then $G_{1}\cup
G_{2}\cup G_{3}\cup G_{4}$ is a $3$-disk with a $K_{4}$-graph on its boundary
formed by vertices $G_{i}\cap G_{j}\cap G_{k}\cap G_{5}$ and edges $G_{i}\cap
G_{j}\cap G_{5}$. There is a point $G_{1}\cap G_{2}\cap G_{3}\cap G_{4}$
inside this disk such that the face structure in the disk $G_{1}\cup G_{2}\cup
G_{3}\cup G_{4}$ is a cone over the face structure on its boundary with this
apex. Also the complement to this disk is the interior of $G_{5}$. Thus,
$\mathcal{C}(P,\Lambda)\simeq\mathcal{C}(n,5)\simeq\partial\Delta^{4}$.

∎

As a corollary we obtain the following theorem.

######  Theorem 3.14.

Let $\Lambda$ be a vector-coloring of rank $r$ of a simple polytope of
dimension $n\leqslant 4$. Then $N(P,\Lambda)\simeq S^{n}$ if and only if
$\mathcal{C}(P,\Lambda)\simeq\mathcal{C}(n,r)$.

###### Proof.

The if part follows from [E24, Construction 5.8]. The only if part follows
from Corollary 3.12 and Lemma 3.13. ∎

##  4\. Hyperelliptic manifolds $N(P,\Lambda)$

Theorem 3.14 implies the following result (see [E24, Construction 8.6] and
[E24, Corollary 10.8]).

######  Corollary 4.1.

Let $\Lambda$ be a vector-coloring of rank $r$ of a simple $n$-polytope $P$,
$n\leqslant 4$. Then an involution $\tau\in\mathbb{Z}_{2}^{r}$ is
hyperelliptic (that is $N(P,\lambda)\simeq S^{n}$) if and only if $\tau$ is
special (that is $\mathcal{C}(P,\Lambda_{\tau})\simeq\mathcal{C}(n,r-1)$).

###  4.1. Hamiltonian subcomplexes

######  Construction 4.2.

For two colorings $c_{1}\colon\\{F_{1},\dots,F_{m}\\}\to[l_{1}]$ and
$c_{2}\colon\\{F_{1},\dots,F_{m}\\}\to[l_{2}]$ we call $\mathcal{C}(P,c_{2})$
a subcomplex in $\mathcal{C}(P,c_{1})$ (and write
$\mathcal{C}(P,c_{2})\subset\mathcal{C}(P,c_{1})$), if there is a mapping
$\pi\colon[l_{1}]\to[l_{2}]$ such that $c_{2}=\pi\circ c_{1}$. Each facet of
$\mathcal{C}(P,c_{2})$ is a union of facets of $\mathcal{C}(P,c_{1})$. More
generally, for any $q\geqslant 0$ each $q$-face of $\mathcal{C}(P,c_{2})$ is a
union of $q$-faces of $\mathcal{C}(P,c_{1})$.

######  Definition 4.3.

We call a subcomplex $\mathcal{C}(P,c_{2})\subset\mathcal{C}(P,c_{1})$
Hamiltonian, if each $q$-skeleton of $\mathcal{C}(P,c_{1})$ belongs to the
$(q+1)$-skeleton of $\mathcal{C}(P,c_{2})$.

######  Lemma 4.4.

A subcomplex $\mathcal{C}(P,c_{2})\subset\mathcal{C}(P,c_{1})$ is Hamiltonian
if and only if one of the following equivalent conditions holds:

  1. (1)

For each nonempty intersection $G_{i_{1}}\cap\dots\cap
G_{i_{k}}\neq\varnothing$ of different facets of $\mathcal{C}(P,c_{1})$ at
least $(k-1)$ of these facets lie in different facets of
$\mathcal{C}(P,c_{2})$.

  2. (2)

For each nonempty intersection $G_{i_{1}}\cap\dots\cap
G_{i_{k}}\neq\varnothing$ of $k=3$ or $k=4$ different facets of
$\mathcal{C}(P,c_{1})$ at least $(k-1)$ of these facets lie in different
facets of $\mathcal{C}(P,c_{2})$.

  3. (3)

The following two conditions hold:

     * •

$G_{i_{1}}\cap G_{i_{2}}\cap G_{i_{3}}=\varnothing$, if
$c_{2}(G_{i_{1}})=c_{2}(G_{i_{2}})=c_{2}(G_{i_{3}})$ and $i_{1}<i_{2}<i_{3}$,

     * •

$G_{i_{1}}\cap G_{i_{2}}\cap G_{i_{3}}\cap G_{i_{4}}=\varnothing$, if
$c_{2}(G_{i_{1}})=c_{2}(G_{i_{2}})\neq c_{2}(G_{i_{3}})=c_{2}(G_{i_{4}})$ and
$i_{1}\neq i_{2}$, $i_{3}\neq i_{4}$;

  4. (4)

Any $(n-4)$-face of $\mathcal{C}(P,c_{1})$ lies in an $(n-3)$-face of
$\mathcal{C}(P,c_{2})$, and any closed manifold among $(n-3)$-faces of
$\mathcal{C}(P,c_{1})$ lies in an $(n-2)$-face of $\mathcal{C}(P,c_{2})$.

###### Proof.

By Lemma 2.10 the facets of $C(P,c)$ intersect locally as coordinate
hyperplanes in $\mathbb{R}^{n}$ (moreover, other facets do not intersect this
local neighbourhood). By definition if
$\mathcal{C}(P,c_{2})\subset\mathcal{C}(P,c_{1})$ is Hamiltonian, then the set
$G_{i_{1}}\cap\dots\cap G_{i_{k}}$ lies in the $(n-k-1)$-skeleton of
$\mathcal{C}(P,c_{2})$. Take any point $p\in G_{i_{1}}\cap\dots\cap G_{i_{k}}$
such that $p\not\in G_{j_{l}}$ for each facet $G_{i_{l}}$ of
$\mathcal{C}(P,c_{1})$ different from $G_{i_{1}},\dots,G_{i_{k}}$ (such a
point exists by the above argument). Then
$p\in\widetilde{G}_{j_{1}}\cap\dots\cap\widetilde{G}_{j_{k-1}}$, where
$\widetilde{G}_{j_{1}},\dots,\widetilde{G}_{j_{k-1}}$ are different facets of
$\mathcal{C}(P,c_{2})$. We have
$\widetilde{G}_{j_{p}}=\bigcup\limits_{G_{q}\subset\widetilde{G}_{j_{p}}}G_{q}$
and

$\widetilde{G}_{j_{1}}\cap\dots\cap\widetilde{G}_{j_{k-1}}=\left(\bigcup\limits_{G_{q_{1}}\subset\widetilde{G}_{j_{1}}}G_{q_{1}}\right)\cap\dots\cap\left(\bigcup\limits_{G_{q_{k-1}}\subset\widetilde{G}_{j_{k-1}}}G_{q_{k-1}}\right)=\bigcup\limits_{q_{1},\dots,q_{k-1}}G_{q_{1}}\cap\dots\cap
G_{q_{k-1}}$

Then $p\in G_{q_{1}}\cap\dots\cap G_{q_{k-1}}$ for some $q_{1},\dots,q_{k-1}$.
Therefore, by the choice of $p$ we have
$\\{q_{1},\dots,q_{k-1}\\}\subset\\{i_{1},\dots,i_{k}\\}$, where
$G_{q_{p}}\subset\widetilde{G}_{j_{p}}$. This implies item (1). On the other
hand, if item (1) holds, then by definition the subcomplex
$\mathcal{C}(P,c_{2})\subset\mathcal{C}(P,c_{1})$ is Hamiltonian.

Item (1) implies item (2). Item (3) is a reformulation of item (2), so these
items are equivalent.

Let item (2) hold. Consider a nonempty intersection $G_{i_{1}}\cap\dots\cap
G_{i_{k}}\neq\varnothing$ of different facets of $\mathcal{C}(P,c_{1})$. For
$k\leqslant 4$ item (1) holds. Assume that $k\geqslant 5$ and item (1) does
not hold. Then $|\\{c_{2}(G_{i_{1}}),\dots,c_{2}(G_{i_{k}})\\}|\leqslant k-2$
and either $c_{2}(G_{i_{p_{1}}})=c_{2}(G_{i_{p_{2}}})=c_{2}(G_{i_{p_{2}}})$
for $p_{1}<p_{2}<p_{3}$, or $c_{2}(G_{i_{p_{1}}})=c_{2}(G_{i_{p_{2}}})\neq
c_{2}(G_{i_{p_{3}}})=c_{2}(G_{i_{p_{4}}})$ for $p_{1}\neq p_{2}$ and
$p_{3}\neq p_{4}$. This contradicts item (3).

Item (2) implies item (4). If item (4) holds, then the argument in the
beginning of the proof implies that item (3) hold for $k=4$ and closed
manifolds for $k=3$. If $G_{i_{1}}\cap G_{i_{2}}\cap G_{i_{3}}$ is not a
closed manifold, then there is a facet $G_{i_{4}}$ of $\mathcal{C}(P,c_{1})$
such that $G_{i_{1}}\cap G_{i_{2}}\cap G_{i_{3}}\cap
G_{i_{4}}\neq\varnothing$. Then by the above argument it is impossible that
$G_{i_{1}}$, $G_{i_{2}}$ and $G_{i_{3}}$ lie in the same facet of
$\mathcal{C}(P,c_{2})$. Hence, item (3) holds. ∎

Lemma 4.4 leads to the following definition.

######  Definition 4.5.

For a subcomplex $\mathcal{C}(P,c_{2})\subset\mathcal{C}(P,c_{1})$ let us call
each connected component of a nonempty intersection $G_{i_{1}}\cap G_{i_{2}}$,
where $G_{i_{1}},G_{i_{2}}$ are facets of $\mathcal{C}(P,c_{1})$ and
$c_{2}(G_{i_{2}})=c_{2}(G_{i_{2}})$, a defining $(n-2)$-face.

######  Proposition 4.6.

A subcomplex $\mathcal{C}(P,c_{2})\subset\mathcal{C}(P,c_{1})$ is Hamiltonian
if and only if any two defining $(n-2)$-facets are disjoint.

###### Proof.

Let any two defining $(n-2)$-facets be disjoint. If
$c_{2}(G_{i_{1}})=c_{2}(G_{i_{2}})=c_{2}(G_{i_{3}})$ and $i_{1}<i_{2}<i_{3}$,
then either one of the intersections $G_{i_{1}}\cap G_{i_{2}}$ and
$G_{i_{1}}\cap G_{i_{3}}$ is empty, or each intersection is a disjoint union
of defining faces and any two faces from different unions are also disjoint.
Then

$G_{i_{1}}\cap G_{i_{2}}\cap G_{i_{3}}=(G_{i_{1}}\cap
G_{i_{2}})\cap(G_{i_{1}}\cap G_{i_{3}})=\varnothing.$

If $c_{2}(G_{i_{1}})=c_{2}(G_{i_{2}})\neq c_{2}(G_{i_{3}})=c_{2}(G_{i_{4}})$
and $i_{1}\neq i_{2}$, $i_{3}\neq i_{4}$, then either one of the intersection
$G_{i_{1}}\cap G_{i_{2}}$ and $G_{i_{3}}\cap G_{i_{4}}$ is empty, or each
intersection is a disjoint union of defining faces and any two faces from
different unions are also disjoint. Then

$G_{i_{1}}\cap G_{i_{2}}\cap G_{i_{3}}\cap G_{i_{4}}=(G_{i_{1}}\cap
G_{i_{2}})\cap(G_{i_{3}}\cap G_{i_{4}})=\varnothing$

Then the subcomplex is Hamiltonian by Lemma 4.4(3).

Now let the subcomplex be Hamiltonian. Consider two defining faces. If they
are connected components of the same intersection $G_{i_{1}}\cap G_{i_{2}}$,
then they are disjoint by definition. If they belong to different
intersections $G_{i_{1}}\cap G_{i_{2}}$ and $G_{i_{3}}\cap G_{i_{4}}$, then
there are two possibilities. First possibility is
$c(G_{i_{1}})=c(G_{i_{2}})=c(G_{i_{3}})=c(G_{i_{4}})$. Then without loss of
generality we can assume that $i_{3}\notin\\{i_{1},i_{2}\\}$. Hence,
$G_{i_{1}}\cap G_{i_{2}}\cap G_{i_{3}}=\varnothing$ by Lemma 4.4(3) and
$(G_{i_{1}}\cap G_{i_{2}})\cap(G_{i_{3}}\cap G_{i_{4}})=\varnothing$. The
second possibility is $c(G_{i_{1}})=c(G_{i_{2}})\neq
c(G_{i_{3}})=c(G_{i_{4}})$. Then $(G_{i_{1}}\cap G_{i_{2}})\cap(G_{i_{3}}\cap
G_{i_{4}})=\varnothing$ by Lemma 4.4(3). This finishes the proof. ∎

######  Proposition 4.7.

A set $\mathcal{S}=\\{M_{1}^{n-2},\dots,M_{q}^{n-2}\\}$ of pairwise disjoint
$(n-2)$-faces of a complex $\mathcal{C}(P,c_{1})$ is a set of defining faces
of a Hamiltonian subcomplex $\mathcal{C}(P,c_{2})\subset\mathcal{C}(P,c_{1})$
if and only if for each two facets $G_{i}\neq G_{j}$ of $\mathcal{C}(P,c_{1})$
admitting a sequence of facets $G_{i}=G_{i_{1}}$, $G_{i_{2}}$, $\dots$,
$G_{i_{p}}=G_{j}$ such that for all $s$ the intersection $G_{i_{s}}\cap
G_{i_{s+1}}$ contains a face from $\mathcal{S}$ either $G_{i}\cap
G_{j}=\varnothing$ or each connected component of $G_{i}\cap G_{j}$ belongs to
$\mathcal{S}$.

###### Proof.

If $\mathcal{S}$ satisfies the above conditions then one can define a coloring
$c_{2}$ by the rule $c_{2}(G_{i})=c_{2}(G_{j})$ if and only if $G_{i}$ and
$G_{j}$ admit a sequence of the above type. Then each connected component of
the intersection of two facets of the same color belongs to $\mathcal{S}$ and
this is indeed the set of defining facets. Since any two of them are disjoint,
the subcomplex is Hamiltonian.

On the other hand, if $\mathcal{S}$ is the set of defining faces of a
Hamiltonian subcomplex, then the facets $G_{i}$ and $G_{j}$ admitting the
above sequence necessarily have the same color. Hence, if $G_{i}\cap
G_{j}\neq\varnothing$, then each connected component of $G_{i}\cap G_{j}$
belongs to $\mathcal{S}$. ∎

###  4.2. Bipartite Hamiltonian subcomplexes and orbit spaces of involutions

######  Definition 4.8.

For a Hamiltonian subcomplex $\mathcal{C}(P,c_{2})\subset\mathcal{C}(P,c_{1})$
and a facet $\widetilde{G}\in\mathcal{C}(P,c_{2})$ define an adjacency graph
$\Gamma(\widetilde{G})$. Its vertices are facets $G_{i}$ of
$\mathcal{C}(P,c_{1})$ lying in $\widetilde{G}$. Its edges bijectively
correspond to connected components of nonempty intersections $G_{i}\cap
G_{j}\neq\varnothing$ of such facets, that is to defining $(n-2)$-faces lying
in $\widetilde{G}$. This graph is connected and may have multiple edges
corresponding to different connected components of $G_{i}\cap
G_{j}\neq\varnothing$. Define an adjacency graph

$\Gamma(P,c_{1},c_{2})=\bigsqcup\limits_{\widetilde{G}\text{ -- a facet of
}\mathcal{C}(P,c_{2})}\Gamma(\widetilde{G}).$

######  Definition 4.9.

We call a Hamiltonian subcomplex
$\mathcal{C}(P,c_{2})\subset\mathcal{C}(P,c_{1})$ bipartite, if the adjacency
graph $\Gamma(P,c_{1},c_{2})$ is bipartite, that is there is a coloring
$\chi\colon\\{G_{1},\dots,G_{M_{1}}\\}\to\\{0,1\\}$ of facets of
$\mathcal{C}(P,c_{1})$ in two colors $0$ and $1$ such that if $G_{i}$ and
$G_{j}$ lie in the same facet of $\mathcal{C}(P,c_{2})$ and $G_{i}\cap
G_{j}\neq\varnothing$, then $\chi(G_{i})\neq\chi(G_{j})$. We will call such
colorings nice.

######  Remark 4.10.

If $\mathcal{C}(P,c_{2})\subset\mathcal{C}(P,c_{1})$ is a proper subcomplex
and $\mathcal{C}(P,c_{2})$ has $M_{2}$ facets, then there are $2^{M_{2}}$ nice
colorings $\chi$.

######  Construction 4.11 (The vector-coloring induced from a bipartite
Hamiltonian subcomplex).

Let $\Lambda\colon\\{F_{1},\dots,F_{m}\\}\to\mathbb{Z}_{2}^{r}$ be a vector-
coloring of rank $r$ of a simple $n$-polytope $P$. Let
$\mathcal{C}(P,\Lambda)\subset\mathcal{C}(P,c)$ be a proper bipartite
Hamiltonian subcomplex and $\chi\colon\\{G_{1},\dots,G_{M_{1}}\\}\to\\{0,1\\}$
be one of its nice colorings. Define the vector-coloring
$\Lambda_{\chi}\colon\\{F_{1},\dots,F_{m}\\}\to\mathbb{Z}_{2}^{r}\times\mathbb{Z}_{2}\simeq\mathbb{Z}_{2}^{r+1}$
of rank $r+1$ as $\Lambda_{\chi}(F_{i})=(\Lambda(F_{i}),\chi(F_{i}))$. We call
$\Lambda_{\chi}$, as well as any vector-coloring obtained from it by a linear
change of coordinates in $\mathbb{Z}_{2}^{r+1}$, a vector-coloring induced
from a bipartite Hamiltonian subcomplex. By definition
$\mathcal{C}(P,c)=\mathcal{C}(P,\Lambda_{\chi})$.

######  Remark 4.12.

It can be shown that in general different nice colorings $\chi_{1}$ and
$\chi_{2}$ may produce vector colorings $\Lambda_{\chi_{1}}$ and
$\Lambda_{\chi_{2}}$ that can not be connected by a linear change of
coordinates in $\mathbb{Z}_{2}^{r+1}$.

######  Example 4.13 (A vector-coloring induced by a bipartite Hamiltonian
subcomplex).

For any complex $C=\mathcal{C}(P,c)$ with $M$ facets $G_{1}$, $\dots$, $G_{M}$
there is a canonical vector-coloring $\Lambda_{C}$ of rank $M$ of the polytope
$P$ defined as $\Lambda_{C}(F_{i})=\boldsymbol{e}_{j}$, where $F_{i}\subset
G_{j}$ and $\boldsymbol{e}_{1}$, $\dots$, $\boldsymbol{e}_{M}$ is the standard
basis in $\mathbb{Z}_{2}^{M}$. (Note that different colorings $c$ may produce
the same complex $C=\mathcal{C}(P,c)$. Therefore, $\Lambda_{C}$ is defined by
the coloring $c$ only if for each color the union of facets of the same color
is connected.) Then if $C=\mathcal{C}(P,c_{2})\subset\mathcal{C}(P,c_{1})$ is
a proper bipartite Hamiltonian subcomplex, there is a canonical vector-
coloring $\Lambda_{c_{2},c_{1}}$ induced by $C$. It is induced from the
canonical vector-coloring $\Lambda_{C}$ described above. Moreover, in this
case for each $i=1,\dots,M$ there is a linear isomorphism defined on basis as

$(\boldsymbol{e}_{1},\dots,\boldsymbol{e}_{i},\dots,\boldsymbol{e}_{M},\boldsymbol{e}_{M+1})\to(\boldsymbol{e}_{1},\dots,\boldsymbol{e}_{i}+\boldsymbol{e}_{M+1},\dots,\boldsymbol{e}_{M},\boldsymbol{e}_{M+1})$

exchanging $\boldsymbol{e}_{i}$ and $\boldsymbol{e}_{i}+\boldsymbol{e}_{M+1}$
and fixing $\boldsymbol{e}_{i}$ and $\boldsymbol{e}_{j}+\boldsymbol{e}_{M+1}$
for all $j\neq i$. Therefore, $\Lambda_{c_{2},c_{1}}$ does not depend on
$\chi$ up to a change of coordinates in $\mathbb{Z}_{2}^{M+1}$ (see also [E24,
Construction 8.6]).

For any vector-coloring $\Lambda$ of rank $r$ each nonzero involution
$\tau\in\mathbb{Z}_{2}^{r}\simeq\mathbb{Z}_{2}^{m}/H(\Lambda)$ corresponds to
a projection
$\Pi_{\tau}\colon\mathbb{Z}_{2}^{r}\to\mathbb{Z}_{2}^{r}/\langle\tau\rangle\simeq\mathbb{Z}_{2}^{r-1}$
and a vector-coloring $\Lambda_{\tau}=\Lambda\circ\Pi_{\tau}$ such that
$N(P,\Lambda)/\langle\tau\rangle\simeq N(P,\Lambda_{\tau})$.

######  Proposition 4.14.

Let $\Lambda$ be a vector-coloring of rank $r$ of a simple $n$-polytope $P$
such that $N(P,\Lambda)$ is a closed topological manifold. Then nonzero
involutions $\tau\in\mathbb{Z}_{2}^{r}\setminus\\{0\\}$ are in bijection with
proper bipartite Hamiltonian subcomplexes
$\mathcal{C}(P,c)\subset\mathcal{C}(P,\Lambda)$ such that one of the following
equivalent conditions holds:

  1. (1)

$\Lambda$ is induced from some vector-coloring of $\mathcal{C}(P,c)$;

  2. (2)

there is $\tau\in\mathbb{Z}_{2}^{r}\setminus\\{0\\}$ such that for any
nonempty intersection of two different facets $G_{i}$ and $G_{j}$ of
$\mathcal{C}(P,\Lambda)$ we have $\Lambda_{i}+\Lambda_{j}=\tau$ if and only if
$c(G_{i})=c(G_{j})$.

###### Proof.

First let us prove that for a proper bipartite Hamiltonian subcomplex
$\mathcal{C}(P,c)\subset\mathcal{C}(P,\Lambda)$ conditions (1) and (2) are
equivalent. Indeed, by Construction 4.11 (1) implies (2). On the other hand,
if (2) holds, then the vector-coloring $\Lambda_{\tau}$ is constant on facets
of $\mathcal{C}(P,c)$ and differs for its adjacent facets. Thus,
$\mathcal{C}(P,c)=\mathcal{C}(P,\Lambda_{\tau})$. Using the change of
coordinates corresponding to an isomorphism
$\mathbb{Z}_{2}^{r}/\langle\tau\rangle\oplus\langle\tau\rangle\simeq\mathbb{Z}_{2}^{r}$
we see that $\Lambda$ is induced from $\Lambda_{\tau}$. Thus, (1) holds.

Now consider a nonzero involution $\tau\in\mathbb{Z}_{2}^{r}$. The subcomplex
$\mathcal{C}(P,\Lambda_{\tau})\subset\mathcal{C}(P,\Lambda)$ is Hamiltonian by
[E24, Proposition 5.12]. Moreover, it is bipartite. Indeed, for any two facets
$G_{i}$, $G_{j}$ of $\mathcal{C}(P,\Lambda)$ lying in the same facet of
$\mathcal{C}(P,\Lambda_{\tau})$ we have
$\Pi_{\tau}(\Lambda_{i})=\Pi_{\tau}(\Lambda_{j})$. Then either
$\Lambda_{i}=\Lambda_{j}$, or $\Lambda_{i}=\Lambda_{j}+\tau$. Moreover, if
$G_{i}\cap G_{j}\neq\varnothing$, then $\Lambda_{i}=\Lambda_{j}+\tau$, since
these vectors are different. Also condition (2) holds by definition of
$\Lambda_{\tau}$.

On the other hand, for a proper bipartite Hamiltonian subcomplex
$\mathcal{C}(P,c)$ the condition (2) uniquely defines the involution $\tau$
such that $\mathcal{C}(P,c)=\mathcal{C}(P,\Lambda_{\tau})$. ∎

######  Proposition 4.15.

Let $\Lambda$ be a vector-coloring of rank $r$ of a simple $n$-polytope $P$
such that $N(P,\Lambda)$ is a closed topological manifold. For an involution
$\tau\in\mathbb{Z}_{2}^{r}\setminus\\{0\\}$ the space
$N(P,\Lambda)/\langle\tau\rangle$ is a closed topological manifold if and only
if

$\displaystyle\tau\notin\\{\Lambda_{i_{1}}+\dots+\Lambda_{i_{k}}\colon k\neq
2,i_{1}<\dots<i_{k},G_{i_{1}}\cap\dots\cap G_{i_{k}}\neq\varnothing\\}=$
$\displaystyle\\{\Lambda_{j_{1}}+\dots+\Lambda_{j_{k}}\colon k\neq
2,j_{1}<\dots<j_{k},F_{j_{1}}\cap\dots\cap
F_{j_{k}}\neq\varnothing,\Lambda_{j_{a}}\neq\Lambda_{j_{b}}\text{ for $a\neq
b$}\\}.$

###### Proof.

By [E24, Theorem 5.1] $N(P,\Lambda)$ is a closed topological manifold of and
only if for any vertex $F_{j_{1}}\cap\dots\cap F_{j_{n}}$ of $P$ different
vectors among $\\{\Lambda_{j_{1}},\dots,\Lambda_{j_{n}}\\}$ are linearly
independent. This is equivalent to the fact that for any collection of indices
$i_{1}<\dots<i_{k}$, $k\geqslant 1$, such that $G_{i_{1}}\cap\dots\cap
G_{i_{k}}\neq\varnothing$ the vectors
$\\{\Lambda_{i_{1}},\dots,\Lambda_{i_{k}}\\}$ are linearly independent and to
the fact that for any collection of indices $j_{1}<\dots<j_{k}$, $k\geqslant
1$, such that $F_{j_{1}}\cap\dots\cap F_{j_{k}}\neq\varnothing$ and
$\Lambda_{j_{a}}\neq\Lambda_{j_{b}}\text{ for $a\neq b$}$ the vectors
$\\{\Lambda_{j_{1}},\dots,\Lambda_{j_{k}}\\}$ are linearly independent.

Let us denote by $[\Lambda_{j}]$ the image of $\Lambda_{j}$ under the
projection $\mathbb{Z}_{2}^{r}\to\mathbb{Z}_{2}^{r}/\langle\tau\rangle$.
Assume that for a collection of indices $j_{1}<\dots<j_{k}$, $k\geqslant 1$,
such that $F_{j_{1}}\cap\dots\cap F_{j_{k}}\neq\varnothing$ we have
$[\Lambda_{j_{a}}]\neq[\Lambda_{j_{b}}]\text{ for $a\neq b$}$. Then
$\Lambda_{j_{a}}+\Lambda_{j_{b}}\notin\\{0,\tau\\}$. For
$\\{[\Lambda_{j_{1}}],\dots,[\Lambda_{j_{k}}]\\}$ to be linearly independent
it is necessary and sufficient that
$\Lambda_{j_{q_{1}}}+\dots+\Lambda_{j_{q_{s}}}\notin\\{0,\tau\\}$ for any
nonempty subset $\\{q_{1},\dots,q_{s}\\}\subset\\{j_{1},\dots,j_{k}\\}$. Since
$N(P,\Lambda)$ is a closed manifold,
$\Lambda_{j_{q_{1}}}+\dots+\Lambda_{j_{q_{s}}}\neq 0$. This finishes the
proof. ∎

######  Proposition 4.16.

Let $\Lambda$ be a vector-coloring of rank $r$ of a simple $n$-polytope $P$
such that $N(P,\Lambda)$ is a closed topological manifold. Let
$\mathcal{C}(P,\Lambda)\subset\mathcal{C}(P,c)$ be a proper bipartite
Hamiltonian subcomplex. Then for any nice coloring $\chi$ the space
$N(P,\Lambda_{\chi})$ is a closed topological manifold.

###### Proof.

Indeed, by Lemma 4.4(1) if $G_{i_{1}}\cap\dots\cap G_{i_{k}}\neq\varnothing$,
then at least $k-1$ of these facets lie in different facets of
$\mathcal{C}(P,\Lambda)$. Then either all of them lie in different facets, or
$k-1$ lie in different facets and two of them lie in the same facet. By
definition of $\Lambda_{\chi}$ the vectors
$\Lambda_{\chi}(G_{i_{1}}),\dots,\Lambda_{\chi}(G_{i_{k}})$ are linearly
independent. ∎

###  4.3. Bipartite Hamiltonian subcomplexes and the four color theorem

Orientable small covers $N(P,\Lambda)$ over $3$-polytopes correspond to
colorings of $P$ in four colors. In this case the image of $\Lambda$ consists
either of three linearly independent vectors
$\boldsymbol{v}_{1},\boldsymbol{v}_{2}$, $\boldsymbol{v}_{3}$, or four vectors
$\boldsymbol{v}_{1}$, $\boldsymbol{v}_{2}$, $\boldsymbol{v}_{3}$,
$\boldsymbol{v}_{4}$ such that each three of them are linearly independent and
$\boldsymbol{v}_{1}+\boldsymbol{v}_{2}+\boldsymbol{v}_{3}+\boldsymbol{v}_{4}=0$.

######  Lemma 4.17.

For $\tau\in\mathbb{Z}_{2}^{3}\setminus\\{0\\}$ the space
$N(P,\Lambda)/\langle\tau\rangle=N(P,\Lambda_{\tau})$ is a closed $3$-manifold
if and only if
$\tau\in\\{\boldsymbol{v}_{1}+\boldsymbol{v}_{2},\boldsymbol{v}_{1}+\boldsymbol{v}_{3},\boldsymbol{v}_{2}+\boldsymbol{v}_{3}\\}$.
Moreover, each manifold $N(P,\Lambda_{\tau})$ is hyperelliptic with the
hyperelliptic involution $\mu=[\boldsymbol{v}_{1}+\boldsymbol{v}_{3}]$,
$[\boldsymbol{v}_{2}+\boldsymbol{v}_{3}]$,
$[\boldsymbol{v}_{1}+\boldsymbol{v}_{2}]$ respectively.

###### Proof.

The first statement is a corollary of Proposition 4.15. For the manifold
$N(P,\Lambda_{\tau})$ we have
$N(P,\Lambda_{\tau})/\langle\mu\rangle=N(P,(\Lambda_{\tau})_{\mu})$, where
$(\Lambda_{\tau})_{\mu}$ is constant on all the facets of $P$. Hence,
$\mathcal{C}(P,(\Lambda_{\tau})_{\mu})\simeq\mathcal{C}(3,1)$, and
$N(P,\Lambda_{\tau})/\langle\mu\rangle\simeq S^{3}$. ∎

######  Lemma 4.18.

For each
$\tau\in\\{\boldsymbol{v}_{1}+\boldsymbol{v}_{2},\boldsymbol{v}_{1}+\boldsymbol{v}_{3},\boldsymbol{v}_{2}+\boldsymbol{v}_{3}\\}$
the complex $\mathcal{C}(P,\Lambda_{\tau})$ is a bipartite Hamiltonian
subcomplex in $P$ defined by a set of disjoint simple edge-cycles containing
all the vertices of $P$.

###### Proof.

The subcomplex is bipartite and Hamiltonian by Proposition 4.14. Since
$N(P,\Lambda_{\tau})$ is a closed manifold and $\Lambda_{\tau}$ has rank $2$,
the complex $\mathcal{C}(P,\Lambda_{\tau})$ has no vertices (at a vertex there
should be three linearly independent vectors). Thus,
$\mathcal{C}^{1}(P,\Lambda_{\tau})$ consists of a disjoint set of circles, and
each vertex of $P$ lies on exactly one circle since
$\mathcal{C}(P,\Lambda_{\tau})\subset\partial P$ is a Hamiltonian subcomplex.
∎

######  Proposition 4.19.

A disjoint set of simple edge-cycles containing all the vertices of $P$
defines a bipartite Hamiltonian subcomplex $C$ in $\partial P$ if and only if
for any cycle $\gamma$ and each connected component of $\partial
P\setminus\gamma$ the number of vertices on $\gamma$ such that the edge of $P$
incident to this vertex and not lying on $\gamma$ lies in the corresponding
component is even.

###### Proof.

Indeed, if $C$ is Hamiltonian, then the facets of $P$ lying in each facet $G$
of $C$ can be nicely colored in two colors (nicely means that adjacent facets
have different colors). Then the colors along each component $\gamma$ of
$\partial G$ alter, therefore there is an even number of vertices on $\gamma$
with the edge going inside $G$. On the other hand, if for each component
$\gamma$ of $\partial G$ there is an even number of such vertices, we can join
these vertices outside $G$ by a set of simple piecewise linear paths. We
obtain a disjoint set of simple piecewise linear closed curves on $\partial
P$. Each curve divides $\partial P$ into two connected components homeomorphic
to disks, hence the adjacency graph of the complement to this set of curves is
a tree. Thus, the complement can be nicely colored in two colors, and this
coloring restricts to a nice coloring of $G$. ∎

######  Corollary 4.20.

For a simple $3$-polytope $P$ the following conditions are equivalent:

  1. (1)

$P$ has a coloring of facets in at most $4$ colors such that adjacent facets
have different colors;

  2. (2)

there is a disjoint set of simple edge-cycles such that

     * •

any vertex of $P$ lies in exactly one cycle;

     * •

for any cycle $\gamma$ from the set and any of the two connected components of
$\partial P\setminus\gamma$ the number of vertices on $\gamma$ with the edge
going inside this component is even.

######  Remark 4.21.

In [B1913] G. D. Birkhoff reduced the four color problem to the family of
simple $3$-polytopes $P$ such that $P$ is different from the simplex, has no
$3$\- and $4$-belts and each $5$-belt surrounds a facet, where a $k$-belt is a
cyclic sequence of $k$ facets such that two facets of this sequence are
adjacent if and only if they are successive and no three facets have a common
vertex. We call such polytopes strongly Pogorelov. Then Corollary 4.20 implies
that the four color theorem is equivalent to the fact that any strongly
Pogorelov polytope satisfies condition (2) of this corollary.

###  4.4. Subspaces of $N(P,\Lambda)$ corresponding to faces of
$\mathcal{C}(P,\Lambda)$

As it was mentioned in Lemma 2.9 any $k$-face $G$ of $\mathcal{C}(P,c)$ is an
orientable topological $k$-manifold, perhaps with a boundary. Moreover, Lemma
2.10 implies that $\partial G$ consists of faces of $\mathcal{C}(P,c)$ lying
in $G$. If $G$ is a connected component of the intersection
$G_{i_{1}}\cap\dots\cap G_{i_{n-k}}$, then $\partial G$ consists of connected
components of nonempty intersections $G\cap G_{j_{1}}\cap\dots\cap G_{j_{l}}$,
$\\{i_{1}<\dots<i_{n-k}\\}\cap\\{j_{1}<\dots<j_{l}\\}=\varnothing$.

######  Lemma 4.22.

Connected components of $G\cap G_{j_{1}}\cap\dots\cap G_{j_{l}}$ are exactly
connected components of $G_{i_{1}}\cap\dots\cap G_{i_{n-k}}\cap
G_{j_{1}}\cap\dots\cap G_{j_{l}}$ lying in $G$.

###### Proof.

Let $G_{i_{1}}\cap\dots\cap G_{i_{n-k}}=\bigsqcup\limits_{s}G_{\alpha_{s}}$,
$G_{j_{1}}\cap\dots\cap G_{j_{l}}=\bigsqcup\limits_{t}G_{\beta_{t}}$, and
$G_{\alpha_{s}}\cap G_{\beta_{t}}=\bigsqcup\limits_{w}G_{s,t,w}$ be the
decompositions into connected components. Then

$G_{i_{1}}\cap\dots\cap G_{i_{n-k}}\cap G_{j_{1}}\cap\dots\cap
G_{j_{l}}=\bigsqcup\limits_{s,t,w}G_{s,t,w}$

is a decomposition into a disjoint union of closed connected sets. Each set in
each union is a union of faces of $P$, hence the number of sets in each union
is finite. Therefore, each set $G_{s,t,w}$ is open in the topology induced to
$G_{i_{1}}\cap\dots\cap G_{i_{n-k}}\cap G_{j_{1}}\cap\dots\cap G_{j_{l}}$ and
it is a connected component of this set. If $G=G_{\alpha_{1}}$, then by the
same argument $G\cap G_{j_{1}}\cap\dots\cap
G_{j_{l}}=\bigsqcup\limits_{t,w}G_{1,t,w}$ is the decomposition into connected
components. This finishes the proof. ∎

######  Lemma 4.23.

For any faces $G^{\prime}$ and $G^{\prime\prime}$ of $\mathcal{C}(P,c)$ if
their intersection is nonempty, then it is a disjoint union of faces
$\mathcal{C}(P,c)$ of equal dimensions.

###### Proof.

Indeed, if $G^{\prime}$ is a connected component of $G_{i_{1}}\cap\dots,\cap
G_{i_{a}}\cap G_{i_{a+1}}\cap\dots\cap G_{i_{b}}$, and $G^{\prime\prime}$ is a
connected component of $G_{i_{1}}\cap\dots\cap G_{i_{a}}\cap
G_{i_{b+1}}\cap\dots\cap G_{i_{c}}$, where $G_{i_{1}}$, $\dots$, $G_{i_{a}}$,
$\dots$, $G_{i_{b}}$, $\dots$, $G_{i_{c}}$ are different facets of
$\mathcal{C}(P,c)$, then $G^{\prime}\cap G^{\prime\prime}\subset
G_{i_{1}}\cap\dots\cap G_{i_{c}}$, where the latter set is a disjoint union of
its connected components – faces of $\mathcal{C}(P,c)$ of the same dimension.
Again, as in the proof of Lemma 4.22 let $G_{i_{1}}\cap\dots,\cap
G_{i_{a}}\cap G_{i_{a+1}}\cap\dots\cap
G_{i_{b}}=\bigsqcup\limits_{s}G_{\alpha_{s}}$, $G_{i_{1}}\cap\dots\cap
G_{i_{a}}\cap G_{i_{b+1}}\cap\dots\cap
G_{i_{c}}=\bigsqcup\limits_{t}G_{\beta_{t}}$, and $G_{\alpha_{s}}\cap
G_{\beta_{t}}=\bigsqcup\limits_{w}G_{s,t,w}$ be the decompositions into
connected components. Then

$G_{i_{1}}\cap\dots\cap G_{i_{c}}=\left(G_{i_{1}}\cap\dots,\cap G_{i_{a}}\cap
G_{i_{a+1}}\cap\dots\cap G_{i_{b}}\right)\cap\left(G_{i_{1}}\cap\dots\cap
G_{i_{a}}\cap G_{i_{b+1}}\cap\dots\cap
G_{i_{c}}\right)=\bigsqcup\limits_{s,t,w}G_{s,t,w}$

is a decomposition into a disjoint union of closed connected sets. Each set in
each union is a union of faces of $P$, hence the number of sets in each union
is finite. Therefore, each set $G_{s,t,w}$ is open in the topology induced to
$G_{i_{1}}\cap\dots\cap G_{i_{c}}$ and it is a connected component of this
set. If $G^{\prime}=G_{\alpha_{1}}$ and $G^{\prime\prime}=G_{\beta_{1}}$ then
by the same argument $G^{\prime}\cap
G^{\prime\prime}=\bigsqcup\limits_{w}G_{1,1,w}$ is the decomposition into
connected components. This finishes the proof.

∎

######  Definition 4.24.

For a $k$-face $G$ of $\mathcal{C}(P,c)$ we will call by facets of $G$
connected components of nonempty intersections of $G\cap G_{j}$, $G\not\subset
G_{j}$, and by faces of $G$ connected components of intersection of its
facets.

######  Corollary 4.25.

Faces of $G$ are exactly faces of $\mathcal{C}(P,c)$ lying in $G$.

The vector-coloring
$\Lambda\colon\\{F_{1},\dots,F_{m}\\}\to\mathbb{Z}_{2}^{r}$ of $P$ induces the
vector-coloring
$\Lambda_{G}\colon\\{\widetilde{G}_{1},\dots,\widetilde{G}_{q}\\}\to\mathbb{Z}_{2}^{r}\to\mathbb{Z}_{2}^{r}/\langle\Lambda_{i_{1}},\dots,\Lambda_{i_{n-k}}\rangle$
of facets of any $k$-face $G\subset\mathcal{C}(P,\Lambda)$: a connected
component of $G\cap G_{j}$ is mapped to $[\Lambda_{j}]$.

######  Definition 4.26.

Define
$V_{G}=\langle\Lambda_{G}(\widetilde{G}_{j}),j=1,\dots,q\rangle\subset\mathbb{Z}_{2}^{r}/\langle\Lambda_{i_{1}},\dots,\Lambda_{i_{n-k}}\rangle$
and

$N(G,\Lambda_{G})=G\times V_{G}/\sim,\text{ where
}(p,t)\sim(q,s)\Leftrightarrow p=q\text{ and
}t-s\in\langle\Lambda_{G}(\widetilde{G}_{j})\colon
p\in\widetilde{G}_{j}\rangle.$

######  Proposition 4.27.

Let $\Lambda$ be a vector-coloring of rank $r$ of a simple $n$-polytope $P$
such that $N(P,\Lambda)$ is a closed topological manifold, and let
$\pi_{\Lambda}\colon N(P,\Lambda)\to P$ be the projection. Then for any
$k$-face $G$ of $\mathcal{C}(P,\Lambda)$ the preimage $\pi_{\Lambda}^{-1}(G)$
is a closed topological $k$-dimensional submanifold (locally defined in some
coordinate system as the intersection of $(n-k)$ coordinate hyperplanes) in
$N(P,\Lambda)$ homeomorphic to a disjoint union of
$|\left(\mathbb{Z}_{2}^{r}/\langle\Lambda_{i_{1}},\dots,\Lambda_{i_{n-k}}\rangle\right)/V_{G}|=2^{r-(n-k)-\dim
V_{G}}$ manifolds $N(G,\Lambda_{G})$.

###### Proof.

Indeed, $N(P,\Lambda)=P\times\mathbb{Z}_{2}^{r}/\sim$, where $(p,a)\sim(q,b)$
if and only if $p=q$ and $a-b\in\langle\Lambda_{i}\colon p\in
F_{i}\rangle=\langle\Lambda_{j}\colon p\in G_{j}\rangle$. Also
$\pi_{\Lambda}^{-1}(G)=G\times\mathbb{Z}_{2}^{r}/\sim$, where $(p,a)\sim(q,b)$
if and only if $p=q$ and $a-b\in\langle\Lambda_{j}\colon p\in G_{j}\rangle$.
But each point $p\in G$ lies in $G_{i_{1}}\cap\dots\cap G_{i_{n-k}}$. Hence,

$\mathbb{Z}_{2}^{r}/\langle\Lambda_{j}\colon p\in
G_{j}\rangle\simeq(\mathbb{Z}_{2}^{r}/\langle\Lambda_{i_{1}},\dots,\Lambda_{i_{n-k}}\rangle)/(\langle\Lambda_{j}\colon
p\in G_{j}\rangle/\langle\Lambda_{i_{1}},\dots,\Lambda_{i_{n-k}}\rangle).$

This space is a disjoint union of cosets modulo $V_{G}$. Starting from the
point $[p,a]\in\pi_{\Lambda}^{-1}(G)$ and moving inside this space we can
reach by a path only the points $[q,b]$, where $a-b\in V_{G}$. Hence,
$\pi_{\Lambda}^{-1}(G)$ is a disjoint union of
$|\left(\mathbb{Z}_{2}^{r}/\langle\Lambda_{i_{1}},\dots,\Lambda_{i_{n-k}}\rangle\right)/V_{G}|$
connected components, and each component is homeomorphic to
$N(G,\Lambda_{G})$. Since $N(P,\Lambda)$ is a closed manifold, for each point
$p\in\partial P$, which belongs to exactly $l$ facets $G_{j_{1}}$, $\dots$,
$G_{j_{l}}$, the vectors $\Lambda_{j_{1}},\dots,\Lambda_{j_{l}}$ are linearly
independent. By Lemma 2.10 $p$ 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 to the neighbourhood $U$
homeomorphic to $\mathbb{R}^{l}\times\mathbb{R}^{n-l}$. Indeed, in $p\times a$
the copies
$P\times(a+\varepsilon_{1}\Lambda_{j_{1}}+\dots+\varepsilon_{l}\Lambda_{j_{l}})$,
$\varepsilon_{s}=\pm 1$, 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 $\Lambda_{j_{s}}$ corresponds to the
operation $y_{s}\to-y_{s}$. At each point of $\pi_{\Lambda}^{-1}(G)$ we may
take $j_{1}=i_{1}$, $\dots$, $j_{n-k}=i_{n-k}$ to see that
$\pi_{\Lambda}^{-1}(G)\cap U$ is defined by equations $y_{1}=\dots=y_{n-k}=0$.
This finishes the proof. ∎

######  Proposition 4.28.

Let $C=\mathcal{C}(P,c_{2})\subset\mathcal{C}(P,c_{1})$ be a proper
Hamiltonian subcomplex and $M_{1}$, $\dots$, $M_{s}$ be the set of its
defining $(n-2)$-faces. Then facets of each $M_{q}$ are connected components
of intersections of $M_{q}$ with facets of $C$ not containing $M_{q}$.
Moreover, each $k$-face $G$ of $M_{q}$, $0\leqslant k\leqslant n-2$, is a
connected component of intersection of $M_{q}$ with a unique $(k+1)$-face
$\widetilde{G}$ of $C$ containing $G$.

###### Proof.

By definition facets of $M_{q}\subset G_{i}\cap G_{j}$ are connected
components of intersections of $M_{q}$ with facets $G_{k}$ of
$\mathcal{C}(P,c_{1})$ different from $G_{i}$ and $G_{j}$. By Lemma 4.4 for
$G_{i}\cap G_{j}\cap G_{k}\neq\varnothing$ the facets $G_{i}$ and $G_{j}$ lie
in the same facet $\widetilde{G}_{a}$ of $\mathcal{C}(P,c_{2})$, and $G_{k}$
lies in another facet $\widetilde{G}_{b}$. Then

$M_{q}\cap G_{k}\subset
M_{q}\cap\widetilde{G}_{b}=M_{q}\cap\left(\bigcup\limits_{G_{l}\subset\widetilde{G}_{b}}G_{l}\right)=\bigcup\limits_{G_{l}\subset\widetilde{G}_{b}\colon
M_{q}\cap G_{l}\neq\varnothing}M_{q}\cap G_{l}.$

The latter union is disjoint since $(M_{q}\cap G_{l_{1}})\cap(M_{q}\cap
G_{l_{2}})\subset G_{i}\cap G_{j}\cap G_{l_{1}}\cap G_{l_{2}}=\varnothing$ by
Lemma 4.4(3). Each $M_{q}\cap G_{l}$ has a finite set of connected components
since it is a union of faces of $P$. Then we have a disjoint finite union of a
finite set of connected components of $M\cap G_{l}$,
$G_{l}\subset\widetilde{G}_{b}$. Then each connected component is also a
connected component of the union. This finishes the proof of the first
statement.

Each $k$-face $G$ of $M_{q}\subset G_{i}\cap G_{j}$ is a connected component
of $M_{q}\cap G_{i_{3}}\cap\dots\cap G_{i_{n-k}}$ by Lemma 4.22. By Lemma 4.4
$G_{i},G_{j}\subset\widetilde{G}_{j_{2}}$,
$G_{i_{3}}\subset\widetilde{G}_{j_{2}}$, $\dots$,
$G_{i_{n-k}}\subset\widetilde{G}_{j_{n-k}}$ for different facets
$\widetilde{G}_{j_{2}}$, $\dots$, $\widetilde{G}_{j_{n-k}}$ of
$\mathcal{C}(P,c_{2})$. Then
$M_{q}\cap\widetilde{G}_{j_{2}}\cap\dots\cap\widetilde{G}_{j_{n-k}}$ is equal
to the intersection of $M_{q}$ with a connected component $\widetilde{G}$ of
$\widetilde{G}_{j_{2}}\cap\dots\cap\widetilde{G}_{j_{n-k}}$ and it is equal to

$\displaystyle
M_{q}\cap\widetilde{G}_{j_{3}}\cap\dots\cap\widetilde{G}_{j_{n-k}}=$
$\displaystyle
M_{q}\cap\left(\bigcup\limits_{G_{l_{3}}\subset\widetilde{G}_{j_{3}}}G_{l_{3}}\right)\cap\dots\cap\left(\bigcup\limits_{G_{l_{n-k}}\subset\widetilde{G}_{j_{n-k}}}G_{l_{n-k}}\right)=\bigcup\limits_{G_{l_{3}}\subset\widetilde{G}_{j_{3}},\dots,G_{l_{n-k}}\subset\widetilde{G}_{j_{n-k}}}M_{q}\cap
G_{l_{3}}\cap\dots\cap G_{l_{n-k}},$

where any two nonempty sets in the union are disjoint, since at some position
$j$ the indices $l_{j}^{\prime}$ and $l_{j}^{\prime\prime}$ are different and
$G_{i}\cap G_{j}\cap G_{l_{j}^{\prime}}\cap
G_{l_{j}^{\prime\prime}}=\varnothing$. Then each connected component of each
set is a connected component of the union. In particular, for $l_{3}=i_{3}$,
$\dots$, $l_{n-k}=i_{n-k}$. Thus, $G$ is a connected component of intersection
of the $(k+1)$-face $\widetilde{G}$ of $C$ with $M_{q}$, and
$G\subset\widetilde{G}$. On the other hand, let $G$ be a connected component
of intersection of some $(k+1)$-face $\widetilde{G}^{\prime}$ of $C$ with
$M_{q}$, where $G\subset\widetilde{G}^{\prime}$. Then $\widetilde{G}^{\prime}$
is a connected component of some intersection
$\widetilde{G}_{u_{2}}\cap\dots\cap\widetilde{G}_{u_{n-k}}$. Then by the
argument in the beginning of the proof of Lemma 4.4 without loss of generality
we may assume that $G_{i}$, $G_{j}\subset\widetilde{G}_{u_{2}}$,
$G_{i_{3}}\subset\widetilde{G}_{u_{3}}$, $\dots$,
$G_{i_{n-k}}\subset\widetilde{G}_{u_{n-k}}$. Then $u_{2}=j_{2}$, $\dots$,
$u_{n-k}=j_{n-k}$, and $\widetilde{G}=\widetilde{G}^{\prime}$. ∎

######  Proposition 4.29.

Let $C=\mathcal{C}(P,c_{2})\subset\mathcal{C}(P,c_{1})$ be a proper
Hamiltonian subcomplex. Then any $k$-face
$\widetilde{G}\subset\mathcal{C}(P,c_{2})$ is a union
$\widetilde{G}=\bigcup_{\alpha}G_{\alpha}$ of $k$-faces of
$\mathcal{C}(P,c_{1})$, where the intersection of any three faces in the union
is empty, while each nonempty intersection of two faces in the union is a
disjoint union of $(q-1)$-faces of defining faces. In particular, the
adjacency graph $\Gamma(\widetilde{G})$ is well-defined. Its vertices are
faces $G_{\alpha}$ and edges correspond to connected components of
intersections of two such faces. The $(k-1)$-faces of defining faces
corresponding to the edges of $\Gamma(\widetilde{G})$ are pairwise disjoint.

###### Proof.

By definition $\widetilde{G}$ is a connected component of the intersection
$\widetilde{G}_{j_{1}}\cap\dots\cap\widetilde{G}_{j_{n-k}}$ of pairwise
different facets. We have

$\widetilde{G}_{j_{1}}\cap\dots\cap\widetilde{G}_{j_{n-k}}=\left(\bigcup\limits_{G_{l_{1}}\subset\widetilde{G}_{j_{1}}}G_{l_{1}}\right)\cap\dots\cap\left(\bigcup\limits_{G_{l_{n-k}}\subset\widetilde{G}_{j_{n-k}}}G_{l_{n-k}}\right)=\bigcup\limits_{G_{l_{1}}\subset\widetilde{G}_{j_{1}},\dots,G_{l_{n-k}}\subset\widetilde{G}_{j_{n-k}}}G_{l_{1}}\cap\dots\cap
G_{l_{n-k}}.$

The union on the right is a union of $k$-faces of $\mathcal{C}(P,c_{1})$,
which are connected components of nonempty intersections. Moreover, if two
such faces intersect, then they are connected components of two different
intersections $G_{l_{1}}\cap\dots\cap G_{l_{n-k}}$ and
$G_{l_{1}^{\prime}}\cap\dots\cap G_{l_{n-k}^{\prime}}$. If $l_{a}\neq
l_{a}^{\prime}$ and $l_{b}\neq l_{b}^{\prime}$, then
$\left(G_{l_{1}}\cap\dots\cap
G_{l_{n-k}}\right)\cap\left(G_{l_{1}^{\prime}}\cap\dots\cap
G_{l_{n-k}^{\prime}}\right)\subset\left(G_{l_{a}}\cap
G_{l_{a}^{\prime}}\right)\cap\left(G_{l_{b}}\cap
G_{l_{b}^{\prime}}\right)=\varnothing$. Thus, if faces intersect, then
$l_{a}=l_{a}^{\prime}$ for all indices $a$ but one, say $b$. Then the
intersection is a disjoint union of $(k-1)$-faces of defining faces lying in
$G_{l_{b}}\cap G_{l_{b}^{\prime}}$. If three faces intersect, then either for
all $a$ but one $l_{a}=l_{a}^{\prime}=l_{a}^{\prime\prime}$, and $l_{b}\neq
l_{b}^{\prime}\neq l_{b}^{\prime\prime}\neq l_{b}$, or there are two indices
$b_{1}\neq b_{2}$ such that $l_{b_{1}}\neq l_{b_{1}}^{\prime}$ and
$l_{b_{2}}\neq l_{b_{2}}^{\prime\prime}$. In the first case the intersection
lies in $(G_{l_{b_{1}}}\cap G_{l_{b_{1}}^{\prime}})\cap(G_{l_{b_{1}}}\cap
G_{l_{b_{1}}^{\prime\prime}})=\varnothing$, and in the second – in
$(G_{l_{b_{1}}}\cap G_{l_{b_{1}}^{\prime}})\cap(G_{l_{b_{2}}}\cap
G_{l_{b_{2}}^{\prime\prime}})=\varnothing$. ∎

######  Corollary 4.30.

Let $C=\mathcal{C}(P,c_{2})\subset\mathcal{C}(P,c_{1})$ be a proper
Hamiltonian subcomplex. If a $k$-face $G\subset\mathcal{C}(P,c_{1})$ does not
intersect defining faces, then $G$ is also a $k$-face of
$\mathcal{C}(P,c_{2})$ of the same combinatorial type.

###### Proof.

Indeed, $G$ lies in some $k$-face $\widetilde{G}$ of $\mathcal{C}(P,c_{2})$.
Since $G$ does not intersect defining faces, we have $G=\widetilde{G}$. By the
same argument each face of $G$ is a face of $\widetilde{G}$. On the other hand
each face of $\widetilde{G}$ is a subset of $G$ not intersecting defining
faces. Hence, it is a face of $G$. ∎

###  4.5. Bipartite Hamiltonian subcomplexes and two-sheeted branched
coverings

Remind that by $\pi_{\Lambda}$ we denote the projection $N(P,\Lambda)\to P$.

######  Proposition 4.31.

Let $\Lambda$ be a vector-coloring of rank $r$ of a simple $n$-polytope $P$
such that $N(P,\Lambda)$ is a closed topological manifold and let
$\tau\in\mathbb{Z}_{2}^{r}\setminus\\{0\\}$ be an involution such that
$N(P,\Lambda)/\langle\tau\rangle=N(P,\Lambda_{\tau})$ is also a closed
topological manifold. Let $M_{1}^{n-2}$, $\dots$, $M_{s}^{n-2}$ be defining
faces of the bipartite Hamiltonian subcomplex
$\mathcal{C}(P,\Lambda_{\tau})\subset\mathcal{C}(P,\Lambda)$ and
$M=M_{1}^{n-2}\sqcup\dots\sqcup M_{s}^{n-2}$. Then for the projection
$\pi\colon N(P,\Lambda)\to N(P,\Lambda_{\tau})$

  * •

the restriction $N(P,\Lambda)\setminus\pi_{\Lambda}^{-1}(M)\to
N(P,\Lambda_{\tau})\setminus\pi_{\Lambda_{\tau}}^{-1}(M)$ is a $2$-sheeted
covering;

  * •

the restriction $\pi_{\Lambda}^{-1}(M)\to\pi_{\Lambda_{\tau}}^{-1}(M)$ it one-
to-one and for each point $x\in\pi_{\Lambda}^{-1}(M)$ and its image $y$ there
are neighbourhoods such that

    * –

each neighbourhood is homeomorphic to $\mathbb{C}\times\mathbb{R}^{n-2}$;

    * –

the preimage of $M$ corresponds to the set $\\{0\\}\times\mathbb{R}^{n-2}$;

    * –

the mapping $N(P,\Lambda)\to N(P,\Lambda_{\tau})$ has the form
$(\boldsymbol{z},\boldsymbol{x})\to(\boldsymbol{z}^{2},\boldsymbol{x})$.

In particular, $\pi$ is a $2$-sheeted branched covering with the branch set
$\pi_{\Lambda_{\tau}}^{-1}(M)$ (see the definition in [G19]). Moreover,
$\pi_{\Lambda_{\tau}}^{-1}(M)\simeq\pi_{\Lambda}^{-1}(M)=\bigsqcup\limits_{q=1}^{s}\pi_{\Lambda}^{-1}(M_{q})$,
where $\pi_{\Lambda}^{-1}(M_{q})$ is a disjoint union of $2^{r-2-\dim
V_{M_{q}}}$ copies of the closed $(n-2)$-manifold $N(M_{q},\Lambda_{M_{q}})$.

###### Proof.

Indeed, for each point $p\notin M$ all the different facets $G_{i}$ of
$\mathcal{C}(P,\Lambda)$ containing $p$ lie in different facets
$\widetilde{G}_{j}$ of $\mathcal{C}(P,\Lambda_{\tau})$. As it was mentioned in
the proof of Proposition 4.27 each point $[p\times a]\in N(P,\Lambda)$ such
that $p$ belongs to exactly $l$ facets $G_{j_{1}}$, $\dots$, $G_{j_{l}}$ has a
neighbourhood $U([p\times a])$ homeomorphic to $\mathbb{R}^{n}$. Moreover,
there are $2^{r-l}$ such neighbourhoods corresponding to cosets in
$\mathbb{Z}_{2}^{r}/\langle\Lambda_{j_{1}},\dots,\Lambda_{j_{l}}\rangle$ and
they can be chosen to be disjoint. For $[p\times[a]]\in N(P,\Lambda_{\tau})$
we have similar $2^{r-1-l}$ neighbourhoods corresponding to cosets in
$(\mathbb{Z}_{2}^{r}/\langle\tau\rangle)/\langle[\Lambda_{j_{1}}],\dots,[\Lambda_{j_{l}}]\rangle$.
Each neighbourhood of $N(P,\Lambda)$ corresponding to a coset
$b+\langle\Lambda_{j_{1}},\dots,\Lambda_{j_{l}}\rangle$ is mapped to the
neighbourhood corresponding to the coset
$[b]+\langle[\Lambda_{j_{1}}],\dots,[\Lambda_{j_{l}}]\rangle$ and this mapping
is a homeomorphism. Moreover, the coset
$[b]+\langle[\Lambda_{j_{1}}],\dots,[\Lambda_{j_{l}}]\rangle$ corresponds to
exactly two cosets $b+\langle\Lambda_{j_{1}},\dots,\Lambda_{j_{l}}\rangle$ and
$(b+\tau)+\langle\Lambda_{j_{1}},\dots,\Lambda_{j_{l}}\rangle$.

Now consider a point $p\in M_{q}^{n-2}\subset G_{j_{1}}\cap G_{j_{2}}$ such
that $G_{j_{1}}$ and $G_{j_{2}}$ lie in the same facet $\widetilde{G}_{i_{2}}$
of $\mathcal{C}(P,\Lambda_{\tau})$ and $p$ lies in exactly $l$ facets
$G_{j_{1}}$, $G_{j_{2}}$, $\dots$, $G_{j_{l}}$. By Lemma 4.4 the sets
$G_{j_{1}}\cup G_{j_{2}}$, $G_{j_{3}}$, $\dots$, $G_{j_{l}}$ lie in $l-1$
different facets $\widetilde{G}_{i_{2}}$, $\widetilde{G}_{i_{3}}$, $\dots$,
$\widetilde{G}_{i_{l}}$. By Lemma 2.10 the point $p$ in $P$ has a
neighbourhood

$U\simeq\mathbb{R}^{l}_{\geqslant}\times\mathbb{R}^{n-l}=\\{(y_{1},\dots,y_{n})\in\mathbb{R}^{n}\colon
y_{1}\geqslant 0,\dots,y_{l}\geqslant 0\\},$

corresponding to $\mathcal{C}(P,\Lambda)$, where the facet $G_{j_{p}}$
corresponds to the hyperplane $y_{p}=0$. Set $\boldsymbol{z}=y_{1}+iy_{2}$ and
$\boldsymbol{x}=(y_{3},\dots,y_{n})$. The mapping
$(\boldsymbol{z},\boldsymbol{x})\to(\boldsymbol{z}^{2},\boldsymbol{x})$
defines a homeomorphism

$\mathbb{R}^{l}_{\geqslant}\times\mathbb{R}^{n-l}\to\mathbb{R}\times\mathbb{R}_{\geqslant}\times\mathbb{R}^{l-2}_{\geqslant}\times\mathbb{R}^{n-l}=\\{(\widetilde{y}_{1},\dots,\widetilde{y}_{n})\in\mathbb{R}^{n}\colon\widetilde{y}_{2}\geqslant
0,\dots,\widetilde{y}_{l}\geqslant
0\\}\simeq\mathbb{R}^{l-1}_{\geqslant}\times\mathbb{R}^{n-l-1},$

and under the composition of homeomorphisms the facets $\widetilde{G}_{i_{p}}$
are mapped to the hyperplanes $\widetilde{y}_{p}=0$, $p=2$, $\dots$, $l$. In
both coordinate systems $M_{q}^{n-2}$ is defined by the condition
$y_{1}=y_{2}=0$ and $\widetilde{y}_{1}=\widetilde{y}_{2}=0$. In $N(P,\Lambda)$
the neighbourhoods $U$ are glued to $2^{r-l}$ neighbourhoods $U([p,a])$
corresponding to cosets in
$\mathbb{Z}_{2}^{r}/\langle\Lambda_{j_{1}},\dots,\Lambda_{j_{l}}\rangle$. In
$N(P,\Lambda_{\tau})$ these neighbourhoods are glued to
$2^{r-l}=2^{(r-1)-(l-1)}$ neighbourhoods $U([p,[a]])$ corresponding to cosets
in
$(\mathbb{Z}_{2}^{r}/\langle\tau\rangle)/\langle[\Lambda_{j_{2}}],\dots,[\Lambda_{j_{l}}]\rangle$.
We have
$\mathbb{Z}_{2}^{r}/\langle\Lambda_{j_{1}},\dots,\Lambda_{j_{l}}\rangle\simeq(\mathbb{Z}_{2}^{r}/\langle\tau\rangle)/\langle[\Lambda_{j_{2}}],\dots,[\Lambda_{j_{l}}]\rangle$
since $\tau=\Lambda_{j_{1}}+\Lambda_{j_{2}}$. Thus, a neighbourhood
corresponding to a coset
$b+\langle\Lambda_{j_{1}},\dots,\Lambda_{j_{l}}\rangle$ is mapped to the
neighbourhood corresponding to the coset
$[b]+\langle[\Lambda_{j_{2}}],\dots,[\Lambda_{j_{l}}]\rangle$ and in the above
coordinates the sets $\pi_{\Lambda}^{-1}(M)$ and
$\pi_{\Lambda_{\tau}}^{-1}(M)$ are defined by the equations $\boldsymbol{z}=0$
and $\widetilde{\boldsymbol{z}}=0$, and the mapping has the form
$(\widetilde{\boldsymbol{z}},\widetilde{\boldsymbol{x}})=(\boldsymbol{z}^{2},\boldsymbol{x})$.
The last statement follows from Proposition 4.27. ∎

###  4.6. $\mathcal{C}(n,k)$-subcomplexes and hyperelliptic involutions

######  Definition 4.32.

We call by a $\mathcal{C}(n,k)$-subcomplex $C$ of $\mathcal{C}(P,c)$ a
subcomplex $C=\mathcal{C}(P,c_{2})\subset\mathcal{C}(P,c)$ such that
$\mathcal{C}(P,c_{2})\simeq\mathcal{C}(n,k)$. A $\mathcal{C}(n,k)$-subcomplex
is Hamiltonian, if $\mathcal{C}(P,c_{2})$ is Hamiltonian.

######  Corollary 4.33.

Any $\mathcal{C}(n,k)$-subcomplex $C\subset\partial P$ corresponds to a
subgroup $H(C)\subset\mathbb{Z}_{2}^{m}$ of rank $m-k$ such that
$\mathbb{R}\mathcal{Z}_{P}/H(C)\simeq S^{n}$. The subgroup is defined in
$\mathbb{Z}_{2}^{m}$ by equations $\sum\limits_{F_{j}\subset G_{i}}x_{j}=0$,
where $G_{i}$, $i=1$, $\dots$, $k$, are facets of the subcomplex.

Theorem 3.14 implies the following result.

######  Corollary 4.34.

For $n\leqslant 4$ the correspondence $C\to H(C)$ is a bijection between
$\mathcal{C}(n,k)$-subcomplexes $C\subset\partial P$ and subgroups
$H\subset\mathbb{Z}_{2}^{m}$ of rank $m-k$ such that
$\mathbb{R}\mathcal{Z}_{P}/H\simeq S^{n}$.

######  Proposition 4.35.

Let $C=\mathcal{C}(P,c_{2})\subset\mathcal{C}(P,c_{1})$ be a proper
Hamiltonian $\mathcal{C}(n,k)$-subcomplex. Then for any facet $\widetilde{G}$
of $C$ the adjacency graph $\Gamma(\widetilde{G})$ is a tree. In particular,
the subcomplex $C$ is bipartite.

###### Proof.

For $n\leqslant 2$ this is trivial. Assume that $n\geqslant 3$. Any facet
$\widetilde{G}$ of $C$ is a topological disk. Consider a defining $(n-2)$-face
$M\subset G_{i}\cap G_{j}$, where $G_{i},G_{j}\subset\widetilde{G}$ are facets
of $\mathcal{C}(P,c_{1})$. $M$ is a connected orientable $(n-2)$-manifold,
perhaps with a boundary. If $M$ has a boundary, then $\partial M$ is a union
of $(n-3)$-faces. Since $C$ is Hamiltonian, any such a face lies in the
$(n-2)$-skeleton of $\mathcal{C}(P,c_{2})$. In particular, $\partial
M\subset\partial G$, which is a topological sphere. Consider a point $p\in M$.
Let $p$ belong to exactly $l$ facets $G_{j_{1}}$, $\dots$, $G_{j_{l}}$, where
$G_{j_{1}}=G_{i}$ and $G_{j_{2}}=G_{j}$. By Lemma 4.4 the sets $G_{j_{1}}\cup
G_{j_{2}}$, $G_{j_{3}}$, $\dots$, $G_{j_{l}}$ lie in $l-1$ different facets
$\widetilde{G}_{i_{2}}$, $\widetilde{G}_{i_{3}}$, $\dots$,
$\widetilde{G}_{i_{l}}$. As in the proof of Proposition 4.31 the point $p$ in
$P$ has a neighbourhood

$U\simeq\\{(\widetilde{y}_{1},\dots,\widetilde{y}_{n})\in\mathbb{R}^{n}\colon\widetilde{y}_{2}\geqslant
0,\dots,\widetilde{y}_{l}\geqslant
0\\}\simeq\mathbb{R}^{l-1}_{\geqslant}\times\mathbb{R}^{n-l-1},$

where the facets $\widetilde{G}_{i_{p}}$ correspond to the hyperplanes
$\widetilde{y}_{p}=0$, $p=2$, $\dots$, $l$, and $M$ is defined by the
equations $\widetilde{y}_{1}=\widetilde{y}_{2}=0$. Thus, $M$ is an orientable
polyhedral $(n-2)$-manifold locally flat embedded to the polyhedral
$(n-1)$-disk $\widetilde{G}$ with $\partial M=M\cap\partial\widetilde{G}$.

######  Lemma 4.36.

The complement $\widetilde{G}\setminus M$ has exactly two connected
components.

###### Proof.

It is easy to see that the complement $\widetilde{G}\setminus M$ has at most
two connected components, since it is valid locally at each point of $M$, and
from each point of the complement we can reach such a point by a simple
piecewise linear path intersecting $M$ at one point.

On the other hand, assume that there is only one connected component. If
$\partial M\neq\varnothing$, then this is valid also for the double manifolds
$DM=M\cup_{\partial M}M\subset
D\widetilde{G}=\widetilde{G}\cup_{\partial\widetilde{G}}\widetilde{G}\simeq
S^{n-1}$. But $DM$ is an orientable $(n-2)$-manifold, hence
$H_{n-2}(DM)=\mathbb{Z}$. By the Alexander duality
$\widetilde{H}^{0}(\widetilde{G}\cup_{\partial\widetilde{G}}\widetilde{G}\setminus
M\cup_{\partial M}M)\simeq\mathbb{Z}$. Hence, there are two connected
components. A contradiction.

If $\partial M=\varnothing$, then $M\cap\partial\widetilde{G}=\varnothing$ and
for the space
$\widetilde{G}^{\prime}=\widetilde{G}/\partial\widetilde{G}\simeq S^{n}$ the
complement $\widetilde{G}^{\prime}\setminus M$ also has one connected
component. But this contradicts the Alexander duality. ∎

######  Corollary 4.37.

We have $M=G_{i}\cap G_{j}$.

###### Proof.

Indeed, by Lemma 4.36 $\widetilde{G}\setminus M$ has two connected components
$C_{1}$ and $C_{2}$, where ${\rm int}\,G_{i}\subset C_{1}$ and ${\rm
int}\,G_{j}\subset C_{2}$. Moreover, $\overline{C_{1}}\cap\overline{C_{2}}=M$
since $M$ is locally flat embedded to $\widetilde{G}$. Then $G_{i}$ and
$G_{j}$ have no common points lying outside $M$. ∎

Lemma 4.36 and Corollary 4.37 imply that the adjacency graph
$\Gamma(\widetilde{G})$ is a tree. Since it is valid for each $\widetilde{G}$,
the subcomplex $C$ is bipartite. ∎

######  Construction 4.38 (A vector-coloring induced by a Hamiltonian
$\mathcal{C}(n,k)$-subcomplex).

Proposition 4.35 and Example 4.13 imply that any proper Hamiltonian
$\mathcal{C}(n,k)$-subcomplex
$C=\mathcal{C}(P,c_{2})\subset\mathcal{C}(P,c_{1})$ induces a vector-coloring
$\widetilde{\Lambda}_{C}$ of $P$ of rank $k+1$ defined up to a linear change
of coordinates in $\mathbb{Z}_{2}^{k+1}$. We have
$\mathcal{C}(P,\widetilde{\Lambda}_{C})=\mathcal{C}(P,c_{1})$ and
$N(P,\widetilde{\Lambda}_{C})$ is a hyperelliptic manifold with the
hyperelliptic involution $(0,0,\dots,1)$ (in terms of Construction 4.11).

Theorem 3.14 implies the following result.

######  Theorem 4.39.

Let $n\leqslant 4$ and $\Lambda$ be a vector-coloring of rank $r$ of a simple
$n$-polytope $P$ such that $N(P,\Lambda)$ is a closed topological manifold.
Then nonzero hyperelliptic involutions
$\tau\in\mathbb{Z}_{2}^{r}\setminus\\{0\\}$ are in bijection with proper
Hamiltonian $\mathcal{C}(n,r-1)$-subcomplexes $C\subset\mathcal{C}(P,\Lambda)$
inducing $\Lambda$ (that is $\Lambda=\widetilde{\Lambda}_{C}$ up to a change
of coordinates).

######  Definition 4.40.

For a complex $C=\mathcal{C}(P,c)$ set $f_{i}(C)$ to be the number of its
$i$-faces. By definition $f_{n}(C)=1$. We have $f_{n-1}(C)=M$.

######  Proposition 4.41.

Let $C=\mathcal{C}(P,c_{2})\subset\mathcal{C}(P,c_{1})$ be a proper
Hamiltonian $\mathcal{C}(n,r)$-subcomplex. If a $k$-face $\widetilde{G}$ of
$C$ is not a circle, then the adjacency graph $\Gamma(\widetilde{G})$ of
$k$-faces $G$ of $\mathcal{C}(P,c_{1})$ lying in $\widetilde{G}$ is a tree,
where we call the faces adjacent if they have a nonempty intersection.

###### Proof.

Indeed, for $k=0$ this is trivial. For $k=1$ if $\widetilde{G}$ is not a
circle, then it is an edge subdivided by vertices of $\mathcal{C}(P,c_{1})$
into edges, and $\Gamma(\widetilde{G})$ is a path. Assume that $k>1$. Each
face of $C$ is either a topological disk or a sphere. Let $\widetilde{G}$ be a
connected component of
$\widetilde{G}_{j_{1}}\cap\dots\cap\widetilde{G}_{j_{n-k}}$.

######  Lemma 4.42.

If the intersection of two $k$-faces $G^{\prime}$ and $G^{\prime\prime}$ of
$\mathcal{C}(P,c_{1})$ lying in $\widetilde{G}$ is nonempty, then
$G^{\prime}\cap G^{\prime\prime}$ is a $(k-1)$-face of $\mathcal{C}(P,c_{1})$
lying in a defining $(n-2)$-face. Moreover,
$\widetilde{G}\setminus(G^{\prime}\cap G^{\prime\prime})$ consists of two
connected components $C_{1}$ and $C_{2}$ such that $G^{\prime}\cap
G^{\prime\prime}=\overline{C_{1}}\cap\overline{C_{2}}$, and
$G^{\prime}\subset\overline{C_{1}}$,
$G^{\prime\prime}\subset\overline{C_{2}}$.

Let $G^{\prime}$ and $G^{\prime\prime}$ be connected components of
$G_{a_{1}}\cap\dots\cap G_{a_{n-k}}$ and $G_{b_{1}}\cap\dots\cap G_{b_{n-k}}$.
Then up to a renumbering of indices
$G_{a_{i}},G_{b_{i}}\subset\widetilde{G}_{j_{i}}$ for all $i$. By Lemma 4.4
$a_{i}\neq b_{i}$ only for one $i$, say for $i=1$. Then $G^{\prime}\cap
G^{\prime\prime}\subset G_{a_{1}}\cap G_{b_{1}}\cap G_{b_{2}}\cap\dots\cap
G_{b_{n-k}}$. In particular, each connected component $G$ of $G^{\prime}\cap
G^{\prime\prime}$ lies in a defining $(n-2)$-face.

The argument in the proof of Proposition 4.35 shows that
$\widetilde{G}\setminus G$ has two connected components $C_{1}$ and $C_{2}$
with $G=\overline{C_{1}}\cap\overline{C_{2}}$ and
$G^{\prime}\subset\overline{C_{1}}$,
$G^{\prime\prime}\subset\overline{C_{2}}$. Then $G^{\prime}\cap
G^{\prime\prime}=G$ and the adjacency graph $\Gamma(\widetilde{G})$ is a tree.
∎

######  Proposition 4.43.

Let $C_{2}=\mathcal{C}(P,c_{2})\subset\mathcal{C}(P,c_{1})=C_{1}$ be a proper
Hamiltonian $\mathcal{C}(n,r)$-subcomplex with the set of defining faces
$M_{1}$, $\dots$, $M_{s}$. Then for $0\leqslant k\leqslant n-1$

(2) $\displaystyle
f_{k}(C_{1})=f_{k}(C_{2})-\delta_{r,n-1}\delta_{k,1}+\sum\limits_{q=1}^{s}\left(f_{k}(M_{q})+f_{k-1}(M_{q})\right)=$
(3) $\displaystyle{r\choose
n-k}+\delta_{r,n}\delta_{k,0}-\delta_{r,n-1}\delta_{k,1}+\sum\limits_{q=1}^{s}\left(f_{k}(M_{q})+f_{k-1}(M_{q})\right).$

where $\delta_{i,j}=1$ if $i=j$, and $\delta_{i,j}=0$ if $i\neq j$.

###### Proof.

Indeed, each $k$-face $G$ of $C_{1}$ either lies in some $M_{q}$ or it does
not lie in defining faces but lies in a unique $k$-face $\widetilde{G}$ of
$C_{2}$. By Proposition 4.41 if $(r,k)\neq(n-1,1)$ the $k$-faces $G$ of the
latter type lying in $\widetilde{G}$ form a tree with edges corresponding to
$(k-1)$-faces of defining faces lying in $\widetilde{G}$. By Proposition 4.28
each $(k-1)$-face of a defining face lies in a unique $k$-face $\widetilde{G}$
of $C_{2}$. Thus, for $(r,k)\neq(n-1,1)$ the number of $k$-faces
$G\subset\widetilde{G}$ is one greater then the number of $(k-1)$-faces inside
$\widetilde{G}$ lying in defining faces. For $(r,k)=(n-1,1)$ the number of
$1$-faces of $C_{1}$ lying in a unique $1$-face of $C_{2}$ (which is a circle)
is equal to the number of vertices of defining faces. Also we have
$f_{k}(C_{2})={r\choose n-k}+\delta_{r,n}\delta_{k,0}$. This finishes the
proof. ∎

######  Proposition 4.44.

Let $C_{2}=\mathcal{C}(P,c_{2})\subset\mathcal{C}(P,c_{1})=C_{1}$ be a proper
Hamiltonian $\mathcal{C}(n,r)$-subcomplex with the set of defining faces
$M_{1}$, $\dots$, $M_{s}$. Then for any defining face $M_{q}$ its facets can
be colored in $(r-1)$ colors in such a way that if the intersection of a set
of facets is nonempty, then their colors are pairwise different.

###### Proof.

Indeed, any defining face $M_{q}$ lies in some facet $\widetilde{G}_{i}$ of
$C_{2}$ and its facets are connected components of intersections of $M_{q}$
with facets $\widetilde{G}_{j}\neq\widetilde{G}_{i}$. Then we can assign to
each connected component of $M_{q}\cap\widetilde{G}_{j}$ the color $j$. If the
intersection of facets of $M_{q}$ is nonempty, then by construction they have
different colors. ∎

###  4.7. Geometric hyperelliptic manifolds $N(P,\Lambda)$

If $P$ is a right-angled $n$-polytope in some geometry $\mathbb{X}$ and
$\Lambda$ is a linearly independent vector-coloring of rank $r$, then
$N(P,\Lambda)$ is a manifold with a geometric structure modelled on
$\mathbb{X}$.

######  Construction 4.45.

Let $P$ be a right-angled $n$-polytope in some geometry $\mathbb{X}$. If
$C\subset P$ is a proper Hamiltonian $C(n,k)$-subcomplex, then there is a
canonical linearly independent vector-coring $\widetilde{\Lambda}_{C}$ of rank
$k+1$ induced by $C$. It is defined up to a linear change of coordinates in
$\mathbb{Z}_{2}^{k+1}$. Then $N(P,\widetilde{\Lambda}_{C})$ is a geometric
hyperelliptic $n$-manifold with a hyperelliptic involution in
$\mathbb{Z}_{2}^{k+1}$ corresponding to $C$.

######  Remark 4.46.

Construction 4.45 is a direct generalization to dimensions $n>3$ of the
construction of geometric hyperelliptic $3$-manifolds in [M90, VM99M, VM99S2]
based on Hamiltonian cycles, theta- and $K_{4}$-subgraphs in right-angled
$3$-polytopes.

######  Remark 4.47.

Theorem 4.39 implies that for $n\leqslant 4$ and any linearly independent
vector-coloring $\Lambda$ of a right-angled $n$-polytope $P$ if $N(P,\Lambda)$
admits a hyperelliptic involution in $\mathbb{Z}_{2}^{r}$, then this
involution, the coloring $\Lambda$ and $N(P,\Lambda)$ can be obtained by
Construction 4.45.

In this section we will discuss which geometries $\mathbb{X}$ admit
hyperelliptic manifolds obtained by Construction 4.45.

######  Corollary 4.48.

Let $n>2$ and $C$ be a Hamiltonian $\mathcal{C}(n,r)$-subcomplex in $\partial
P$. Then $n-1\leqslant r\leqslant n+1$ and the vertex set of $P$ is a disjoint
union of vertices of $C$ (their number $V_{n,r}$ is: $V_{n,n-1}=0$,
$V_{n,n}=2$, $V_{n,n+1}=n+1$) and vertices of defining $(n-2)$-faces $M_{1}$,
$\dots$, $M_{s}$. The number of defining faces is equal to $m-r$. Moreover,
for each $0\leqslant k\leqslant n-1$

$f_{k}(P)={r\choose
n-k}+\delta_{r,n}\delta_{k,0}-\delta_{r,n-1}\delta_{k,1}+\sum\limits_{q=1}^{s}\left(f_{k}(M_{q})+f_{k-1}(M_{q})\right).$

###### Proof.

This follows directly from Proposition 4.43.

In particular, each vertex of $P$ either is a vertex of $C$ or lies on its
$1$-face. Thus, $C$ should have $1$-faces. This is possible only for
$k\in\\{n-1,n,n+1\\}$. Moreover, $\mathcal{C}^{1}(n,n-1)$ is a circle,
$\mathcal{C}^{1}(n,n-1)$ consists of two vertices and $n$ multiple edges, and
$\mathcal{C}^{1}(n,n-1)$ is the complete graph on $n$ vertices (see [E24,
Example 8.17]). If a vertex $v$ of $P$ is a vertex of $C$, then it is the
intersection of $n$ different facets of $C$, and it does not lie in defining
faces. If $v$ lies in a $1$-skeleton of $C$, then two facets of $P$ containing
$v$ lie in the same facet of $C$. Then their intersection is a defining face.

For $k=n-1>1$ we have $m=f_{n-1}(P)={r\choose
1}+\sum\limits_{q=1}^{s}f_{n-2}(M_{q})=r-s$. Thus, $s=m-r$ is the total number
of defining faces. ∎

######  Corollary 4.49.

The $n$-cube $I^{n}$ does not admit Hamiltonian
$\mathcal{C}(n,k)$-subcomplexes for $n>3$. In particular, there are no locally
Euclidean hyperelliptic $n$-manifolds $N(P,\Lambda)$, $n>3$, obtained by
Construction 4.45.

###### Proof.

The cube has $2^{n}$ vertices and $2n$ facets. Any $(n-2)$-face is $I^{n-2}$
and has $2^{n-2}$ vertices. If $I^{n}$ has a Hamiltonian
$\mathcal{C}(n,k)$-subcomplex, then $2^{n}=V_{k}+(2n-k)2^{n-2}$, where
$n-1\leqslant k\leqslant n+1$. We have $2n-k\geqslant n-1$. Thus, the right
part is at least $V_{k}+(n-1)2^{n-2}$. If $n\geqslant 5$, then it is at least
$V_{k}+4\cdot 2^{n-2}=V_{k}+2^{n}\geqslant 2^{n}$, where the equality holds
only if $n=5$ and $k=n+1$. But in this case $V_{k}=n+1=6$ and we obtain a
contradiction. If $n=4$, then we have $16=V_{k}+(8-k)\cdot 4$. For $k=3$ we
have $16=0+5\cdot 4=20$, for $k=4$ we have $16=2+4\cdot 4=18$, and for $k=5$
we have $16=5+3\cdot 4=17$. A contradiction. ∎

######  Remark 4.50.

For $n\leqslant 3$ the cube $I^{n}$ admits Hamiltonian
$\mathcal{C}(n,k)$-subcomplexes.

######  Corollary 4.51.

Let and $C$ be a Hamiltonian $\mathcal{C}(n,n-1)$-subcomplex in $\partial P$.
Then any its defining face is an $(n-2)$-face of $P$ admitting a coloring in
$(n-2)$ colors such that adjacent facets have different colors.

######  Remark 4.52.

As is was proved in [J01] a simple $n$-polytope $P$ admits a coloring of its
facets in $n$ colors such that adjacent facets have different colors if and
only if any $2$-face of $P$ has an even number of edges.

######  Corollary 4.53.

Let and $C$ be a Hamiltonian $\mathcal{C}(n,n-1)$-subcomplex in $\partial P$.
Then the vertices of $P$ lie on a disjoint set of even-gonal $2$-faces of
defining faces.

###### Proof.

Indeed, by Corollary 4.51 and Remark 4.52 each defining face $M_{q}$ has a
coloring in $(n-2)$ colors such that adjacent facets have different colors and
each its $2$-gonal face has an even number of edges. Moreover, for each choice
of $(n-4)$ colors any nonempty intersection of facets of these colors is an
even-gonal $2$-face of $M_{q}$, and any vertex of $M_{q}$ lies on exactly one
such a $2$-face. ∎

######  Corollary 4.54.

If a small cover $N(P,\Lambda)$ over a $4$-polytope $P$ has a hyperelliptic
involution in $\mathbb{Z}_{2}^{4}$, then this involution corresponds to a
Hamiltonian $\mathcal{C}(4,3)$-subcomplex inducing $\Lambda$ and the vertices
of $P$ lie on a disjoint set of $(m-3)$ defining even-gons.

######  Remark 4.55.

The first example of a small cover $N(P,\Lambda)$ over a $4$-polytope $P$ with
a hyperelliptic involution in $\mathbb{Z}_{2}^{4}$ was build by Alexei
Koretskii, see [K24].

######  Proposition 4.56.

If a $4$-polytope $P$ admits a Hamiltonian $\mathcal{C}(4,r)$-subcomplex, then
$P$ has at least one triangular or quadrangular $2$-face. In particular, there
are no hyperbolic ($\mathbb{X}=\mathbb{L}^{4}$) hyperelliptic $4$-manifolds
$N(P,\Lambda)$ obtained by Construction 4.45. Moreover, for all $n\geqslant 4$
there are no hyperbolic ($\mathbb{X}=\mathbb{L}^{n}$) $n$-manifolds
$N(P,\Lambda)$ obtained by Construction 4.45.

###### Proof.

Let $p_{k}$ be the number of $k$-gonal $2$-faces of $P$, and $f_{i}$ be the
total number of its $i$-faces (in particular, $f_{3}=m$). We have the Euler
formula $f_{0}-f_{1}+f_{2}-f_{3}=0$. Also $4f_{0}=2f_{1}$ since $P$ is simple.
Also $\sum\limits_{k}kp_{k}={4\choose 2}f_{0}=6f_{0}$. Then
$f_{2}=f_{1}-f_{0}+f_{3}=f_{0}+f_{3}=f_{0}+m$, and

$\sum\limits_{k}p_{k}=f_{2}=f_{0}+m=\frac{1}{6}\sum\limits_{k}kp_{k}+m.$

Thus, $\sum\limits_{k}6p_{k}=\sum\limits_{k}kp_{k}+6m$,
$\sum\limits_{k}(k-6)p_{k}+6m=0$, and

(4) $3p_{3}+2p_{4}+p_{5}=6m+\sum\limits_{k\geqslant 7}(k-6)p_{k}$

Assume that $P$ has no triangles and quadrangles. Then
$p_{5}=6m+\sum_{k\geqslant 7}(k-6)p_{k}$ and $f_{2}=\sum\limits_{k\geqslant
5}p_{k}=6m+\sum_{k\geqslant 6}(k-5)p_{k}$. Let $k_{i}$ be the number of edges
of $i$-th defining $2$-face. Then

$f_{0}=V_{4,r}+\sum_{i=1}^{m-r}k_{i}=f_{2}-f_{3}=5m+\sum_{k\geqslant
6}(k-5)p_{k}.$

Thus,

$\sum_{i=1}^{m-r}(k_{i}-5)=5m+\sum_{k\geqslant
6}(k-5)p_{k}-5(m-r)-V_{4,r}=5r-V_{4,r}+\sum_{k\geqslant 6}(k-5)p_{k}\geqslant
5r-V_{4,r}+\sum_{i=1}^{m-r}(k_{i}-5).$

Then $V_{4,r}\geqslant 5r$. For $r=3$ we have $0\geqslant 15$, for $r=4$ we
have $2\geqslant 20$, and for $r=5$ we have $5\geqslant 25$. In all cases this
is a contradiction.

Since any right-angled hyperbolic $4$-polytope has no triangular and
quadrangular $2$-faces, it does not admit Hamiltonian
$\mathcal{C}(4,r)$-subcomplexes. Moreover, it follows from the paper [N82] by
V.V. Nikulin that for $n>4$ in the hyperbolic $n$-space $\mathbb{L}^{n}$ there
are no compact right-angled polytopes. This finishes the proof. ∎

######  Proposition 4.57.

If a $4$-polytope $P=Q\times I$ admits a Hamiltonian
$\mathcal{C}(4,r)$-subcomplex, then at least one of the defining $2$-faces of
$P$ is a triangle. In particular, there are no hyperelliptic $4$-manifolds
$N(P,\Lambda)$ with geometries $\mathbb{L}^{3}\times\mathbb{R}$,
$\mathbb{L}^{2}\times\mathbb{R}^{2}$, and $\mathbb{R}^{4}$ obtained by
Construction 4.45.

###### Proof.

Let $q$ be the number of facets of $Q$. Then $f_{0}(Q)=2(q-2)$,
$m=f_{3}(P)=q+2$, and $f_{0}(P)=2f_{0}(Q)=4(m-4)$. Ley $k_{i}$ be the number
of edges of the $i$-th defining $2$-face. We have
$f_{0}(P)=V_{4,r}+\sum_{i=1}^{m-r}k_{i}=4(m-4)=4m-16$. Then the average number
of edges in defining $2$-faces is

$\frac{\sum\limits_{i=1}^{m-r}k_{i}}{m-r}=\frac{4m-16-V_{4,r}}{m-r}=\frac{4(m-r)+4r-16-V_{4,r}}{m-r}=4-\frac{16+V_{4,r}-4r}{m-r}=\begin{cases}4-\frac{4}{m-3}&\text{if
}r=3;\\\ 4-\frac{2}{m-4}&\text{if }r=4;\\\ 4-\frac{1}{m-5}&\text{if
}r=5;\end{cases}$

This number is strictly less then $4$. Therefore, at least one defining face
is a triangle. ∎

######  Proposition 4.58.

For $n\geqslant 4$ the simplex $\Delta^{n}$ up to symmetries has a unique
proper Hamiltonian $\mathcal{C}(n,r)$-subcomplex defined by a single
$(n-2)$-face $\Delta^{n-2}$. For this subcomplex $r=n$. It corresponds to a
hyperelliptic involution on the manifold
$\mathbb{R}\mathcal{Z}_{\Delta^{n}}\simeq S^{n}$. In particular, the geometry
$\mathbb{S}^{n}$ arises in Construction 4.45. The branch set of the covering
$S^{n}\to S^{n}$ is the sphere $S^{n-2}$.

######  Remark 4.59.

For $n=3$ the simplex $\Delta^{3}$ admits also a Hamiltonian
$\mathcal{C}(n,n-1)$-subcomplex corresponding to a Hamiltonian cycle.

###### Proof.

Indeed, for $n\geqslant 4$ any two $(n-2)$-faces of $\Delta^{n}$ intersect.
Also by definition any $(n-2)$-face of $\Delta^{n}$ is a defining face a
Hamiltonian $\mathcal{C}(n,n)$-subcomplex. The structure of the branch set
follows from the following fact.

######  Lemma 4.60.

Let $F$ be one of the defining $(n-2)$-faces of a Hamiltonian
$\mathcal{C}(n,k)$-subcomplex $C\subset\partial P$. If the facets of $F$
correspond to different facets of $C$ (equivalently, if $F\cap F_{i}$ and
$F\cap F_{j}$ are facets of $F$, then $F_{i}$ and $F_{j}$ lie in different
facet of $C$), then in the branch set of the covering
$N(P,\widetilde{\Lambda}_{C})\to S^{n}$ the preimage of $F$ is a disjoint
union of $2^{k-1-m_{F}}$ copies of $\mathbb{R}\mathcal{Z}_{F}$, where $m_{F}$
is the number of facets of $F$.

###### Proof.

This follows from Proposition 4.31. ∎

∎

######  Proposition 4.61.

For $n\geqslant 4$ the prism $\Delta^{n-1}\times I$ up to symmetries admits
exactly $3$ Hamiltonian $\mathcal{C}(n,r)$-subcomplexes: a unique
$\mathcal{C}(n,n+1)$-subcomplex and two $\mathcal{C}(n,n)$-subcomplexes. In
particular, the geometry $\mathbb{S}^{n-1}\times\mathbb{R}$ arises in
Construction 4.45. For the $\mathcal{C}(n,n+1)$-subcomplex and the
$\mathcal{C}(n,n)$-subcomplexes the branch sets of the coverings
$S^{n-1}\times S^{1}\to S^{n}$ and $N(P,\widetilde{\Lambda}_{C})\to S^{n}$ are
disjoint unions of two spheres $S^{n-2}$.

###### Proof.

Indeed, up to symmetries $\Delta^{n-1}\times I$ has two types of
$(n-2)$-faces: $\Delta^{n-3}\times I$ and $\Delta^{n-2}$. In the Hamiltonian
subcomplex corresponding to $\Delta^{n-3}\times I$ the facets corresponding to
$\Delta^{n-1}\times 0$ and $\Delta^{n-1}\times 1$ do not intersect. Therefore,
this subcomplex is not equivalent to $\mathcal{C}(n,r)$ for all $r$. The face
$\Delta^{n-2}$ corresponds to a Hamiltonian $\mathcal{C}(n,n+1)$-subcomplex.

Any face $\Delta^{n-3}\times I$ intersects any face of the type
$\Delta^{n-2}$. For $n\geqslant 5$ it also intersects any other face of the
type $\Delta^{n-3}\times I$. Hence it can not be a defining face of a
Hamiltonian $\mathcal{C}(n,r)$-subcomplex. For $n=4$ and any face $I\times I$
there is a unique face $I\times I$ such that these faces are disjoint. In the
corresponding Hamiltonian subcomplex the facets corresponding to
$\Delta^{3}\times 0$ and $\Delta^{3}\times 1$ do not intersect. Therefore,
this subcomplex is not equivalent to $\mathcal{C}(n,r)$ for all $r$. Any face
$\Delta^{n-2}\times\boldsymbol{x}$ intersects any other face
$\Delta^{n-2}\times\boldsymbol{x}$. Two faces $\Delta^{n-2}_{1}\times 0$ and
$\Delta^{n-2}_{2}\times 1$ do not intersect. Up to symmetries there can be two
types of such faces. In the first case $\Delta^{n-2}_{1}=\Delta^{n-2}_{2}$. In
the second case $\Delta^{n-2}_{1}\cap\Delta^{n-2}_{2}$ is an $(n-3)$-simplex.
Both these pairs of faces define Hamiltonian $\mathcal{C}(n,n)$-subcomplexes.
From the above argument there can not be more than two defining faces. The
structure of the branch set follows from Lemma 4.60. ∎

######  Construction 4.62 (Cutting off a face).

Let $P=\\{\boldsymbol{a}_{i}\boldsymbol{x}+b_{i}\geqslant
0,i=1,\dots,m\\}\subset\mathbb{R}^{n}$ be a simple $n$-polytope and
$G=F_{i_{1}}\cap\dots\cap F_{i_{k}}$ be its $(n-k)$-face, $k>1$. Set
$\boldsymbol{a}_{G}=\sum\limits_{s=1}^{k}\boldsymbol{a}_{i_{s}}$ and
$b_{G}=\sum\limits_{s=1}^{k}b_{i_{s}}$. Take the halfspace
$\mathcal{H}_{\geqslant\varepsilon}=\\{\boldsymbol{a}_{G}\boldsymbol{x}+b_{G}\geqslant\varepsilon\\}$.
For small $\varepsilon\geqslant 0$ the intersection
$P_{G,\varepsilon}=P\cap\mathcal{H}_{\geqslant\varepsilon}$ is a simple
$n$-polytope. It has an additional facet $F$ corresponding to the intersection
of $P$ with the hyperplane
$\mathcal{H}_{\varepsilon}=\partial\mathcal{H}_{\geqslant\varepsilon}$. We
have $F\cap F_{j_{1}}\cap\dots\cap F_{j_{l}}\neq\varnothing$ if and only if
$F_{j_{1}}\cap\dots\cap F_{j_{l}}$ is a face of $P$ intersecting $G$ and not
lying in $G$. That is, $F_{j_{1}}\cap\dots\cap F_{j_{l}}\cap
F_{i_{1}}\cap\dots\cap F_{i_{k}}\neq\varnothing$ and
$\\{i_{1},\dots,i_{k}\\}\not\subset\\{j_{1},\dots,j_{l}\\}$.

######  Lemma 4.63.

The facet $F$ of $P_{G,\varepsilon}$ is combinatorially equivalent to
$G\times\Delta^{k-1}$.

###### Proof.

Indeed, facets of $F$ are in bijection with facets of $P$ intersecting $G$.
These are $F_{i_{1}}$, $\dots$, $F_{i_{k}}$, and facets $F_{j}$ such that
$F_{j}\cap G$ is a facet of $G$. The facets $F\cap F_{i_{a_{1}}}$, $\dots$,
$F\cap F_{i_{a_{l}}}$, $F\cap F_{j_{1}}$, $\dots$, $F\cap F_{j_{s}}$ intersect
if and only if $\\{i_{a_{1}},\dots,i_{a_{l}}\\}\neq\\{i_{1},\dots,i_{k}\\}$,
and $G\cap F_{j_{1}}\cap\dots\cap F_{j_{s}}=(G\cap
F_{j_{1}})\cap\dots\cap(G\cap F_{j_{s}})\neq\varnothing$. Thus, $F\simeq
G\times\Delta^{k-1}$. ∎

######  Lemma 4.64.

The other part $P\cap\mathcal{H}_{\leqslant\varepsilon}$ of the polytope $P$
is combinatorially equivalent to $G\times\Delta^{k}$.

###### Proof.

Indeed, by the same argument as above the facets of the polytope
$P\cap\mathcal{H}_{\leqslant\varepsilon}$ are in bijection with the facet $F$,
facets $F_{i_{1}}$, $\dots$, $F_{i_{k}}$, and facets $F_{j}$ such that
$F_{j}\cap G$ is a facet of $G$. Moreover, its collection of facets
$\\{F,F_{i_{a_{1}}},\dots,F_{i_{a_{l}}},F_{j_{1}},\dots,F_{j_{s}}\\}$
intersects if any only if
$\\{i_{a_{1}},\dots,i_{a_{l}}\\}\neq\\{i_{1},\dots,i_{k}\\}$ and $G\cap
F_{j_{1}}\cap\dots\cap F_{j_{s}}=(G\cap F_{j_{1}})\cap\dots\cap(G\cap
F_{j_{s}})\neq\varnothing$, while for
$\\{F_{i_{a_{1}}},\dots,F_{i_{a_{l}}},F_{j_{1}},\dots,F_{j_{s}}\\}$ the
criterion of intersection is just $G\cap F_{j_{1}}\cap\dots\cap
F_{j_{s}}\neq\varnothing$. ∎

######  Lemma 4.65.

Any $(n-2)$-face $F\cap F_{i_{q}}\simeq G\times\Delta^{k-2}$ of
$P_{G,\varepsilon}$ defines a Hamiltonian subcomplex $C\simeq\partial P$.

###### Proof.

Let us rotate the hyperplane $\mathcal{H}_{\varepsilon}$ around the
$(n-2)$-plane containing the face $F\cap F_{i_{q}}$. The rotating plane
$\mathcal{H}(\varphi)$ is defined by the formula

$\cos\varphi(\boldsymbol{a}_{G}\boldsymbol{x}+b_{G}-\varepsilon)+\sin\varphi(\boldsymbol{a}_{i_{q}}\boldsymbol{x}+b_{i_{q}})=0.$

We have $\mathcal{H}(0)=\mathcal{H}_{\varepsilon}={\rm aff}(F)$ and
$\mathcal{H}(\frac{\pi}{2})={\rm aff}(F_{i_{q}})$. Moreover, for small
$\delta>0$ at each moment $-\delta\leqslant\varphi<\frac{\pi}{2}$ the polytope
$P\cap\mathcal{H}_{\geqslant 0}(\varphi)$ has the same combinatorial type
$P_{\varepsilon}$ and for any
$-\delta\leqslant\varphi_{1}<\varphi_{2}\leqslant\frac{\pi}{2}$ the polytope
$P\cap\mathcal{H}_{\leqslant
0}(\varphi_{1})\cap\mathcal{H}_{\geqslant}(\varphi_{2})$ has the same
combinatorial type $G\times\Delta^{k}\simeq
P\cap\mathcal{H}_{\leqslant\varepsilon}$. In particular, the polytope
$P\cap\mathcal{H}_{\leqslant 0}(-\delta)\cap\mathcal{H}_{\geqslant 0}(0)$ is
combinatorially equivalent to $P\cap\mathcal{H}_{\leqslant
0}(-\delta)\cap\mathcal{H}_{\geqslant}(\frac{\pi}{2})$. Then the piecewise
linear homeomorphism defined using the barycentric subdivisions preserves the
face structures and is identical on $P\cap\mathcal{H}(-\delta)$. This
homeomorphism together with the identical mapping on
$P\cap\mathcal{H}_{\geqslant 0}(-\delta)$ defines the desired equivalence
$C\simeq\partial P$. ∎

######  Corollary 4.66.

For any set $G_{1}$, $\dots$, $G_{k}$ of pairwise disjoint faces of the
simplex $\Delta^{n}$ the polytope obtained from $\Delta^{n}$ by cutting off
all these faces by different hyperplanes has a Hamiltonian
$\mathcal{C}(n,n+1)$-subcomplex given by a set of $k$ defining $(n-2)$-faces
lying in different cutting hyperplanes.

######  Example 4.67.

Any hyperplane separating a face $\Delta^{k}$ of the simplex $\Delta^{n}$ from
the other vertices leaves a face $\Delta^{n-k-1}$ on the other side. Then
Lemma 4.64 implies that if we cut off $\Delta^{k}$ we obtain combinatorially
the polytope $\Delta^{n-k-1}\times\Delta^{k+1}$. Moreover, the defining face
$F\cap F_{i_{q}}$ from Lemma 4.65 has the form
$\Delta^{n-k-2}\times\Delta^{k}$.

######  Corollary 4.68.

For any $p,q\geqslant 1$ the face $\Delta^{p-1}\times\Delta^{q-1}$ defines a
Hamiltonian $\mathcal{C}(n,n+1)$-subcomplex in
$P^{n}=\Delta^{p}\times\Delta^{q}$. In particular, for $p,q\geqslant 2$ the
geometry $\mathbb{S}^{p}\times\mathbb{S}^{q}$ arises in Construction 4.45.
(For $p=1$ or $q=1$ we obtain the polytope $\Delta^{n-1}\times I$ and the
geometry $\mathbb{S}^{n-1}\times\mathbb{R}$ already covered by Proposition
4.61.) For $p,q\geqslant 2$ the branch set of the coverings $S^{p}\times
S^{q}\to S^{n}$ is homeomorphic to $S^{p-1}\times S^{q-1}$.

######  Remark 4.69.

The structure of the branch set follows from Lemma 4.60. Moreover, this
implies that in Corollary 4.66 the branch set is a disjoint union of the
products $S^{n-k-2}\times S^{k}$ corresponding to faces $\Delta^{k}$,
$0\leqslant k\leqslant n-2$.

######  Proposition 4.70.

If an $n$-polytope $P$ has a Hamiltonian $\mathcal{C}(n,n+1)$-subcomplex such
that its defining faces $\\{M_{1},\dots,M_{s}\\}$ do not intersect a facet
$F_{i}\simeq\Delta^{n-1}$, then for any $k\geqslant 1$ the polytope
$\Delta^{k}\times P$ has a Hamiltonian $\mathcal{C}(n+k,n+k+1)$-subcomplex
with defining faces $\Delta^{k-1}\times F_{i}$ and $\Delta^{k}\times M_{1}$,
$\dots$, $\Delta^{k}\times M_{q}$.

###### Proof.

Indeed, the faces $\Delta^{k}\times M_{1}$, $\dots$, $\Delta^{k}\times M_{q}$
define the subcomplex equivalent to $\partial(\Delta^{k}\times\Delta^{n})$,
and the face $\Delta^{k-1}\times F_{i}\simeq\Delta^{k-1}\times\Delta^{n-1}$
transforms it to $\partial\Delta^{n+k}$. ∎

######  Corollary 4.71.

For any $p,q\geqslant 1$ the polytope $P^{n}=\Delta^{p}\times\Delta^{q}\times
I$ has a Hamiltonian $C(n,n+1)$-subcomplex defined by two faces
$\Delta^{p-1}\times\Delta^{q}\times\\{0\\}$ and
$\Delta^{p}\times\Delta^{q-1}\times\\{1\\}$. In particular, the geometries
$\mathbb{S}^{p}\times\mathbb{S}^{q}\times\mathbb{R}$, $p,q\geqslant 2$,
$\mathbb{S}^{p}\times\mathbb{R}^{2}$, $p\geqslant 2$, and $\mathbb{R}^{3}$
arise in Construction 4.45. The branch set of the covering
$N(P,\widetilde{\Lambda}_{C})\to S^{n}$ is $S^{p-1}\times S^{q}\sqcup
S^{p}\times S^{q-1}$.

###### Proof.

Indeed, $\Delta^{q-1}\times\\{1\\}\cap\Delta^{q}\times\\{0\\}=\varnothing$.
Hence, we can apply Proposition 4.70 to $P=\Delta^{q}\times I$. The structure
of the branch set follows from Lemma 4.60. ∎

######  Proposition 4.72.

If an $n$-polytope $P$ has a Hamiltonian $\mathcal{C}(n,n+1)$-subcomplex $C$
such that its defining faces $\\{M_{1},\dots,M_{s}\\}$ do not intersect an
$(n-2)$-face $G\subset P$, then $G\simeq\Delta^{n-2}$, and the faces
$\\{M_{1},\dots,M_{s},G\\}$ define a Hamiltonian $\mathcal{C}(n,n)$-subcomplex
in $\partial P$.

###### Proof.

By Corollary 4.30 $G$ is an $(n-2)$-face of $C$, hence $G\simeq\Delta^{n-2}$
and it defines in $C$ the Hamiltonian $\mathcal{C}(n,n)$-subcomplex. Then the
faces $\\{M_{1},\dots,M_{s},G\\}$ define the corresponding Hamiltonian
$\mathcal{C}(n,n)$-subcomplex in $\partial P$. ∎

######  Corollary 4.73.

Let $\Delta^{2}={\rm conv}\\{v_{1},v_{2},v_{3}\\}$. Then the faces ${\rm
conv}\\{v_{1},v_{2}\\}\times\Delta^{n-3}$ and $v_{3}\times\Delta^{n-2}$ define
a Hamiltonian $\mathcal{C}(n,n)$-subcomplex $C$ in
$\Delta^{2}\times\Delta^{n-2}$ for $n\geqslant 3$. The branch set of the
covering $N(P,\widetilde{\Lambda}_{C})\to S^{n}$ is a disjoint union of
$S^{n-2}$ and $S^{1}\times S^{n-3}$.

###### Proof.

The structure of the branch set over $v_{3}\times\Delta^{n-2}$ follows from
Lemma 4.60. For the face $F={\rm conv}\\{v_{1},v_{2}\\}\times\Delta^{n-3}$ its
facets corresponding to $\Delta^{n-3}$ lie in different facets of $C$, while
the facets $v_{1}\times\Delta^{n-3}$ and $v_{2}\times\Delta^{n-3}$ lie in the
same facet of $C$ different from the above facets. Hence, by Proposition 4.31
the branch set over $F$ is homeomorphic to $S^{1}\times S^{n-3}$. ∎

######  Corollary 4.74.

Let $P_{k}$ be a $k$-gon, $k\geqslant 3$. Then for $p\geqslant 1$ the polytope
$P^{n}=\Delta^{p}\times P_{k}$ admits Hamiltonian $\mathcal{C}(n,n)$\- and
$\mathcal{C}(n,n+1)$-subcomplexes $C_{n}$ and $C_{n+1}$. In particular, taking
a right-angled triangle in $\mathbb{S}^{2}$, a square in $\mathbb{R}^{2}$ and
a right-angled $k$-gon, $k\geqslant 5$, in $\mathbb{L}^{2}$ we see that the
geometries $\mathbb{S}^{p}\times\mathbb{S}^{2}$,
$\mathbb{S}^{p}\times\mathbb{R}^{2}$ and $\mathbb{S}^{p}\times\mathbb{L}^{2}$,
$p\geqslant 2$, arise in Construction 4.45. The branch set of the covering
$N(P,\widetilde{\Lambda}_{C_{n+1}})\to S^{n}$ is a disjoint union of $2(k-3)$
copies of $S^{n-2}$ and one $S^{n-3}\times S^{1}$, and for the covering
$N(P,\widetilde{\Lambda}_{C_{n}})\to S^{n}$ it is a disjoint union of
$(k-2)$-copies of $S^{n-2}$ and one $S^{n-3}\times S^{1}$.

###### Proof.

Indeed, let $P_{k}={\rm conv}\\{v_{1},\dots,v_{k}\\}$. Then the vertices
$v_{1}$, $\dots$, $v_{k-3}$ define a Hamiltonian $\mathcal{C}(2,3)$-subcomplex
in $P_{k}$, and the edge ${\rm conv}\\{v_{k-2},v_{k-1}\\}$ does not intersect
these vertices. By Proposition 4.70 the faces $\Delta^{p}\times v_{1}$,
$\dots$, $\Delta^{p}\times v_{k-3}$, $\Delta^{p-1}\times{\rm
conv}\\{v_{k-2},v_{k-1}\\}$ define a Hamiltonian
$\mathcal{C}(n,n+1)$-subcomplex $C_{n+1}$. By Proposition 4.72 the addition of
the face $\Delta^{p}\times v_{k}$ defines a Hamiltonian
$\mathcal{C}(n,n)$-subcomplex $C_{n}$. For each defining face of $C_{n+1}$ its
facets correspond to different facets of $C_{n+1}$, hence the structure of the
branch set follows from Lemma 4.60. For defining faces $\Delta^{p}\times
v_{j}$ of $C_{n}$ the same argument works. For the defining face
$\Delta^{p-1}\times{\rm conv}\\{v_{k-2},v_{k-1}\\}$ of $C_{n}$ its facets
corresponding to $\Delta^{p-1}$ lie in different facets of $C_{n}$, while the
facets $\Delta^{p-1}\times v_{k-2}$ and $\Delta^{p-1}\times v_{k-1}$ lie in
the same facet of $C_{n}$ different from the above facets. Hence, by
Proposition 4.31 the branch set over this facet is homeomorphic to
$S^{p-1}\times S^{1}$. ∎

######  Example 4.75.

On Fig. 1 we show a set of five disjoint quadrangles defining a Hamiltonian
$\mathcal{C}(4,5)$-subcomplex in $P_{5}\times P_{5}$. In particular, the
geometry $\mathbb{L}^{2}\times\mathbb{L}^{2}$ arises in Construction 4.45. It
follows from Lemma 4.60 that the branch set of the covering
$N(P,\widetilde{\Lambda}_{C})\to S^{4}$ is a disjoint union of five tori
$\mathbb{T}^{2}=S^{1}\times S^{1}$.

Figure 1. A set of disjoint quadrangles defining a Hamiltonian
$\mathcal{C}(4,5)$-subcomplex in $P_{5}\times P_{5}$

In dimension $n=4$ all the products of Euclidean, spherical and hyperbolic
geometries (including these geometries) are $10$ geometries: $\mathbb{S}^{4}$,
$\mathbb{R}^{4}$, $\mathbb{L}^{4}$, $\mathbb{S}^{3}\times\mathbb{R}$,
$\mathbb{S}^{2}\times\mathbb{S}^{2}$, $\mathbb{S}^{2}\times\mathbb{R}^{2}$,