# Some hyperbolic three-manifolds that bound geometrically

Alexander Kolpakov, Bruno Martelli, Steven Tschantz

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Alexander Kolpakov, Bruno Martelli, Steven Tschantz. Some hyperbolic three-manifolds that bound geometrically. Proceedings of the American Mathematical Society, 2015, 143, pp.4103-4111. 10.1090/proc/12520 . hal-01287540v2

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<figure><img src="image_1.png" /></figure>
# SOME HYPERBOLIC THREE-MANIFOLDS THAT BOUND GEOMETRICALLY

ALEXANDER KOLPAKOV, BRUNO MARTELLI, AND STEVEN TSCHANTZ

ABSTRACT. A closed connected hyperbolic $n$-manifold *bounds geometri-cally* if it is isometric to the geodesic boundary of a compact hyperbolic ($n+1$)-manifold. A. Reid and D. Long have shown by arithmetic meth-ods the existence of infinitely many manifolds that bound geometrically in every dimension.

We construct here infinitely many explicit examples in dimension $n = 3$ using right-angled dodecahedra and 120-cells and a simple colouring technique introduced by M. Davis and T. Januszkiewicz. Namely, for every $k \ge 1$, we build an orientable compact closed 3-manifold tessellated by 16$k$ right-angled dodecahedra that bounds a 4-manifold tessellated by 32$k$ right-angled 120-cells.

A notable feature of this family is that the ratio between the volumes of the 4-manifolds and their boundary components is constant and, in particular, bounded.

## 1. INTRODUCTION

The study of hyperbolic manifolds that bound geometrically dates back to the works of D. Long, A. Reid [11, 12] and B. Niemershiem [17], motivated by a preceding work of M. Gromov [7, 8] and a question by F. Farrell and S. Zdravkovska [4]. This question is also related to hyperbolic instantons, as described by J. Ratcliffe and S. Tschantz [19, 20]. In particular, the following problems are of particular interest:

**Question 1.1.** Which compact orientable hyperbolic $n$-manifold $\mathcal{N}$ can represent the totally geodesic boundary of a compact orientable hyperbolic ($n+1$)-manifold $\mathcal{M}$?

**Question 1.2.** Which compact orientable flat $n$-manifold $\mathcal{N}$ can represent the cusp section of a single-cusped orientable hyperbolic ($n+1$)-manifold $\mathcal{M}$?

Once there exist such manifolds $\mathcal{N}$ and $\mathcal{M}$, we say that $\mathcal{N}$ *bounds* $\mathcal{M}$ *geometrically*. In this note, we shall concentrate on Question 1.1, devoted to compact geometric boundaries. The recent progress on Question 1.2, that involves cusp sections, is indicated by [10, 12, 14, 15]. However, this is still

<footnote>The first named author is supported by the SNSF researcher scholarship P300P2-151316. The second named author was supported by the Italian FIRB project "Geometry and topology of low-dimensional manifolds", RBFR10GHHH.</footnote>
an open problem in dimensions $\ge 5$. On the other hand, by a result of M. Stover [21], an arithmetic orbifold in dimension $\ge 30$ cannot have a single cusp.

In [11], D. Long and A. Reid have shown that many closed hyperbolic 3-manifolds do not bound geometrically: a necessary condition is that the eta invariant of the 3-manifold must be an integer. The first known closed hyperbolic 3-manifold that bounds geometrically was constructed by J. Ratcliffe and S. Tschantz in [19] and has volume of order 200.

Then, D. Long and A. Reid produced in [13], by arithmetic techniques, infinitely many orientable hyperbolic $n$-manifolds $\mathcal{N}$ that bound geometrically an $(n+1)$-manifold $\mathcal{M}$, in every dimension $n \ge 2$. Every such manifold $\mathcal{N}$ is obtained as a cover of some $n$-orbifold $O_{\mathcal{N}}$ geodesically immersed in a suitable $(n+1)$-orbifold $O_{\mathcal{M}}$.

In this paper, we construct an explicit infinite family in dimension $n = 3$, via a similar covering technique where the roles of $O_{\mathcal{N}}$ and $O_{\mathcal{M}}$ are played by the right-angled dodecahedron $\mathcal{D}$ and 120-cell $\mathcal{L}$. These two compact Coxeter right-angled regular polytopes exist in $\mathbb{H}^3$ and $\mathbb{H}^4$ respectively, and the first is a facet of the second. The existence of suitable finite covers is guaranteed here by assigning appropriate colourings to their facets, following A. Vesnin [23, 24], M. Davis and T. Januszkiewicz [3], I. Izmestiev [9].

A colouring determines a manifold covering, and the main factual observation is that a colouring of the dodecahedron $\mathcal{D}$ can be enhanced in a suitable way to a colouring of the right-angled hyperbolic Coxeter 120-cell $\mathcal{L}$. We produce in this way a degree-32 orientable cover of $\mathcal{L}$ that contains four copies of a non-orientable degree-8 cover of $\mathcal{D}$. By cutting along one such non-orientable geodesic submanifold we get a hyperbolic four-manifold $\mathcal{N}_1$ with connected geodesic boundary $\mathcal{M}_1 = \partial\mathcal{N}_1$.

The colouring technique applied to a single polytope can produce only finitely many manifolds. To get infinitely many examples we assemble $n$ copies of $\mathcal{D}$ and $\mathcal{L}$ to get more complicated right-angled polytopes, to which the above construction easily extends. We finally obtain the following. Let $V_{\mathcal{D}} \approx 4.3062...$ and $V_{\mathcal{L}} = \frac{34}{3}\pi^2$ be the volumes of $\mathcal{D}$ and $\mathcal{L}$, respectively.

**Theorem 1.3.** For every $n \ge 1$ there exists an orientable compact hyperbolic three-manifold $\mathcal{N}_n$ of volume $16nV_{\mathcal{D}}$ which bounds geometrically an orientable compact hyperbolic four-manifold $\mathcal{M}_n$ of volume $32nV_{\mathcal{L}}$.

The manifolds $\mathcal{N}_n$ and $\mathcal{M}_n$ are tessellated respectively by 16n right-angled dodecahedra and 32n right-angled 120-cells.

An interesting feature of this construction is that it provides manifolds $\mathcal{N}_n$ and $\mathcal{M}_n$ of controlled volume. In particular, we deduce the following.

**Corollary 1.4.** There are infinitely many hyperbolic three-manifolds $\mathcal{N}$ that bound geometrically some hyperbolic $\mathcal{M}$ with constant ratio:

$$ \frac{\mathrm{Vol}(\mathcal{M})}{\mathrm{Vol}(\mathcal{N})} = \frac{2V_{\mathcal{L}}}{V_{\mathcal{D}}} < 53. $$
The manifold $M_1$ has volume $16V_G \approx 68.8992$ and is to our knowledge the smallest closed hyperbolic 3-manifold known to bound geometrically.

**Structure of the paper.** In Section 2 we introduce right-angled polytopes as orbifolds, and a simple colouring technique from [3] that produces manifold coverings of small degree. Then we show how this colouring technique passes easily from dimension $n$ to $n+1$ and conversely, and may be used to produce $n$-manifolds that bound geometrically when $n=3$. In Section 3 we assemble dodecahedra and 120-cells to produce the manifolds $N_k$ and $M_k$ of Theorem 1.3.

## 2. COLORINGS AND COVERS OF COXETER ORBIFOLDS

A right-angled hyperbolic polytope $\mathcal{P} \subset \mathbb{H}^n$ may be interpreted as an orbifold with mirror boundary, where the mirrors correspond to its facets. As such an orbifold, it has a plenty of manifold coverings. A few of them may be constructed by colouring appropriately the facets of $\mathcal{P}$ as shown in [3, 6].

**2.1. Colourings and manifold covers.** Let $\mathcal{P} \subset \mathbb{H}^n$ be a convex compact right-angled polytope. Such objects exist only if $2 \le n \le 4$, see [18]; the two important basic examples we consider here are the right-angled dodecahedron $\mathcal{D} \subset \mathbb{H}^3$ and the right-angled 120-cell $\mathcal{X} \subset \mathbb{H}^4$.

We consider $\mathcal{P}$ as an orbifold $\mathbb{H}^n/\Gamma$. The group $\Gamma$ is a right-angled Coxeter group that may be presented as

$$ \Gamma = \langle r_F | r_F^2, [r_F, r_{F'}] \rangle $$

where $F$ varies over all the facets of $\mathcal{P}$ and the pair $F, F'$ varies over all the pairs of adjacent facets. The isometry $r_F \in \text{Isom}(\mathbb{H}^n)$ is a reflection in the hyperplane containing $F$.

A right-angled polyhedron $\mathcal{P}$ is simple [25, Theorem 1.8], which means that it looks combinatorially at every vertex $v$ like the origin of an orthant in $\mathbb{R}^n$. In particular, $v$ is the intersection of exactly $n$ facets.

Let $V$ be a finite-dimensional vector space over $\mathbb{F}_2$, thus isomorphic to $\mathbb{F}_2^s$ for some $s$. A $V$-colouring (or simply, a colouring) $\lambda$ is the assignment of a vector $\lambda_F \in V$ to each facet $F$ of $\mathcal{P}$ (called its colour) such that the following holds: at every vertex $v$, the $n$ colours assigned to the $n$ adjacent facets around $v$ are linearly independent vectors in $V$.

A colouring induces a group homomorphism $\lambda: \Gamma \to V$, defined by sending $r_F$ to $\lambda_F$ for every facet $F$. Its kernel $\Gamma_\lambda = \ker \lambda$ is a subgroup of $\Gamma$ which determines an orbifold $M_\lambda = \mathbb{H}^n/\Gamma_\lambda$ covering $\mathcal{P}$.

**Proposition 2.1.** The orbifold $M_\lambda$ is a manifold.

*Proof.* We follow [23, Lemma 1]. A torsion element in $\Gamma$ fixes some face $F$ of the tessellation of $\mathbb{H}^n$ obtained by reflecting $\mathcal{P}$ in its own facets. Up to conjugacy, we can suppose that $F \subset \mathcal{P}$. The stabilizer of $F$ is generated by the reflections in the facets containing $F$ (see [2, Theorem 12.3.4]) and
is hence mapped injectively into $V$ by $\lambda$. Thus, $\Gamma_{\lambda}$ is torsion-free and $M_{\lambda}$ is a manifold. $\square$

At each vertex $v$ the colours $\lambda_F$ of the $n$ incident facets $F$ are independent: therefore the image of $\lambda$ has dimension at least $n$ and the covering $M_{\lambda} \to \mathcal{P}$ has degree $|\Gamma : \Gamma_{\lambda}| \ge 2^n$; if the equality holds the manifold $M_{\lambda}$ is called a *small cover* of $\mathcal{P}$. The manifold coverings of $\mathcal{P}$ of smallest degree are precisely its small covers, see [6, Proposition 2.1].

We say that the colouring spans $V$ if the vectors $\lambda_F$ span $V$ as $F$ varies, which is equivalent to the map $\lambda: \Gamma \to V$ being surjective. In that case the covering $M_{\lambda} \to \mathcal{P}$ has degree $|\Gamma : \Gamma_{\lambda}| = 2^{\dim V}$.

**2.2. k-colourings.** Here, we give an example: recall that a *k-colouring* of a polytope is the assignment of a colour from the set $\{1, \dots, k\}$ to each facet so that two adjacent facets have distinct colours. A *k-colouring* for $\mathcal{P}$ produces an $\mathbb{F}_2^k$-colouring that spans $\mathbb{F}_2^k$: simply replace each colour $i \in \{1, \dots, k\}$ with the element $e_i$ of the canonical basis for $\mathbb{F}_2^k$.

**Example 2.2.** The dodecahedron has precisely one four-colouring, up to symmetries. This induces an $\mathbb{F}_2^3$-colouring (of lower dimension 3 rather than 4) on the hyperbolic right-angled dodecahedron $\mathcal{D}$ described in [23] and hence a manifold covering having degree $2^3 = 8$.

**Example 2.3.** The 120-cell has a five-colouring (in fact, ten five-colourings up to symmetries [5]). Each produces a manifold covering of the hyperbolic right-angled 120-cell $\mathcal{X}$ of degree $2^5 = 32$.

**2.3. Orientable coverings.** The following lemma gives an orientability criterion analogous to that for small covers in [16] or Löbell manifolds in [23, Lemma 2]. Let $\lambda$ be a $V$-colouring of a right-angled polytope $\mathcal{P}$.

**Lemma 2.4.** *Suppose $\lambda$ spans $V$. The manifold $M_\lambda$ is orientable if and only if, for some isomorphism $V \cong \mathbb{F}_2^s$, each colour $\lambda_F$ has an odd number of 1's.*

*Proof.* Let $\Gamma^+ \triangleleft \Gamma$ be the index two subgroup consisting of orientation-preserving isometries. Then $\Gamma^+$ is the kernel of the homomorphism $\phi: \Gamma \to \mathbb{F}_2$ that sends $r_F$ to 1 for every facet $F$.

The manifold $M_\lambda$ is orientable if and only if $\Gamma_\lambda$ is contained in $\Gamma^+$, and this in turn holds if and only if there is a homomorphism $\chi: V \to \mathbb{F}_2$ such that $\phi = \chi \circ \lambda$. The latter is equivalent to the existence of an isomorphism $V \cong \mathbb{F}_2$ that transforms $\lambda_F$ into a vector with an odd number of 1's for each $F$. Indeed, if such an isomorphism exists, then $\chi$ can be taken to be the sum of the coordinates of a vector.

Conversely, suppose such an isomorphism does not exist. Since the vectors $\lambda_F$ span $V$ we may take some of them as a basis for $V$ and write them as $e_1 = (1, 0, 0, \dots, 0)$, $e_2 = (0, 1, 0, \dots, 0)$, ..., $e_s = (0, \dots, 0, 0, 1)$. By hypothesis, there exists a facet $F$ such that $\lambda_F$ has an even number of 1's.
Up to reordering, we may write $\lambda_F = \sum_{i=1}^{2k} e_i$ for some $k$. Now we can see that the homomorphism $\chi$ does not exist, since its existence would imply $1 = \phi(r_F) = \sum_{i=1}^{2k} \phi(e_i) = \sum_{i=1}^{2k} 1 = 0$. $\square$

**Corollary 2.5.** Let there be facets $F, F'$ and $F''$ of $\mathcal{P}$ such that $\lambda_F + \lambda_{F'} + \lambda_{F''} = 0$. Then $\lambda$ is a non-orientable colouring.

*Proof.* For a vector $v = (v_1, v_2, \dots, v_s) \in \mathbb{F}_2^s$ let $\epsilon(v) := \sum_{i=1}^s v_i$. Suppose that $\lambda$ is orientable, so there exists an isomorphism $V \cong \mathbb{F}_2^s$, such that each $\lambda_F$ has an odd number of 1's. Then we arrive at a contradiction, since $0 = \epsilon(0) = \epsilon(\lambda_F + \lambda_{F'} + \lambda_{F''}) = \epsilon(\lambda_F) + \epsilon(\lambda_{F'}) + \epsilon(\lambda_{F''}) = 1$. $\square$

**Example 2.6.** The manifolds in Examples 2.2 and 2.3 are orientable.

**Example 2.7.** Consider the 25 small covers of the hyperbolic right-angled dodecahedron $\mathcal{D}$ found by A. Garrison and R. Scott in [6]. The list is complete, up to isometries between the corresponding manifolds. Using the orientability criterion one sees immediately from [6, Table 1] that 24 of them are non-orientable and exactly 1 is orientable and corresponds to Example 2.2. Another example carried out in [6] is a small cover of the hyperbolic right-angled 120-cell $\mathcal{L}$. This cover is again non-orientable. There is no classification of small covers of $\mathcal{L}$ known at present.

**2.4. Induced colouring.** A facet $F$ of a $n$-dimensional right-angled polytope $\mathcal{P} \subset \mathbb{H}^n$ is a $(n-1)$-dimensional right-angled polytope. A $V$-colouring $\lambda$ of $\mathcal{P}$ induces a $W$-colouring $\mu$ of $F$ with $W = V/\langle\lambda_F\rangle$: simply assign to every face of $F$ the colour of the facet of $\mathcal{P}$ that is incident to it. The following lemma generalises [6, Proposition 2.3].

**Lemma 2.8.** The manifold $M_\mu$ is contained in $M_\lambda$ as a totally geodesic sub-manifold, so that the cover $M_\lambda \to \mathcal{P}$ restricts to the cover $M_\mu \to F$.

*Proof.* Let $\Gamma$ be the Coxeter group of $\mathcal{P}$ and $\mathbb{H}^{n-1} \subset \mathbb{H}^n$ be the hyperplane containing $F$. We regard $F$ as the orbifold $\mathbb{H}^{n-1}/\Gamma_F$ where $\Gamma_F$ is the Coxeter group of $F$. The following natural diagram commutes:

<figure><img src="image_2.png" /></figure>

The first line is an exact sequence. We deduce easily that $f$ restricts to an isomorphism

$f: \ker \lambda \cap \text{Stab}(\mathbb{H}^{n-1}) \to \ker \mu$.

Hence $M_\mu = \mathbb{H}^{n-1}/\Gamma_\lambda \cap \text{Stab}(\mathbb{H}^{n-1})$ is naturally contained in $M_\lambda = \mathbb{H}^n/\Gamma_\lambda$. $\square$

The pre-image of $F$ in $M_\lambda$ with respect to the regular covering $M_\lambda \to \mathcal{P}$ consists of possibly several copies of $M_\mu$.
2.5. **Extended colouring.** Conversely, we can also extend a colouring from a facet to the whole polytope. We say that two colourings $\lambda$ and $\lambda'$ on $\mathcal{P}$ are equivalent if they have isomorphic kernels $\Gamma_\lambda \cong \Gamma_{\lambda'}$ (cf. the definition before [6, Proposition 2.4]).

**Proposition 2.9.** Let $F$ be a facet of a compact right-angled polytope $\mathcal{P} \subset \mathbb{H}^n$. Every colouring of $F$ is equivalent to one induced by an orientable colouring of $\mathcal{P}$.

*Proof.* Let $\lambda$ be a $V$-colouring of the facet $F$. Fix an isomorphism $V \cong \mathbb{F}_2^s$. Define $W = \mathbb{F}_2 \oplus \mathbb{F}_2^s \oplus \mathbb{F}_2^f$ where $f$ is the number of facets of $\mathcal{P}$ that are not adjacent to $F$. We define a $W$-colouring $\mu$ of $\mathcal{P}$ as follows:

* set $\mu_F = (1, 0, 0)$;

* set $\mu_G = (\epsilon(\lambda_{G \cap F}) + 1, \lambda_{G \cap F}, 0)$ where $\epsilon(v) = \sum_{i=1}^s v_i$, for every facet $G$ adjacent to $F$;

* set $\mu_{G_i} = (0, 0, e_i)$ for the remaining facets $G_1, \dots, G_f$.

Indeed, the map $\mu$ is a colouring: the linear independence condition is satisfied at each vertex. Moreover, each vector $\mu_F, \mu_G, \mu_{G_i}$ has an odd number of 1's. Finally, by construction $\mu|_F$ is equivalent to $\lambda$. $\square$

We call the colouring $\mu$ an *extension* of $\lambda$.

2.6. **A more efficient extension.** Proposition 2.9 shows how to extend a $V$-colouring of a facet $F$ to an orientable $W$-colouring of the polytope $\mathcal{P}$. The proof shows that the dimension of $W$ can grow considerably during the process, since $\dim W = \dim V + 1 + f$ where $f$ is the number of facets of $\mathcal{P}$ not adjacent to $F$.

We may use Proposition 2.9 with $\mathcal{P}$ being the right-angled 12-cell and $F$ its dodecahedral facet. However, in this case we could find examples where both $V$ and $W$ have smaller dimension via computer. The following was proved by using “Mathematica”.

**Proposition 2.10.** Each of the 24 non-orientable $\mathbb{F}_2^3$-colourings of $\mathcal{D}$ from [6, Table 1] is equivalent to one induced by an orientable $\mathbb{F}_2^5$-colouring of $\mathcal{L}$.

*Proof.* The Mathematica program code given in [22] takes a non-orientable $\mathbb{F}_2^3$-colouring of $\mathcal{D}$ and produces an orientable $\mathbb{F}_2^5$-colouring of $\mathcal{L}$, as required.

Each vector $v = (v_1, v_2, \dots, v_s) \in \mathbb{F}_2^s$ is encoded by the binary number $n_v = v_1 \cdot 2^0 + v_2 \cdot 2^1 + \dots + v_s \cdot 2^{s-1}$. Let $P_0 := \mathcal{D}$, be the right-angled dodecahedron. We enumerate its faces exactly as shown in [6, Figure 3] and the corresponding 12-tuple of numbers encodes its colouring. Let $P_i$, $i = 1, \dots, 12$ be the dodecahedral facets incident to $P_0$ at the respective faces $F_i$, $i = 1, \dots, 12$. We start extending the colouring of $P_0 := \mathcal{D}$ as follows:

* set $\lambda_{P_0} = (0, 0, 0, 0, 1)$;

* if $\mu_{F_i} = v = (v_1, v_2, v_3)$, then $\lambda_{P_i} := (v_1, v_2, v_3, 0, \epsilon(v) + 1)$.
We obtain a 13-tuple, which is the initial segment of the colouring of $\mathcal{L}$. Then the Mathematica code [22] attempts to produce an orientable 120-tuple, which encodes the entire colouring. $\square$

We were not able to find any orientable $\mathbb{F}_2^4$-colouring of $\mathcal{L}$ extending a non-orientable $\mathbb{F}_2^3$-colouring of $\mathcal{D}$: our examples are not small covers.

Let now $M_\lambda$ be the manifold obtained by one $\mathbb{F}_2^5$-colouring $\lambda$ of $\mathcal{L}$: it covers $\mathcal{L}$ with degree $2^5 = 32$. The colouring $\lambda$ restricts to a non-orientable colouring $\mu$ of the facet $\mathcal{D}$, which gives rise, by Lemma 2.8, to a non-orientable codimension-1 geodesic submanifold $M_\mu \subset M_\lambda$, covering $\mathcal{D}$ with degree $2^3 = 8$.

In complete analogy to [13, Lemma 3.2], by cutting $M_\lambda$ along $M_\mu$ we get a compact orientable hyperbolic manifold tessellated by 32 right-angled 120-cells having a geodesic boundary isometric to a connected manifold $\tilde{M}_\mu$ that double-covers $M_\mu$ and is hence tessellated by $2 \cdot 2^3 = 16$ right-angled dodecahedra.

To prove Theorem 1.3 it only remains to extend this argument from one 120-cell to an appropriate assembling of $n \ge 2$ distinct 120-cells.

### 3. ASSEMBLING RIGHT-ANGLED DODECAHEDRA AND 120-CELLS

Here, we assemble right-angled dodecahedra and 120-cells in order to construct more complicated convex compact right-angled convex compact polytopes.

**3.1. Connected sum of polytopes.** Let $\mathcal{P}_1$ and $\mathcal{P}_2$ be two right-angled polytopes in $\mathbb{H}^n$. If there is an isometry between two facets of $\mathcal{P}_1$ and $\mathcal{P}_2$, we may use it to glue them: the result is a new right-angled polytope in $\mathbb{H}^n$ which we call a *connected sum* of $\mathcal{P}_1$ and $\mathcal{P}_2$ along these facets.

**3.2. Assembling.** An *assembling* of right-angled dodecahedra (or 120-cells) is a right-angled polytope constructed from a finite sequence

$$ \mathcal{P}_1 \# \mathcal{P}_2 \# \mathcal{P}_3 \# \dots \# \mathcal{P}_k $$

of connected sums performed from the left to the right, where each $\mathcal{P}_i$ is a right-angled dodecahedron (or 120-cell).

**Lemma 3.1.** An assembling of $k$ right-angled dodecahedra is a facet of an assembling of $k$ right-angled 120-cells.

*Proof.* Consider $\mathbb{H}^3$ inside $\mathbb{H}^4$ as a geodesic hyperplane. Consider $\mathcal{D} \subset \mathbb{H}^3$ as a facet of $\mathcal{L} \subset \mathbb{H}^4$. Every time we attach a new copy $\mathcal{D}_i$ of $\mathcal{D}$ in $\mathbb{H}^3$, we correspondingly attach a new copy $\mathcal{Z}_i$ of $\mathcal{L}$ having $\mathcal{D}_i$ as a facet. $\square$
<figure><img src="image_3.png" /><figcaption>FIGURE 1. A non-orientable colouring of $\mathcal{D}$ from [6, Table 1].
Face colours are encoded by binary numbers</figcaption></figure>

3.3. **Proof of Theorem 1.3.** We have described all the ingredients necessary to prove Theorem 1.3.

By Proposition 2.10, pick an orientable $\mathbb{F}_2^5$-colouring of the right-angled 120-cell $\mathcal{L}$ that induces a non-orientable $\mathbb{F}_2^3$-colouring of one dodecahedral facet $\mathcal{D}$.

Then assemble $n$ copies of $\mathcal{D}$ as

$$ \mathcal{P} = \mathcal{D}_1\#\mathcal{D}_2\#\dots\#\mathcal{D}_n $$

and consider, as in Lemma 3.1, the resulting right-angled polyhedron $\mathcal{P}$ as a facet of a right-angled polytope $\mathcal{D}$ made of $n$ copies of the right-angled 120-cell, each having a $\mathcal{D}_i$ as a facet.

Every time we assemble a new copy of $\mathcal{D}_i$, we give $\mathcal{D}_i$ the colouring of the adjacent dodecahedron, mirrored along the glued pentagonal face, and we do the same with each corresponding new 120-cell $\mathcal{L}_i$. The resulting polytope $\mathcal{D}$ inherits an orientable $\mathbb{F}_2^5$-colouring $\lambda$ that induces a non-orientable $\mathbb{F}_2^3$-colouring $\mu$ of $\mathcal{P}$, if $\mathcal{P}$ is assembled appropriately. Indeed, each of the 24 non-orientable colourings of $\mathcal{D}$ has three faces $F$, $F'$ and $F''$ satisfying the conditions of Corollary 2.5. Moreover, we can find a fourth face $F^*$ which is disjoint from each of them. These properties are easily verifiable by using [6, Table 1]. Then, we start assembling $\mathcal{D}_i$'s by forming a connected sum along $F^*$. Then the colouring of the resulting polytope $\mathcal{P}$ again satisfies Corollary 2.5 and hence is non-orientable, as required.

By cutting $M_\lambda$ along $M_\mu$ we get a compact orientable hyperbolic manifold tessellated by $32n$ right-angled 120-cells having a geodesic boundary that is isometric to a connected manifold $\tilde{M}_\mu$ that double-covers $M_\mu$ and is hence tessellated by $2 \cdot 2^3n = 16n$ right-angled dodecahedra. $\square$

We conclude the paper by providing an example of the construction carried in the proof above.
**Example 3.2.** Let us choose a non-orientable colouring $\mu$ of the dodecahedron $\mathcal{D}$ from [6, Table 1], say the one having the maximal symmetry group $(\mathbb{Z}_2 \times \mathbb{Z}_2) \times \mathbb{Z}_6$. This is an $\mathbb{F}_2^3$-colouring depicted in Fig. 1, with face colours encoded by binary numbers in the decimal range 1, ..., 7. Here, the grey shaded faces of $\mathcal{D}$ are exactly the faces $F, F'$ and $F''$, satisfying the conditions of Corollary 2.5. The face $F^*$, along which we take a connected sum in the proof of Theorem 1.3, has a chequerboard shading. Now, we take a connected sum of $\mathcal{D}_1 := \mathcal{D}$ along $F^*$ with its isometric copy $\mathcal{D}_2$, having the same colouring. Then we can choose a face of $\mathcal{D}_1\#\mathcal{D}_2$, distinct from any of $F, F'$ or $F''$ and continue assembling until we use all $n$ given copies of $\mathcal{D}$, which produce a polyhedron $\mathcal{P} = \mathcal{D}_1\#\dots\#\mathcal{D}_n$. The faces $F, F'$ and $F''$ of $\mathcal{D}_1$ still remain among those of $\mathcal{P}$. Thus, the colouring of $\mathcal{P}$ is again non-orientable by Corollary 2.5.

Finally, by applying Proposition 2.9 and Lemma 3.1, we obtain a four-dimensional polytope $\tilde{\mathcal{P}}$ with an orientable $\mathbb{F}_2^5$-colouring, that induces a non-orientable colouring on one of its facets, which is isometric to $\mathcal{P}$. Indeed, the non-orientable $\mathbb{F}_2^5$-colouring of each $\mathcal{D}_i$ can be extended by Proposition 2.9 to an orientable $\mathbb{F}_2^5$-colouring of a 120-cell $\mathcal{L}_i$. Then, by Lemma 3.1, $\mathcal{P} = \mathcal{D}_1\#\dots\#\mathcal{D}_n$ is a facet of a polytope $\tilde{\mathcal{P}} = \mathcal{L}_1\#\dots\#\mathcal{L}_n$. Since the colourings of $\mathcal{D}_i$'s match under taking connected sums, the colourings of $\mathcal{L}_i$'s also match. The polytope $\tilde{\mathcal{P}}$ gives rise to a covering manifold with totally geodesic boundary, as described in the proof of Theorem 1.3.

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UNIVERSITY OF TORONTO, DEPARTMENT OF MATHEMATICS, 40 ST. GEORGE STREET, TORONTO ON, M5S 2E4, CANADA

Email address: kolpakov dot alexander at gmail dot com

UNIVERSITÀ DI PISA, DIPARTIMENTO DI MATEMATICA “TONELLI”, LARGO PONTE-CORVO 5, 56127 PISA, ITALY

Email address: martelli at dm dot unipi dot it

VANDERBILT UNIVERSITY, DEPARTMENT OF MATHEMATICS, 1326 STEVENSON CENTER, NASHVILLE, TN 37240, USA

Email address: steven dot tschantz at vanderbilt dot edu