# Moduli space of simple polygons

Ahtziri González and Manuel Sedano-Mendoza

###### Abstract

In this work we study the space $S(n)$ of positively oriented, simple $n$-gons
with labeled vertices up to oriented similarity and ${\mathcal{M}}(n)$ the
moduli space of simple $n$-gons as a quotient of $S(n)$. We give a local
description of $S(n)$ and ${\mathcal{M}}(n)$ around the regular $n$-gon and
describe completely the two spaces when $n=4$ using a topological Morse
function. We also provide an asymptotic description of the compactifications
$\overline{S(4)}$ and $\overline{{\mathcal{M}}(4)}$ up to a complicated set at
their boundaries.

Keywords:  Polygons; Similarity; Quadrilaterals; Moduli space;
Compactification.

##  1 Introduction

Spaces of polygons have been widely studied since they capture much of the
technical difficulties in different areas of mathematics. Examples of this are
the difference between fine and coarse moduli spaces and its generalization to
moduli stacks in algebraic geometry [1], the existence or non-existence of
global regularizing flows in differential geometry [13], [28], [29], geometric
constructions on spaces of polyhedrons [8] and [17], compactifications of
other Moduli spaces [27], [20]. In addition to this, the study of spaces of
polygons have consecuences on physics and engineering, with applications such
as robotic motion planing and pattern recognition [25], [26]. The study of the
geometry and topology of spaces of polygons with fixed properties is
interesting by itself. Examples of this are the study of spaces of polygons
with fixed side lengths [19], [20], fixed angles [23] and star shaped polygons
[24].

In this work we study the space of simple $n$-gons, with labeled vertices,
modulo oriented similarity. More precisely, the set of simple $n$-gons is
naturally embedded in the complex projective space
$\mathbb{C}\mathbb{P}^{n-2}$, having two homeomorphic connected components,
corresponding to positively and negatively oriented $n$-gons respectively, see
Section 2. The connected component of positively oriented polygons is denoted
here as $S(n)$ and as an open subset of $\mathbb{C}\mathbb{P}^{n-2}$, $S(n)$
has a structure of an analytic manifold. We propose the following conjecture

######  Conjecture 1.1.

$S(n)$ is homeomorphic to $\mathbb{R}^{2n-4}$.

There is a parallelism between the space $S(n)$ and the concept of Teichmüller
space of Riemann surfaces, and between its symmetry group $D_{n}$ (isomorphic
to the dihedral group) and the modular group of the corresponding surface, see
Remark 2.6. Motivated by this, we refer to the quotient
${\mathcal{M}}(n)=S(n)/D_{n}$ as the moduli space of simple polygons. If
$C(X)=\big{(}X\times(0,1]\big{)}/(X\times\\{1\\})$ denotes the open cone over
a topological space $X$, then by linearizing the action of the dihedral group
around the regular polygon (i.e. the polygon whose sides and angles are all
equal), we can recover the local result

######  Theorem 1.2.

There is a neighborhood $U\subset{\mathcal{M}}(n)$ around the regular $n$-gon
such that $U$ is homeomorphic to
$C(\mathbb{S}^{2n-5}/\langle\Sigma,T\rangle)$, where

$(z_{1},z_{2},\dots,z_{n-2})\overset{\Sigma}{\longrightarrow}(e^{4\pi
i/n}z_{1},e^{6\pi i/n}z_{2},\dots,e^{2\pi i(n-1)/n}z_{n-2}),\quad\text{and}$
$(z_{1},z_{2},\dots,z_{n-2})\overset{T}{\longrightarrow}-(e^{-4\pi
i/n}\overline{z}_{1},e^{-6\pi i/n}\overline{z}_{2},\dots,e^{-2\pi
i(n-1)/n}\overline{z}_{n-2}).$

We conjecture that such result holds globally.

######  Conjecture 1.3.

${\mathcal{M}}(n)$ is homeomorphic to
$C(\mathbb{S}^{2n-5}/\langle\Sigma,T\rangle)$.

General strategy. Conjectures 1.1 and 1.3 can be proven independently, but a
strategy to prove both simultaneously is to find a differentiable Morse
function in $S(n)$ with a unique singular point at the regular polygon and
whose level sets are invariant under the action of the symmetry group $D_{n}$.
In fact, we can observe that proving the cone structure on $S(n)$ and
constructing a Morse function with a unique singular point are equivalent
problems in the differentiable category. Moreover, the existence of such Morse
function implies the existence of a regularizing flow on $S(n)$ (its gradient
flow), that is, a flow taking each polygon onto the regular one withouth
loosing the properties of being simple and withouth collapsing vertices. To
the knowledge of the authors, the existence of such regularizing flow remains
open besides numerous attempts.

In this work, we propose a height function
$h:\overline{S(4)}\subset\mathbb{C}\mathbb{P}^{2}\rightarrow[0,1]$, which is a
Morse function in the topological sense (see [11], [12] and [9]), which has a
unique maximum $h=1$ at the regular quadrilateral (square) and whose level
sets are homeomorphic to $\mathbb{S}^{3}$ and are invariant under the action
of the dihedral group, leading to the following

######  Theorem 1.4.

There is a homeomorphism between $S(4)$ and the open cone
$C(\mathbb{S}^{3})\cong\mathbb{R}^{4}$, such that the height function $h$
corresponds to the function $C(\mathbb{S}^{3})\rightarrow(0,1]$, induced from
the projection $\mathbb{S}^{3}\times(0,1]\rightarrow(0,1]$.

In [3] it was already proven that $S(4)$ is homeomorphic to $\mathbb{R}^{4}$
and a description of the boundary $\partial
S(4)\subset\mathbb{C}\mathbb{P}^{2}$ was given by gluing a polyhedron, making
evident that the closure $\overline{S(4)}$ is complicated, in particular it is
not a manifold. The advantage of the approach given here, is that the cone
structure on $S(4)$ is invariant under the action of the dihedral group,
leading to the following result which was out of reach with the techniques of
[3].

######  Corolary 1.5.

Conjectures 1.1 and 1.3 are true for $n=4$.

Difficulties. Here, we are calling $h$ a “Morse function”, however, it is not
easy to prove this statement. In fact, it seems to the authors that it is
equally difficult to prove that the height function $h$ is a Morse function
than to prove Theorem 1.4 directly and then conclude that $h$ is a Morse
function from there. The most problematic regions are the neighborhoods of the
square, where there are a lot of quadrilaterals with different symmetries and
where the height function loses differentiability (a phenomenon present in
other constructions were symmetries cause problems, such as in [1]). To prove
Theorem 1.4, we may attempt to give a family of homeomorphisms
$f_{s}:h^{-1}(s)\rightarrow\mathbb{S}^{3}$ which vary continuously on the
parameter $s$, and then construct the homeomorphism $S(4)\rightarrow
C(\mathbb{S}^{3})$ as $Z\mapsto[f_{h(Z)}(Z),h(Z)]$. This strategy is hard to
carry out and it seems to the authors that the construction of such a
continuous family of homeomorphisms equals the difficulty of the construction
of a regularizing flow, which theoretically exists precisely by Theorem 1.4.
We overcome this difficulty by splitting $S(4)$ into subsets where the height
function $h$ behaves well (see Remark 5.6 for instance) and where we are able
to construct the continuous family of homeomorphisms, thus obtaining a cone
structure on every such set. With the previous partition, the problem
translates into understanding the combinatorics of the intersections of the
sets, which we give explicitly. The last main difficulty we have to face is on
the compactifications $\overline{S(4)}$ and $\overline{{\mathcal{M}}(4)}$,
where there are some bad convergence phenomena. We face this difficulty by
identifying the problematic region which gives bad behaviour at the boundary
and prove that we have a very good asymptotic behaviour outside of it

######  Theorem 1.6.

There exists a subspace ${\mathcal{D}}\subset\overline{S(4)}$ such that
$\overline{S(4)}\smallsetminus{\mathcal{D}}\cong\mathbb{H}^{4}\sqcup\mathbb{H}^{4}$,
where $\mathbb{H}^{4}$ denotes the fourth-dimensional, closed upper half-
space. The space ${\mathcal{D}}$ is an open cone over the two sphere
${\mathcal{D}}\cap S(4)\cong C(\mathbb{S}^{2})$, compactified with a space
${\mathcal{D}}\cap\partial S(4)$, which is obtained as a union of a wedge of
spheres $\mathbb{S}^{2}\vee\mathbb{S}^{2}$ and a closed disc $\mathbb{D}^{2}$.

As these constructions are invariant under the action of the dihedral group,
we obtain the asymptotic description of the Moduli space

######  Theorem 1.7.

There is a subspace
${\mathcal{D}}^{\prime}\subset\overline{{\mathcal{M}}(4)}=\overline{S(4)}/D_{4}$,
that is homeomorphic to an open cone over the $2$-disc
${\mathcal{D}}^{\prime}\cap{\mathcal{M}}(4)\cong C(\mathbb{D}^{2})$,
compactified with a topological disc
${\mathcal{D}}^{\prime}\cap\partial{\mathcal{M}}(4)\cong\mathbb{D}^{2}$, so
that the complement
$\overline{{\mathcal{M}}(4)}\smallsetminus{\mathcal{D}}^{\prime}$ is
homeomorphic to $\mathbb{H}^{4}$.

As the subset ${\mathcal{D}}^{\prime}$ in Theorem 1.7 is capturing the bad
asymptotic behaviour, the compactification of the open cone
${\mathcal{D}}^{\prime}\cap{\mathcal{M}}(4)$ with the topological disc
${\mathcal{D}}^{\prime}\cap\partial{\mathcal{M}}(4)$ is complicated, see
Remark 7.19.

It is worth noting that there have been previous uses of Morse functions on
spaces of polygons, by considering parameters such as side lengths [18] or
diagonals [19]. Those functions don’t work here, because they don’t capture
the asymptotic behaviour to the boundary $\partial S(n)$. There is however a
Morse function on Teichmüller spaces of Riemann surfaces whose behaviour
resembles much more to the one presented here, which is the systole [10].

Structure of the paper. In Section 2, we define precisely the spaces $S(n)$
and ${\mathcal{M}}(n)$, give the examples for $n=3$, and conclude with an
index of notations. In Section 3, we give the local description of
${\mathcal{M}}(n)$, prove Theorem 1.2, and in Example 3.4 we give a local,
linearized version of Theorem 1.4. In Section 4, we introduce the height
function $h$ for quadrilaterals and give some basic properties, we also give
the example of an analogous height function on $S(3)$ and explain how it
determines its topology. Section 5 is devoted to give a good decomposition of
$S(4)$ into subsets on which the height function behaves well and describe the
combinatorics of their intersections; we finish this section by proving that
the level sets of $h$ in $S(4)$ are homeomorphic to the sphere
$\mathbb{S}^{3}$. In Section 6, we prove Theorem 1.4 and in the process, we
prove Conjectures 1.1 and 1.3. In Section 7, we study the compactifications
$\overline{S(4)}$ and $\overline{{\mathcal{M}}(4)}$, and as it is the most
technical section of the paper, we split it into four subsections. In
Subsection 7.1 we describe the set ${\mathcal{D}}$ mentioned in Theorem 1.6,
which is the set of quadrilaterals presenting bad asymptotic phenomena at the
boundary. In Subsection 7.2, we give a decomposition of the boundary $\partial
S(4)\smallsetminus{\mathcal{D}}$, with equivalent combinatorics of the
decompositions of every other level set
$h^{-1}(s)\smallsetminus{\mathcal{D}}$. In Subsection 7.3, we use all the
previous constructions to prove Theorem 1.6. In Subsection 7.4 we study the
decomposition in $\overline{{\mathcal{M}}(4)}$ induced by the previous
construction and prove Theorem 1.7. We present in Section 8 a list of
concluding remarks and open questions. Finally, we add two appendices:
Appendix A, were we give two technical theorems on gluing two topological
closed balls and two topological closed half-spaces, and Appendix B, were we
give a homeomorphism between two quotients, one with a group and the other as
a double quotient by a normal subgroup.

### Acknowledgements

We would like to thank Gerardo Arizmendi Echegaray for fruitful conversations
on an early stage of the project, as well as for suggesting a modification to
the height function which decreased the complexity of the combinatorics in the
decomposition of its level sets, c.f. Definition 5.3. Manuel Sedano would like
to thank CONACyT grant CB2016-283960 as well as DGAPA-UNAM Postdoctoral
Scholarship.

##  2 Preliminaries

A polygon with $n$ sides (or simply an $n$-gon) in the complex plane with
consecutive vertices at $z_{1},z_{2},\dots,z_{n}\in\mathbb{C}$, is completely
determined by the point $(z_{1},z_{2},\dots,z_{n})\in\mathbb{C}^{n}$. In this
way, the set of $n$-gons in $\mathbb{C}$ with labeled vertices is equipped
with the topology of $\mathbb{C}^{n}$. For all
$Z=(z_{1},\dots,z_{n})\in\mathbb{C}^{n}$, $\mathfrak{c}(Z)$ denotes the
polygonal curve
$\overline{z_{1}z_{2}}\cup\overline{z_{2}z_{3}}\cup\cdots\cup\overline{z_{n-1}z_{n}}\cup\overline{z_{n}z_{1}}$,
where $\overline{ab}\subset\mathbb{C}$ is the segment from $a$ to $b$.

######  Definition 2.1.

Let $Z\in\mathbb{C}^{n}$ be a $n$-gon.

  1. (a)

We say that $Z$ is simple if its vertices are different and $\mathfrak{c}(Z)$
is a Jordan’s curve. $\mathbb{S}(n)\subset\mathbb{C}^{n}$ denotes the set of
simple $n$-gons.

  2. (b)

If $Z$ is simple, then we say that $Z$ is positively (negatively) oriented if
the increasing order in the vertices of $\mathfrak{c}(Z)$ is in counter-
clockwise (clockwise).

In [2] it is proved that $\mathbb{S}(n)$ is an open subset and has two
homeomorphic connected components, corresponding to positively and negatively
oriented $n$-gons. The group of complex affine transformations on
$\mathbb{C}$, denoted here as ${\mathcal{A}}_{\mathbb{C}}$, acts diagonally on
$\mathbb{C}^{n}$ as

$(az+b,(z_{1},\dots,z_{n}))\mapsto(az_{1}+b,\dots,az_{n}+b),$

preserving angles, labeled vertices and proportional side lengths in the
$n$-gons. The quotient
$P(n):=\left(\mathbb{C}^{n}\smallsetminus\\{(z,z,\dots,z)\\}\right)/{\mathcal{A}}_{\mathbb{C}}$
is the space of shapes of $n$-gons with labeled vertices up to oriented
similarity as considered in [8]. Notice that two equivalent $n$-gons in the
linear subspace $V_{n}=\\{(0,z_{2},\dots,z_{n})\in\mathbb{C}^{n}\\}$ differ by
just a complex factor, so the space $P(n)$ is a complex projective space

$P(n)\cong\mathbb{P}_{\mathbb{C}}V_{n}\cong\mathbb{C}\mathbb{P}^{n-2}.$

The action of ${\mathcal{A}}_{\mathbb{C}}$ preserves orientation and the
property of being simple in $n$-gons, so if
$\eta\colon\mathbb{C}^{n}\smallsetminus\\{(z,z,\dots,z)\\}\to P(n)$ denotes
the natural projection, then $\eta\left(\mathbb{S}(n)\right)\subset P(n)$ is
open and has two connected components, corresponding to positively and
negatively oriented $n$-gons. Denote by $S(n)\subset\eta(\mathbb{S}(n))$ the
connected component of positively oriented shapes of $n$-gons and denote by
$\eta(z_{1},z_{2},\dots,z_{n})=[z_{1},z_{2},\dots,z_{n}]\in S(n)$ the
corresponding class of an $n$-gon.

The linear isomorphism
$\widehat{\sigma}(z_{1},z_{2},\dots,z_{n})=(z_{2},\dots,z_{n},z_{1})$, re-
enumerates the vertices of an $n$-gon in a cyclic way. Notice that two
$n$-gons $Z,W\in\mathbb{C}^{n}$ have the same shape if and only if
$\widehat{\sigma}(Z)$ and $\widehat{\sigma}(W)$ have the same shape, so there
exists a function $\sigma\colon P(n)\to P(n)$ such that
$\sigma\circ\eta=\eta\circ\widehat{\sigma}$.

######  Remark 2.2.

We may observe that, as the diagonal $\\{(z,z,\dots,z)\\}$ is not considered
in $P(n)$, at least two vertices of a polygon in $P(n)$ must be different. So
if we consider the set $U_{12}=\\{[z_{1},z_{2},\dots,z_{n}]\in P(n)\colon
z_{1}\neq z_{2}\\}$ and the coordinate chart $\phi_{12}\colon
U_{12}\to\mathbb{C}^{n-2}$, given by

$\phi_{12}\left([z_{1},z_{2},\dots,z_{n}]\right)=\left(\frac{z_{3}-z_{1}}{z_{2}-z_{1}},\dots,\frac{z_{n}-z_{1}}{z_{2}-z_{1}}\right),$

then for every $Z\in P(n)$, $\sigma^{k}(Z)\in U_{12}$, for some
$k\in\mathbb{N}$. That is, up to re-enumeration of the vertices, $P(n)$ is
contained in $U_{12}$. Through the paper, we will mostly work with
representatives $[0,1,z_{3},\dots,z_{n}]\in U_{12}$.

######  Example 2.3 (Triangles).

The space $P(3)$ is biholomorphic to the Riemann sphere. There are three
options for a triangle $[0,1,x+iy]\in U_{12}$:

$(a)~{}~{}y>0,\qquad\quad(b)~{}~{}y<0,\qquad\text{~{}~{}and~{}~{}}\qquad~{}(c)~{}~{}y=0.$

In the plane $\phi_{12}(U_{12})$, the case $(a)$ is the upper half plane and
corresponds to the set $S(3)$ of positively oriented triangles; the case $(b)$
is the lower half plane and corresponds to the set of negatively oriented
triangles; and the case $(c)$, which corresponds to triangles degenerated to a
segment, compactifies to a circle in $P(3)$ with the degenerated triangle
$[0,0,1]$. Thus, the set of degenerated triangles is a circle that splits the
sphere $P(3)$ into two discs, corresponding to positively and negatively
oriented triangles.

The involution $\tau\colon U_{12}\to U_{12}$, which reflects the last
$(n-2)$-vertices trough the line $\textrm{Re}(z)=\frac{1}{2}$, changes the
order in which the interior angles are presented but without changing the
orientation and can be written as

$\tau([0,1,z_{3},\dots,z_{n}])=[0,1,1-\bar{z}_{n},\dots,1-\bar{z}_{3}].$

Since $U_{12}\subset P(n)$ is dense (Remark 2.2), then $\tau$ can be
continuously extended to an involution of $P(n)$.

######  Proposition 2.4.

The functions $\sigma,\tau\colon P(n)\to P(n)$ are diffeomorphisms and the
group generated by them $\langle\sigma,\tau\rangle$ is isomorphic to the
dihedral group of order $2n$.

###### Proof.

As the funciton $\sigma$ is defined by the property
$\sigma\circ\eta=\eta\circ\widehat{\sigma}$, with
$\widehat{\sigma}(z_{1},z_{2},\dots,z_{n})=(z_{2},\dots,z_{n},z_{1})$ and
$\eta\colon\mathbb{C}^{n}\smallsetminus\\{(z,z,\dots,z)\\}\to P(n)$ the
natural projection, the differentiability of $\sigma$ follows from the
differentiability of $\widehat{\sigma}$. Analogously, if we define
$\widehat{\tau}\colon\mathbb{C}^{n}\to\mathbb{C}^{n}$ as
$\widehat{\tau}(z_{1},\dots,z_{n})=-(\overline{z_{2}},\overline{z_{1}},\overline{z_{n}},\dots,\overline{z_{3}})$,
then $\sigma\circ\tau=\eta\circ\widehat{\tau}$ and the differentiability of
$\tau$ follows from the differentiability of $\widehat{\tau}$. Finally, it is
straightforward to check that $\sigma^{n}=\tau^{2}=(\tau\sigma)^{2}=1$. Then
$\langle\sigma,\tau\rangle$ is isomorphic to the dihedral group of order $2n$.
∎

Just as the action of the group ${\mathcal{A}}_{\mathbb{C}}$ in
$\mathbb{C}^{n}$ eliminates translations, rotations and homotheties in
$n$-gons, the action of the dihedral group $\langle\sigma,\tau\rangle$ in
$P(n)$, removes the labels at the vertices and identifies $n$-gons equivalent
under reflections.

######  Definition 2.5.

Let $\overline{S(n)}\subset P(n)$ denote the closure of $S(n)$. Then

  1. (a)

the quotient ${\mathcal{M}}(n)=S(n)/\langle\sigma,\tau\rangle$ is the Moduli
Space of $n$-gons.

  2. (b)

the quotient
$\overline{{\mathcal{M}}(n)}=\overline{S(n)}/\langle\sigma,\tau\rangle$ is a
compactification of ${\mathcal{M}}(n)$.

######  Remark 2.6.

A different description of the space $S(n)$, is to consider it as the set of
graph isomorphisms $\mathfrak{R}_{n}\rightarrow Z$ (where $\mathfrak{R}_{n}$
is the regular $n$-gon and $Z$ is an arbitrary simple polygon), with the
action of the dihedral group acting via precomposition. This approach creates
a parallelism between the concept of Teichmüller space and $S(n)$, and between
the modular group of a surface and the dihedral group as a symmetry group of
$S(n)$ (see [31] for this and a similar construction for the group of outer
automorphisms $Out(F_{n})$). Moreover, by construction, the action of the
dihedral group extends to the closure
$\overline{S(n)}\subset\mathbb{C}\mathbb{P}^{n-2}$, which is a well behaved
property for this parallelism and a good justification to consider
$\overline{S(n)}/\langle\sigma,\tau\rangle$ as the compactification of
${\mathcal{M}}(n)$, see [30].

######  Example 2.7.

From Example 2.3 we know that $S(3)\subset U_{12}$ is a copy of the upper half
plane. In the representatives $[0,1,z_{3}]$, the functions
$\sigma([0,1,z_{3}])=[0,1,\frac{1}{1-z_{3}}]$ and
$\tau([0,1,z_{3}])=[0,1,1-\bar{z}_{3}]$ are a hyperbolic rotation of angle
$\frac{2\pi}{3}$, centered at the equilateral triangle
$\mathfrak{R}_{3}=[0,1,\frac{1+i\sqrt{3}}{2}]$, and a reflection through the
line $[0,1,\frac{1}{2}+iy]$, respectively. A fundamental domain of the
$\langle\sigma,\tau\rangle$-action in $S(3)$ is

${\mathcal{M}}(3)=\left\\{[0,1,z_{3}]\in U_{12}\colon|z_{3}|\leq
1,Im(z_{3})>0,Re(z_{3})\geq\frac{1}{2}\right\\}.$

The compactification $\overline{{\mathcal{M}}(3)}$ is obtained by adding to
${\mathcal{M}}(3)$ the degenerated triangles $[0,1,s]$ with
$s\in[\frac{1}{2},1]$, see Figure 1.

$\partial
S(3)\cong\mathbb{R}\cup\\{\infty\\}$$S(3)$$[0,1,-1]$$\bullet$$[0,1,0]$$\bullet$$[0,1,1]$$\bullet$$[0,1,2]$$\bullet$$\bullet$$[0,1,\frac{1}{2}]$$\bullet$$\bullet$$\big{[}0,1,\frac{1}{2}\big{]}$$\bullet$$[0,1,1]$$\bullet$Equilateral$\big{[}0,1,\frac{1+i\sqrt{3}}{2}\big{]}$Isosceles$[0,1,\frac{1+it}{2}]$Isosceles$[0,1,e^{i\theta}]$$[0,1,s]$Degenerated
Figure 1: Different types of triangles in the compactification
$\overline{{\mathcal{M}}(3)}$.

From Example 2.7 it follows that ${\mathcal{M}}(3)$ is an hyperbolic orbifold
of dimension two, with boundary $\partial\overline{{\mathcal{M}}(3)}$ at
infinite distance with respect to this metric. This is the only case where the
moduli space ${\mathcal{M}}(n)$ is a geometric quotient.

### Index of notations

We collect here some of the notations that will be used throughout the paper.

  1. 1.

Open cone over a topological space $X$:
$C(X)=\big{(}X\times(0,1]\big{)}/\\{\sim\\}$, where $(x,1)\sim(x^{\prime},1)$,
for all $x\in X$.

  2. 2.

Suspension over a topological space $X$:
$\Sigma(A)=\big{(}X\times[-1,1]\big{)}/{\sim}$, where
$(x,1)\sim(x^{\prime},1)$ or $(x,-1)\sim(x^{\prime},-1)$, for all $x\in X$.

  3. 3.

Closed half spaces
$\mathbb{H}^{n}=\\{(x_{1},\cdots,x_{n})\in\mathbb{R}^{n}:x_{n}\geq 0\\}$

  4. 4.

Closed ball
$\mathbb{D}^{n}=\\{(x_{1},\cdots,x_{n})\in\mathbb{R}^{n}:x_{1}^{2}+\cdots+x_{n}^{2}=1\\}$,

  5. 5.

The distance $d(z,A)=\min\\{|z-a|:z\in A\\}$, from $z$ to a subset
$A\subset\mathbb{C}$.

  6. 6.

Segment $\overline{z_{1}z_{2}}=\\{(1-t)z_{1}+tz_{2}:0\leq t\leq
1\\}\subset\mathbb{C}$, between two points $z_{1},z_{2}\in\mathbb{C}$. Not to
be confused with conjugation $\overline{x+iy}=x-iy$, as we won’t ever use
complex conjugation on a multiplication of two complex numbers.

  7. 7.

Polygonal curve
$\mathfrak{c}(Z)=\overline{z_{1}z_{2}}\cup\overline{z_{2}z_{3}}\cup\cdots\cup\overline{z_{n-1}z_{n}}\cup\overline{z_{n}z_{1}}$
defined by a polygon $Z=(z_{1},\cdots,z_{n})\in\mathbb{C}^{n}$.

  8. 8.

Polygonal curve
$\mathcal{L}_{j}(Z)=\overline{z_{j+1}z_{j+2}}\cup\overline{z_{j+2}z_{j+3}}\cup\cdots\overline{z_{j-2}z_{j-1}}$,
opposite to a vertex $z_{j}$ of a polygon
$Z=(z_{1},\cdots,z_{n})\in\mathbb{C}^{n}$. We take the cyclic convention on
indices mod $n$.

  9. 9.

The shape $[z_{1},\cdots,z_{n}]\in S(n)$ as the class of the polygon
$(z_{1},\cdots,z_{n})\in\mathbb{C}^{n}$.

  10. 10.

The regular polygon
$\mathfrak{R}_{n}=[1,e^{2i\pi/n},\dots,e^{2i\pi(n-1)/n}]\in S(n)$. We call
$\mathfrak{R}_{4}$ “the square” as well.

  11. 11.

The closure $\overline{A}\subset\mathbb{C}\mathbb{P}^{n-2}$ of a subset
$A\subset S(n)\subset\mathbb{C}\mathbb{P}^{n-2}$.

  12. 12.

The restricted closure $\textrm{cl}(A)=\overline{A}\cap S(n)$ of a subset
$A\subset S(n)\subset\mathbb{C}\mathbb{P}^{n-2}$.

  13. 13.

We will use the word “side” por the edges of a polygon, including bigons (c.f.
Definición 5.9). On the other hand we will use the word “edge” for the one-
dimensional sides of polyhedrons the corresponding graphs (such as in Figure 8
and Figure 16).

We clarify on two more concepts widely used in the text:

######  Remark 2.8.

If we consider the trivial $I$-bundle $\mathbb{D}^{2}\times\mathbb{D}^{1}$,
which is homeomorphic to $\mathbb{D}^{3}$ and collapse some fibers on a closed
interval of the boundary $I\subset\partial\mathbb{D}^{2}\cong\mathbb{S}^{1}$:
$(x,t)\sim(x,t^{\prime})$ for all $t\in\mathbb{D}^{1}$ and $x\in I$, then the
quotient

$(\mathbb{D}^{2}\times\mathbb{D}^{1})/{\sim}$

is still homeomorphic to $\mathbb{D}^{3}$. We will call this kind of quotients
an “$I$-bundle with collapsed fibers”. We can further observe that the we can
collapse fibers in a series of disjoint intervals $I_{1}\sqcup\cdots\sqcup
I_{k}\subset\partial\mathbb{D}^{2}$ with analogous behaviour.

######  Remark 2.9.

We will mostly use Remark 2.8 on $I$-bundles with collapsed fibers, where the
bases space is a closed subspace of $\\{z\in\mathbb{C}:|z|\leq 1,\
\textrm{Im}(z)\geq 0\\}$ homeomorphic to $\mathbb{D}^{2}$. We will be using
mostly the notation $\Omega,\Omega^{\prime},\Omega^{\prime\prime},...$ with
some added sub-indices. We don’t have a unified notation for this as we will
be using different subsets for different constructions and the precise
description of the domain is subtle and necessary. We will define each domain
where needed.

##  3 Local description of ${\mathcal{M}}(n)$

In this section we describe the topology of ${\mathcal{M}}(n)$ around the
regular $n$-gon and explain in detail the construction for triangles and
quadrilaterals.

Let $\mathfrak{R}_{n}=[1,e^{2i\pi/n},\dots,e^{2i\pi(n-1)/n}]\in S(n)$ be the
shape of the regular $n$-gon. It is straightforward to check that
$\mathfrak{R}_{n}=\tau(\mathfrak{R}_{n})=\sigma(\mathfrak{R}_{n})$, and in
fact, $\mathfrak{R}_{n}$ is the only element in $S(n)$ that is fixed by
$\sigma$.

######  Lemma 3.1.

Let $D_{\mathfrak{R}_{n}}\sigma$ and $D_{\mathfrak{R}_{n}}\tau$ be the
differentials of $\sigma$ and $\tau$ at $\mathfrak{R}_{n}$. Then there is a
basis $\\{\zeta_{2},\dots,\zeta_{n-1}\\}$ of the complex vector space
$T_{\mathfrak{R}_{n}}S(n)$ such that

$D_{\mathfrak{R}_{n}}\sigma\Big{(}\\!\sum_{k=2}^{n-1}s_{k}\zeta_{k}\\!\Big{)}\\!=\\!\sum_{k=2}^{n-1}e^{2\pi
ik/n}s_{k}\zeta_{k},~{}\text{and}~{}~{}D_{\mathfrak{R}_{n}}\tau\Big{(}\\!\sum_{k=2}^{n-1}s_{k}\zeta_{k}\\!\Big{)}\\!=\\!\sum_{k=2}^{n-1}-e^{-2\pi
ik/n}\overline{s_{k}}\zeta_{k}\,.$

###### Proof.

Let $B(Z,W)=\sum_{j}z_{j}\overline{w}_{j}$ denotes the standard Hermitian
product in $\mathbb{C}^{n}$. Consider the basis
$\beta=\big{\\{}\widehat{\zeta}_{k}=(1,e^{2\pi ik/n},(e^{2\pi
ik/n})^{2},\dots,(e^{2\pi ik/n})^{n-1})\big{\\}}$ with $k=0,1,\dots,n-1$, of
$\mathbb{C}^{n}$. The products between these vectors are

$B(\widehat{\zeta}_{k},\widehat{\zeta}_{l})=\sum_{m=0}^{n-1}\left(e^{\frac{2\pi
i(k-l)}{n}}\right)^{m}=\left\\{\begin{array}[]{lcr}n&\text{if}&k=l\\\
0&\text{if}&k\neq l\end{array}\right.$

and therefore, $\beta$ is an orthogonal basis. Let
$V=\\{(z_{1},z_{2},\dots,z_{n})\in\mathbb{C}^{n}\colon
z_{1}+z_{2}+\cdots+z_{n}=0\\}$ be the complex vector subspace of $n$-gons that
have center of mass at the origin. Note that
$V=\langle\widehat{\zeta}_{1},\widehat{\zeta}_{2},\dots,\widehat{\zeta}_{n-1}\rangle$,
$\mathbb{C}^{n}=\langle\widehat{\zeta}_{0}\rangle\oplus V$,
$\eta(V)=\mathbb{P}_{\mathbb{C}}V=P(n)$ and
$\eta(\widehat{\zeta}_{1})=\mathfrak{R}_{n}$. Since
$\langle\widehat{\zeta}_{1}\rangle$ is the kernel of the differential
$D_{\widehat{\zeta}_{1}}\eta\colon V\to T_{\mathfrak{R}_{n}}P(n)$, then the
restriction

$D_{\widehat{\zeta}_{1}}\eta\colon\langle\widehat{\zeta_{2}},\cdots,\widehat{\zeta}_{n-1}\rangle\to
T_{\mathfrak{R}_{n}}S(n)$

is a linear isomorphism.

Let $\widehat{\tau}\colon\mathbb{C}^{n}\to\mathbb{C}^{n}$ be the function
$(z_{1},\dots,z_{n})\mapsto-(\overline{z_{2}},\overline{z_{1}},\overline{z_{n}},\dots,\overline{z_{3}})$.
Since $\widehat{\sigma}(\widehat{\zeta}_{k})=e^{2ik\pi/n}\widehat{\zeta}_{k}$
and $\widehat{\tau}(\widehat{\zeta}_{k})=-e^{-2ik\pi/n}\widehat{\zeta}_{k}$,
then $V$ and $\langle\widehat{\zeta_{2}},\dots,\widehat{\zeta}_{n-1}\rangle$
are invariant under $\widehat{\sigma}$ and $\widehat{\tau}$, moreover,
$\widehat{\sigma}$ and $\widehat{\tau}$ satisfy
$\eta\circ\widehat{\sigma}=\sigma\circ\eta$ and
$\eta\circ\widehat{\tau}=\tau\circ\eta$. Differentiating in
$\widehat{\zeta}_{1}$, we obtain

$D_{\widehat{\zeta}_{1}}\eta\circ\widehat{\sigma}=D_{\mathfrak{R}_{n}}\sigma\circ
D_{\widehat{\zeta}_{1}}\eta\,,\quad\text{and}\quad
D_{\widehat{\zeta}_{1}}\eta\circ\widehat{\tau}=D_{\mathfrak{R}_{n}}\tau\circ
D_{\widehat{\zeta}_{1}}\eta\,.$

The proof is finished by taking
$\zeta_{k}=D_{\widehat{\zeta}_{1}}\eta(\widehat{\zeta_{k}})$, for
$k=2,3,\dots,n-1$. ∎

Let $\Sigma,T\colon\mathbb{C}^{n-2}\to\mathbb{C}^{n-2}$ denote the orthogonal
transformations:

$(z_{1},z_{2},\dots,z_{n-2})\overset{\Sigma}{\longrightarrow}(e^{4\pi
i/n}z_{1},e^{6\pi i/n}z_{2},\dots,e^{2\pi i(n-1)/n}z_{n-2}),\quad\text{and}$
$(z_{1},z_{2},\dots,z_{n-2})\overset{T}{\longrightarrow}-(e^{-4\pi
i/n}\overline{z}_{1},e^{-6\pi i/n}\overline{z}_{2},\dots,e^{-2\pi
i(n-1)/n}\overline{z}_{n-2}).$

The projection of the regular $n$-gon into ${\mathcal{M}}(n)$ is also denoted
by $\mathfrak{R}_{n}$. The following result provides a local topological
description of ${\mathcal{M}}(n)$ around $\mathfrak{R}_{n}$.

######  Theorem 3.2.

Local neighbourhoods of $\mathfrak{R}_{n}\in{\mathcal{M}}(n)$ are homeomorphic
to the cone over the quotient $\mathbb{S}^{2n-5}/\langle\Sigma,T\rangle$.

###### Proof.

Since the maps $\widehat{\sigma},\widehat{\tau}\colon V\to V$ are orthogonal
with respect to the Euclidean product, then the projections $\sigma,\tau\colon
P(n)\to P(n)$ are Riemannian isometries of the Fubini-Study metric in $P(n)$.
The exponential map around $\mathfrak{R}_{n}$, locally conjugates the actions
of $\sigma$ and $\tau$ in $S(n)$ with their differential maps at
$T_{\mathfrak{R}_{n}}S(n)$. Then, around $\mathfrak{R}_{n}$, the moduli space
${\mathcal{M}}(n)=S(n)/\langle\sigma,\tau\rangle$ is homeomorphic to the cone
over the quotient of the unit sphere in $T_{\mathfrak{R}_{n}}S(n)$ with the
action of the dihedral group $\langle
D_{\mathfrak{R}_{n}}\sigma,D_{\mathfrak{R}_{n}}\tau\rangle$. From Lemma 3.1,
it follows that the actions of $\langle
D_{\mathfrak{R}_{n}}\sigma,D_{\mathfrak{R}_{n}}\tau\rangle$ in
$T_{\mathfrak{R}_{n}}S(n)$ and $\langle\Sigma,T\rangle$ in $\mathbb{C}^{n-2}$
are conjugated. ∎

######  Example 3.3.

For $n=3$, $\Sigma(z)=e^{4\pi i/3}z$ and $T(z)=-e^{-4\pi i/3}\overline{z}$.
Since $T$ is a reflection through the line with argument $-\frac{\pi}{6}$,
then a fundamental domain in the circle $\mathbb{S}^{1}$ for the action of
$\langle\Sigma,T\rangle$, is the arc
$\\{e^{i\theta}\colon-\frac{\pi}{6}\leq\theta\leq 0\\}$. In Example 2.7 we
showed that the compactification $\overline{{\mathcal{M}}(3)}$ is homeomorphic
to the closed cone over a copy of this arc, this is an exceptional case, as
the higher dimensional compactifications $\overline{{\mathcal{M}}(n)}$ will be
much more involved than a closed cone over the quotient
$\mathbb{S}^{2n-5}/\langle\Sigma,T\rangle$.

######  Example 3.4.

For $n=4$, the quotient $\mathbb{S}^{3}/\langle\Sigma,T\rangle$ is
homeomorphic to $\mathbb{S}^{3}$. The homeomorphism can be described as
follows: For $\theta\in[0,2\pi]$, the set
$S^{+}_{\theta}:=\\{(z_{1},te^{i\theta})\in\mathbb{S}^{3}\colon t\geq 0\\}$ is
a copy of the closed upper hemisphere in $\mathbb{S}^{2}$. Notice that

$\mathbb{S}^{3}=\bigcup_{\theta\in[0,2\pi]}S^{+}_{\theta}\,,\qquad{\mathcal{S}}:=\bigcap_{\theta\in[0,2\pi]}S^{+}_{\theta}=\\{(z_{1},0)\in\mathbb{S}^{3}\\}\cong\mathbb{S}^{1}$

is the boundary circle of all the hemispheres $S^{+}_{\theta}$, and for a
generic interval $[\theta_{1},\theta_{2}]$, the union
$\cup_{\theta\in[\theta_{1},\theta_{2}]}S_{\theta}^{+}$ is a copy of the
closed disc $\mathbb{D}^{3}$ with boundary sphere $S_{\theta_{1}}^{+}\cup
S_{\theta_{2}}^{+}\cong\mathbb{S}^{2}$ (Figure 2 illustrates the union
$\cup_{\theta\in[-\frac{\pi}{4},0]}S_{\theta}^{+}$). As
$\Sigma(z,w)=(-z,e^{3\pi i/2}w)$ and $T(z,w)=(\overline{z},e^{-\pi
i/2}\overline{w})$, then a fundamental domain for the action of
$\langle\Sigma,T\rangle$ in $\mathbb{S}^{3}$, is the angular region
$\cup_{\theta\in[-\frac{\pi}{4},0]}S_{\theta}^{+}$, homeomorphic to a closed
ball $\mathbb{B}^{3}$. We can split this domain as its interior
$\bigcup_{\theta\in(-\frac{\pi}{4},0)}S_{\theta}^{+}\smallsetminus{\mathcal{S}}$,
where no two elements are identified with an elemen of
$\langle\Sigma,T\rangle$ and its boundary $S_{0}^{+}\cup
S_{-\pi/4}^{+}\cong\mathbb{S}^{2}$, where $T$ acts in $S_{-\pi/4}^{+}$ as
$(z,w)\mapsto(\overline{z},w)$, and $T\circ\Sigma$ acts in $S_{0}^{+}$ as
$(z,w)\mapsto(-\overline{z},w)$. It is not hard to see that taking the
quotient of $\cup_{\theta\in[-\frac{\pi}{4},0]}S_{\theta}^{+}$ under the
identifications given by $T$ gives us a space homeomorphic to $\mathbb{B}^{3}$
and then taking the quotient of this space under the identifications given by
$T\circ\Sigma$ gives us a space homeomorphic to $\mathbb{S}^{3}$.

$\bullet$$\bullet$$S_{0}^{+}$$S_{-\frac{\pi}{4}}^{+}$$\mathbb{S}^{3}$$\cup_{\theta\in[-\frac{\pi}{4},0]}S_{\theta}^{+}$
Figure 2: A schematic representation of the fundamental domain for the
$\langle\Sigma,T\rangle$-action on the sphere $\mathbb{S}^{3}$.

######  Remark 3.5.

The constructions for quadrilaterals in Example 3.4, can be done analogously
for all $n$. It is necessary to analyze the quotient of the
$(2k-1)$-dimensional waist sphere
$S^{2k-1}:=\\{(z_{1},\dots,z_{k},0,\dots,0)\in\mathbb{S}^{2n-5}\\}$ by the
action of the dihedral group $\langle\Sigma,T\rangle$. In this way we
construct a fundamental domain in $\mathbb{S}^{2n-5}$ for the action of
$\Sigma$, which is invariant by the action of $T$. Such fundamental domain
makes evident that there is a homeomorphism between
$\mathbb{S}^{2n-5}/\langle\Sigma,T\rangle$ and the double quotient

$\big{\\{}\mathbb{S}^{2n-5}/\langle\Sigma\rangle\big{\\}}/\langle
T\rangle\cong L(n:2,3,\dots,n-1)/\langle T\rangle,$

where $L(n:2,3,\dots,n-1)=\mathbb{S}^{2n-5}/\langle\Sigma\rangle$ is an
Orbifold Lens Space.

The reason behind the fact that $\mathbb{S}^{2n-5}/\langle\Sigma,T\rangle$ is
homeomophic to the double quotient
$\big{\\{}\mathbb{S}^{2n-5}/\langle\Sigma\rangle\big{\\}}/\langle T\rangle$,
is that the cyclic group $\langle\Sigma\rangle$ is normal in the dihedral
group $\langle\Sigma,T\rangle$, see Proposition B.1.

##  4 Height function in quadrilaterals

In this section, we define the height function in the space of simple
quadrilaterals, which will be used to describe its topology, and give some of
its properties. We also describe the analogous function in the case of
triangles and use it to describe its topology.

######  Definition 4.1.

For every $Z=(z_{1},z_{2},z_{3},z_{4})\in\mathbb{C}^{4}$,

  1. (a)

the longest side function is $\ell\colon\mathbb{C}^{4}\to\mathbb{R}$,
$\ell(Z)=\max_{1\leq j\leq 4}|z_{j}-z_{j+1}|$, and

  2. (b)

the minimum distance function is $r\colon\mathbb{C}^{4}\to\mathbb{R}$,
$r(Z)=\min_{1\leq j\leq 4}\\{r_{j}(Z)\\}$, where
$r_{j}(Z):=d(z_{j},{\mathcal{L}}_{j})$ with
${\mathcal{L}}_{j}:=\overline{z_{j+1}z_{j+2}}\cup\overline{z_{j+2}z_{j+3}}$.

Notice that the functions $\ell$ and $r$ are continuous and non-negative.
Moreover, $r(Z)\leq\ell(Z)$ for all $Z\in\mathbb{C}^{4}$, and $\ell(Z)=0$ if
and only if $Z=(z,z,z,z)$, for some $z\in\mathbb{C}$. Thus, the function
$h\colon\mathbb{C}^{4}\smallsetminus\\{(z,z,z,z)\\}\to\mathbb{R}$ defined as
$h(Z)=\frac{r(Z)}{\ell(Z)}$ is continuous and $0\leq h(Z)\leq 1$. For all
$Z\in\mathbb{C}^{4}\smallsetminus\\{(z,z,z,z)\\}$ and $a,b\in\mathbb{C}$ with
$a\neq 0$, the properties

$r(aZ+b)=|a|r(Z)\quad\textrm{and}\quad\ell(aZ+b)=|a|\ell(Z)$

hold, so that $h$ is invariant under the action of the complex affine group
${\mathcal{A}}_{\mathbb{C}}$ on $\mathbb{C}^{4}$. We conclude that there is a
well defined and continuous function $h\colon P(4)\to\mathbb{R}$ in the space
of shapes of quadrilaterals.

######  Definition 4.2.

The height function in the space of shapes of simple and positively oriented
quadrilaterals is the restriction $h\colon S(4)\to\mathbb{R}$, defined by

$h[z_{1},z_{2},z_{3},z_{4}]=\frac{r(z_{1},z_{2},z_{3},z_{4})}{\ell(z_{1},z_{2},z_{3},z_{4})}.$

######  Proposition 4.3.

The height function $h\colon S(4)\to\mathbb{R}$ is positive and has a global
maximum in the square $\mathfrak{R}_{4}=[0,1,1+i,i]$, where $h$ equals $1$.

###### Proof.

It is straightforward to check from Definition 4.1 that
$h(\mathfrak{R}_{4})=1$ and $h(Z)>0$, for every $Z\in S(4)$. If $Z$ has one
side strictly smaller than the others, say $|z_{j}-z_{j+1}|$ for some
$j\in\\{1,2,3,4\\}$, then $r(Z)\leq r_{j}(Z)\leq|z_{j}-z_{j+1}|<\ell(Z)$, and
therefore, $h(Z)<1$. If all of the sides of $Z\in S(4)$ have equal length (we
may choose a representative whose sides are equal to 1), then it is a rhombus.
If $Z\neq\mathfrak{R}_{4}$, then there exists $j\in\\{1,2,3,4\\}$, such that
the interior angle at the vertex $z_{j}$ satisfies
$0<\theta_{j}<\frac{\pi}{2}$ which implies that
$h(Z)=r(Z)=r_{j+1}(Z)=\sin(\theta_{j})<1$. ∎

######  Proposition 4.4.

The height function $h\colon S(4)\to\mathbb{R}$ is continuously extended to
the closure $\overline{S(4)}$ such that $h(Z)=0$ if and only if $Z\in\partial
S(4)$.

###### Proof.

For every $Z=[z_{1},z_{2},z_{3},z_{4}]\in\overline{S(4)}$, we have that
$h(Z)=0$ if and only if there is an index $j\in\\{1,2,3,4\\}$, such that
$r_{j}(Z)=d(z_{j},{\mathcal{L}}_{j})=0$, with
${\mathcal{L}}_{j}=\overline{z_{j+1}z_{j+2}}\cup\overline{z_{j+2}z_{j+3}}$ the
opposite polygonal to the vertex $z_{j}$. Thus $h(Z)=0$ if and only if
$z_{j}\in{\mathcal{L}}_{j}$, for some $j\in\\{1,2,3,4\\}$, and this is true if
and only if $Z\in\partial S(4)$. We may observe that the property $\ell(Z)>0$
holds for every $Z\in\overline{S(4)}$. ∎

######  Remark 4.5.

We can observe that $h(Z)=h(\sigma(Z))=h(\tau(Z))$, for all $Z\in P(4)$, so
that the level sets of the height function $h\colon S(4)\to\mathbb{R}$ are
invariant under the action of the dihedral group $\langle\sigma,\tau\rangle$.

######  Example 4.6.

The analogous height function for the case of triangles is given by $h\colon
S(3)\to\mathbb{R}$, $h(Z)=\frac{r(Z)}{\ell(Z)}$, with $Z=[z_{1},z_{2},z_{3}]$,
$r(Z)=\min_{j}d(z_{j},\overline{z_{j+1}z_{j+2}})$ and
$\ell(Z)=\max_{j}|z_{j}-z_{j+1}|$. In this case the function $h$ also has a
global maximum at the equilateral triangle
$\mathfrak{R}_{3}=[0,1,(1+i\sqrt{3})/2]$, where it attains the value
$\sqrt{3}/2$, and can be continuously extended by the zero function in
$\partial S(3)=\mathbb{R}\cup\\{\infty\\}$. If we consider the transverse
curve $\mu:[0,\infty)\rightarrow S(3)$ given by
$\mu(t)=[0,1,1/2+ie^{t}\sqrt{3}/2]$, then the level curves of $h$ together
with the curve $\mu(t)$ provide polar coordinates to the space $S(3)$, giving
us the homeomorphism with the open cone $S(3)\cong
C(\mathbb{S}^{1})\cong\mathbb{R}^{2}$, see Figure 3.

$[0,1,0]$$[0,1,1]$$\partial
S(3)\cong\mathbb{R}\cup\\{\infty\\}$$S(3)\cong\mathbb{H}$$\bullet$$\bullet$$\bullet$$\mathfrak{R}_{3}$
Figure 3: Three level curves of the height function $h\colon
S(3)\to\mathbb{R}$, from the inside out, these correspond to
$h^{-1}(\sin(\frac{\pi}{4}))$, $h^{-1}(\sin(\frac{\pi}{6}))$, and
$h^{-1}(\sin(\frac{\pi}{12}))$.

######  Remark 4.7.

The three regions bounded by the vertical line and the two arcs of circles of
radius $1$ passing through $0,1\in\mathbb{R}\cup\\{\infty\\}\cong\partial
S(3)$ in Figure 3, correspond to the triangles with fixed largest side, that
is, those regions are the sets

$\mathrm{cl}({\mathcal{U}}_{j})=\\{Z\in S(3)\colon\ell(Z)=\ell_{j}(Z)\\}.$

The intersection of each one of these sets with a level set is a topological
segment $h^{-1}(s)\cap\mathrm{cl}({\mathcal{U}}_{j})\cong\mathbb{D}^{1}$ and
thus, $h^{-1}(s)$ is a union of three segments with pairwise intersection in a
single point. This trivial fact on the space of triangles generalizes to the
space of quadrilaterals in a more complicated combinatorial decomposition.

##  5 Level sets of the height function

In this section we prove that for all $s\in(0,1)$, the level set $h^{-1}(s)$
is homeomorphic to the sphere $\mathbb{S}^{3}$. As this section is much more
technical than the previous ones, before we start, we summarize the main
points of the proof in a general strategy:

  1. 1.

Introduce the sets in $S(4)$, where $\ell$ and $r$ are uniquely reached, this
is Definition 5.3. In $S(3)$ there are three such sets, completely determined
by $\ell$, as seen in Remark 4.7.

  2. 2.

Show that for all $s\in(0,1)$, the level set $h^{-1}(s)$, intersected with the
closure of each of the sets in the previous step, is homeomorphic to the
closed ball $\mathbb{D}^{3}$, this is Lemma 5.7. In the case of $S(3)$, it is
clear that the corresponding intersections are topological segments, Figure 3.

  3. 3.

Describe the decomposition of $h^{-1}(s)$ into closed balls, by codifying the
combinatorics of the intersections in a graph, this is Lemma 5.10. The
corresponding behaviour in $S(3)$ is the fact that $h^{-1}(s)$, as a union of
closed segments, is a topological triangle, and the non-trivial intersection
of the segments are described by the dual polygon which is again a triangle.

  4. 4.

Conclude that $h^{-1}(s)\cong\mathbb{S}^{3}$ by constructing a decomposition
of $\mathbb{S}^{3}$ into closed balls which is equivalent to the one given for
$h^{-1}(s)$. This is given in Theorem 5.12.

######  Definition 5.1.

Let $Z=(z_{1},z_{2},z_{3},z_{4})\in\mathbb{S}(4)$. Then

  1. (a)

we say that $\ell(Z)$ is unique if there exists $j\in\\{1,2,3,4\\}$, such that
$\ell(Z)=|z_{j}-z_{j+1}|$ and for $k\neq j$,
$|z_{k}-z_{k+1}|<|z_{j}-z_{j+1}|$.

  2. (b)

we say that $r(Z)$ is unique if there exist $j\in\\{1,2,3,4\\}$ and
$w\in{\mathcal{L}}_{j}$, such that $r(Z)=|z_{j}-w|$ and
$r(Z)<|z_{k}-w^{\prime}|$, for every $k\in\\{1,2,3,4\\}$ and
$w^{\prime}\in{\mathcal{L}}_{k}$ such that
$\\{z_{k},w^{\prime}\\}\neq\\{z_{j},w\\}$.

Notice that in a quadrilateral, $r$ cannot be realized as the distance between
two opposite vertices, because if two non-adjacent vertices are close, then
$r$ is reached between one of them and a side non-adjacent to it. However, $r$
could be reached from a vertex to a two distinct non-adjacent sides.

######  Proposition 5.2.

If $Z\in S(4)$ is such that $\ell(Z)=|z_{j}-z_{j+1}|$ is unique, then
$d(z_{j},\overline{z_{j+1}z_{j+2}})>r(Z)$ and
$d(z_{j+1},\overline{z_{j+3}z_{j}})>r(Z)$.

###### Proof.

We can assume that $j=1$ and prove that $d(z_{2},\overline{z_{4}z_{1}})>r(Z)$,
as the other cases are analogous. We work with the representative
$Z=[0,1,z_{3},z_{4}]$.

If the interior angle at the vertex $z_{1}$, denoted by $\theta_{1}$ here,
satisfies $\theta_{1}\geq\frac{\pi}{2}$, then
$d(z_{2},\overline{z_{4}z_{1}})=|z_{2}-z_{1}|=\ell(Z)=1>r(Z)$, by Proposition
4.3.

For $\theta_{1}<\frac{\pi}{2}$. If $\measuredangle
z_{1}z_{4}z_{2}\geq\frac{\pi}{2}$, then $r(Z)\leq
d(z_{4},\overline{z_{1}z_{2}})=Im(z_{4})<|z_{2}-z_{4}|=d(z_{2},\overline{z_{4}z_{1}})$.
If $\measuredangle z_{1}z_{4}z_{2}<\frac{\pi}{2}$, then there exists a point
$w\in\overline{z_{4}z_{1}}$, such that $\measuredangle
z_{1}wz_{2}=\frac{\pi}{2}$. We conclude that
$d(z_{2},\overline{z_{4}z_{1}})=d(z_{2},w)=\ell(Z)\sin\theta_{1}>|z_{4}-z_{1}|\sin\theta_{1}=d(z_{4},\overline{z_{1}z_{2}})\geq
r(Z)$. ∎

######  Definition 5.3.

Let $j\in\\{1,2,3,4\\}$. The set of quadrilaterals $Z\in S(4)$ such that
$\ell(Z)=|z_{j}-z_{j+1}|$ and $r(Z)$ are unique, is divided in the next three
subsets (Figure 4):

  1. (a)

${\mathcal{U}}_{j}:=\\{r(Z)=d(z_{j+1},\overline{z_{j+2}z_{j+3}})\text{ or to
}r(Z)=d(z_{j+2},\overline{z_{j}z_{j+1}})\\}.$

  2. (b)

${\mathcal{V}}_{j}:=\\{r(Z)=d(z_{j+3},\overline{z_{j}z_{j+1}})\text{ or
}r(Z)=d(z_{j},\overline{z_{j+2}z_{j+3}})\\}$.

  3. (c)

${\mathcal{W}}_{j}:=\\{r(Z)=d(z_{j+2},\overline{z_{j+3}z_{j}})\text{ or
}r(Z)=d(z_{j+3},\overline{z_{j+1}z_{j+2}})\\}.$

For all $j\in\\{1,2,3,4\\}$, the sets ${\mathcal{U}}_{j}$, ${\mathcal{V}}_{j}$
and ${\mathcal{W}}_{j}$ are disjoint and every simple and positively oriented
quadrilateral with $\ell$ and $r$ unique, belongs to the union $\bigcup_{1\leq
j\leq 4}({\mathcal{U}}_{j}\cup{\mathcal{V}}_{j}\cup{\mathcal{W}}_{j})\subset
S(4)$ (Proposition 5.2).

$\ell$$r$$\bullet$$\bullet$$\bullet$$\bullet$$z_{1}$$z_{2}$$z_{3}$$z_{4}$$(a)$$\ell$$r$$\bullet$$\bullet$$\bullet$$\bullet$$z_{1}$$z_{2}$$z_{3}$$z_{4}$$(b)$$\ell$$r$$\bullet$$\bullet$$\bullet$$\bullet$$z_{1}$$z_{2}$$z_{3}$$z_{4}$$(c)$
Figure 4: Three quadrilaterals with largest side uniquely reached at
$\overline{z_{1}z_{2}}$. The quadrilaterals in $(a)$, $(b)$ and $(c)$ belong
to ${\mathcal{U}}_{1}$, ${\mathcal{V}}_{1}$ and ${\mathcal{W}}_{1}$
respectively.

######  Remark 5.4.

The sets ${\mathcal{U}}_{j},{\mathcal{V}}_{j}$ and ${\mathcal{W}}_{j}$ satisfy
$\sigma^{k}({\mathcal{U}}_{j})={\mathcal{U}}_{j+k}$,
$\sigma^{k}({\mathcal{V}}_{j})={\mathcal{V}}_{j+k}$,
$\sigma^{k}({\mathcal{W}}_{j})={\mathcal{W}}_{j+k}$,
$\tau({\mathcal{U}}_{1})={\mathcal{V}}_{1}$, and
$\tau({\mathcal{W}}_{1})={\mathcal{W}}_{1}$. In particular, we have that
$\sigma^{j}\circ\tau\circ\sigma^{-j}({\mathcal{U}}_{j})={\mathcal{V}}_{j}$ and
$\sigma^{j}\circ\tau\circ\sigma^{-j}({\mathcal{W}}_{j})={\mathcal{W}}_{j}$.

######  Proposition 5.5.

The set
$\bigcup_{j=1}^{4}\left({\mathcal{U}}_{j}\cup{\mathcal{V}}_{j}\cup{\mathcal{W}}_{j}\right)$
is open and dense in $S(4)$.

###### Proof.

Observe that every simple quadrilateral can be approximated by a quadrilateral
with $\ell$ and $r$ unique, so that
$\bigcup_{j=1}^{4}\left({\mathcal{U}}_{j}\cup{\mathcal{V}}_{j}\cup{\mathcal{W}}_{j}\right)\subset
S(4)$ is dense.

Since the functions $r_{j}$ and $\ell_{k}$ are continuous, then if the
inequalities

$\ell_{j}(Z)<\ell_{k}(Z),\qquad r_{j}(Z)<r_{k}(Z)$

hold in a quadrilateral $Z$, then the same inequalities remain valid in an
open neighborhood of $Z$. If the minimum satisfies $r(Z)=r_{j}(Z)=r_{j+1}(Z)$,
i.e. $r(Z)=|z_{j}-z_{j+1}|$, then it can be verified that $r$ remains unique
in a neighbourhood of $Z$ and the result follows. ∎

######  Remark 5.6.

We may observe that the functions $r$ and $\ell$ have unique expressions as
$r_{k}$ and $\ell_{l}$, for some $k,l\in\\{1,2,3,4\\}$, when restricted to one
of the sets
$\\{\mathrm{cl}({\mathcal{U}}_{j}),\mathrm{cl}({\mathcal{V}}_{j}),\mathrm{cl}({\mathcal{W}}_{j}):j\in\\{1,2,3,4\\}\\}$
and this implies in particular that the height function $h$ is differentiable
in such restrictions.

######  Lemma 5.7.

If $j\in\\{1,2,3,4\\}$ and $s\in(0,1)$, then the closures
$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{j})$,
$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{V}}_{j})$ and
$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{j})$ are homeomorphic to the closed
disc $\mathbb{D}^{3}$.

###### Proof.

From Remarks 4.5 and 5.4, it is enough to show the result for
$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})$ and
$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1})$. We will construct these sets
separately by working on the representatives $Z=[0,1,z_{3},z_{4}]$. By
Definition 5.3, in both cases the constrains $\ell(Z)=1$, $r(Z)=s$,
$s\leq|z_{4}|\leq 1$ and $s\leq|z_{3}-1|\leq 1$ must be satisfied.

For each set $\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})$ and
$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1})$, the proof will be carried out
in three steps: 1 - Show that the set $\Omega$ of possible points for $z_{4}$
is homeomorphic to $\mathbb{D}^{2}$ by giving a precise description of it. 2 -
For a fixed $z_{4}\in\Omega$, determine the curve $\gamma_{z_{4}}$ of possible
values for $z_{3}$. 3 - Prove that steps 1 and 2 endow the set with a
structure of an $I$-bundle over $\Omega$ (with some collapsed fibers), which
will imply the homeomorphism with $\mathbb{D}^{3}$.

Construction of $\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})$. Suppose that
$[0,1,z_{3},z_{4}]\in\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})$.

Step 1. By Definition 5.3.$(a)$, $s=d(z_{3},\overline{01})$ or
$s=d(1,\overline{z_{3}z_{4}})$, then either the vertex $z_{3}$ belongs to the
segment $\overline{(is)(1+is)}$ or to the circle
${\mathcal{C}}_{1}=\\{|z-1|=s\\}$, or the side $\overline{z_{3}z_{4}}$ is
tangent to ${\mathcal{C}}_{1}$. These conditions and the inequalities
$d(0,\overline{z_{3}z_{4}})\geq s$ and $d(z_{4},\overline{01})\geq s$, imply
that $\Omega_{{\mathcal{U}}s}=\\{z\in\mathbb{D}^{2}\colon Im(z)\geq s\\}$ is
the set of possible points for $z_{4}$, see Figure 5.

Step 2. Fix a point $z_{4}\in\Omega_{{\mathcal{U}}s}$. Let
$p_{z_{4}}\in{\mathcal{C}}_{1}$ be the point satisfying the condition
$Re(p_{z_{4}})\geq 1$ and such that $\overline{p_{z_{4}}z_{4}}$ is tangent to
${\mathcal{C}}_{1}$. Denote by
${\mathcal{A}}_{z_{4}}\subset{\mathcal{C}}_{1}$, the arc in the negative sense
from $1+is$ to $p_{z_{4}}$, and
${\mathcal{R}}_{z_{4}}=\\{t(p_{z_{4}}-z_{4})+p_{z_{4}}\colon t\geq 0\\}$ the
ray starting at $p_{z_{4}}$. The set $\gamma_{z_{4}}$ of possible values for
$z_{3}$ is

$\\{z\in\overline{(is)(1+is)}\cup{\mathcal{A}}_{z_{4}}\cup{\mathcal{R}}_{z_{4}}\colon
d(z_{4},\overline{1z})\geq s,\ d(z,\overline{z_{4}0})\geq
s,~{}\text{and}~{}~{}|z-z_{4}|\leq 1\\}.$

Observe that for $z_{4}\in\Omega_{{\mathcal{U}}s}$ such that
$|z_{4}-1|=\sqrt{2}s$ (equivalently the circles ${\mathcal{C}}_{1}$ and
$\\{|z-z_{4}|=s\\}$ are orthogonal), $d(z_{4},\overline{1z})=s$ if and only if
$d(z,\overline{z_{4}0})=s$, and therefore, the constraints
$d(z_{4},\overline{1z})\geq s$ and $d(z,\overline{z_{4}0})\geq s$ are
equivalent. Then, for the case $|z_{4}-1|\geq\sqrt{2}s$ it is satisfied that

$\gamma_{z_{4}}=\\{z\in\overline{(is)(1+is)}\cup{\mathcal{A}}_{z_{4}}\cup{\mathcal{R}}_{z_{4}}\colon
d(z,\overline{z_{4}0})\geq s,~{}\text{and}~{}~{}|z-z_{4}|\leq 1\\},$

(illustrated by the construction on the points $z_{4}$ and
$z_{4}^{\prime\prime}$ in Figure 5) and for the case $|z_{4}-1|\leq\sqrt{2}s$,
it is satisfied that

$\gamma_{z_{4}}=\\{z\in\overline{(is)(1+is)}\cup{\mathcal{A}}_{z_{4}}\cup{\mathcal{R}}_{z_{4}}\colon
d(z_{4},\overline{1z})\geq s,~{}\text{and}~{}~{}|z-z_{4}|\leq 1\\},$

(illustrated by the construction on the point $z_{4}^{\prime}$ in Figure 5).
As the curve $\gamma_{z_{4}}$ is constructed by an arc and two segments
tangent to it, it is a $C^{1}$-curve whose construction only depends on the
point $z_{4}$ and the parameter $s\in(0,1)$.

$z^{\prime}_{4}$$\bullet$$\bullet$$\bullet$$\bullet$$\gamma_{z^{\prime}_{4}}$$\\{|z\\!-\\!z^{\prime}_{4}|\\!=\\!s\\}$$s$$\bullet$$z_{4}$$\bullet$$\bullet$$\bullet$$p_{z_{4}}$$\gamma_{z_{4}}$$\bullet$$z^{\prime\prime}_{4}$$s$$\gamma_{z^{\prime\prime}_{4}}$$\bullet$$\bullet$$is$$\bullet$$\bullet$$q_{{}_{1}}$$\bullet$$q_{{}_{2}}$$\bullet$$q_{{}_{1}}\\!\\!+\\!1$$-1$$\bullet$$i$$\bullet$$0$$\bullet$$1$$\bullet$$2$$\bullet$$\Omega_{{\mathcal{U}}s}$$\bullet$$1\\!\\!+\\!\\!s\sqrt{2}$$\bullet$$1\\!\\!-\\!\\!s$
Figure 5: The shaded region is the set $\Omega_{{\mathcal{U}}s}$. For the
points $z_{4},z^{\prime}_{4},z^{\prime\prime}_{4}\in\Omega_{{\mathcal{U}}s}$,
the curves $\gamma_{z_{4}},\gamma_{z^{\prime}_{4}}$ and
$\gamma_{z^{\prime\prime}_{4}}$ of possible values for the vertex $z_{3}$ are
represented with the darkest lines.

Step 3. As noted above, for every $z_{4}\in\Omega_{{\mathcal{U}}s}$,
$\gamma_{z_{4}}$ is a $C^{1}$-curve and so, it can be parametrized by a
rescaling of the arc-length as a continuous function
$\gamma_{z_{4}}\colon[0,1]\rightarrow\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})$.
Observe that for the points $q_{1}=e^{i\arcsin(s)}$ and
$q_{2}=-e^{-i\arcsin(s)}$ (Figure 5), we have that
$\gamma_{q_{1}}=\\{q_{1}+1\\}$ and $\gamma_{q_{2}}=\\{q_{2}+1\\}$, and these
are the only points with collapsed curves. The continuous function

$\Omega_{{\mathcal{U}}s}\times[0,1]\rightarrow\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1}),\qquad(z_{4},t)\rightarrow[0,1,\gamma_{z_{4}}(t),z_{4}],$

is a homeomorphism when restricted to
$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})\smallsetminus\\{[0,1,q_{1}+1,q_{1}],[0,1,q_{2}+1,q_{2}]\\}$.
This map endows the set $\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})$ with the
structure of an $I$-bundle with two collapsed fibers at the points $q_{1}$ and
$q_{2}$ in the boundary of $\Omega_{{\mathcal{U}}s}$. Then, we conclude that
$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})$ is homeomorphic to
$\mathbb{D}^{3}$.

Construction of $\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1})$. Suppose that
$[0,1,z_{3},z_{4}]\in\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1})$.

Step 1. Since $s=d(z_{3},\overline{z_{4}0})$ or $s=d(z_{4},\overline{1z_{3}})$
(Definition 5.3.$(c)$), then the vertices $z_{3}$ and $z_{4}$ have analogous
roles. As $d(z_{4},\overline{01})\geq s$, then for $0\leq Re(z_{4})\leq 1$,
necessarily $Im(z_{4})\geq s$. In order to determine the possible values for
$z_{4}$ with $Re(z_{4})\leq 0$, we analyze the constraints of $z_{4}$ when
belonging to the radii $r_{\alpha}=\\{te^{i\alpha}\colon 0\leq t\leq 1\\}$
with argument $\alpha\geq\frac{\pi}{2}$. We denote by
${\mathcal{C}}_{0}=\\{|z|=s\\}$ the circle of radius $s$ centered at $0$, and
by $L_{\alpha}=\\{(t-is)e^{i\alpha}\colon t\in\mathbb{R}\\}$ the line parallel
to $r_{\alpha}$, at distance $s$ which is tangent to ${\mathcal{C}}_{0}$ at
$-ise^{i\alpha}$. The tangent to ${\mathcal{C}}_{0}$ at $se^{i\alpha}$
intersects $L_{\alpha}$ in the point $p_{\alpha}=(s-is)e^{i\alpha}$ (Figure
6.$(a)$). There are three cases:

  1. i.

If $|p_{\alpha}-1|\leq 1$, then
$[0,1,p_{\alpha},se^{i\alpha}]\in\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1})$.
The set of possible values for $z_{4}$ in $r_{\alpha}$ is
$\\{te^{i\alpha}\colon s\leq t\leq 1\\}$. This case occurs for arguments
$\frac{\pi}{2}\leq\alpha\leq\alpha_{0}$, with

$\alpha_{0}=\arccos\left(-\sqrt{\frac{1-s\sqrt{2-s^{2}}}{2}}\right),$

the angle such that $|p_{\alpha_{0}}-1|=1$ holds (see Figure 6.$(a)$).

  2. ii.

If $|p_{\alpha}-1|>1$, then $p_{\alpha}$ is not a valid value for $z_{3}$
anymore. Let $w_{\alpha}\in L_{\alpha}\cap\\{|z-1|=1\\}$ be the intersection
point with $\textrm{Im}(w_{\alpha})>0$, and $q_{\alpha}\in r_{\alpha}$ so that
the segment $\overline{q_{\alpha}w_{\alpha}}$ is tangent to
${\mathcal{C}}_{0}$ (Figure 6.$(b)$). If

$\lambda_{\alpha}=\cos(\alpha)+\sqrt{\cos(\alpha)^{2}+s(2\sin(\alpha)-s)},$

then $w_{\alpha}=(\lambda_{\alpha}-is)e^{i\alpha}$ and
$q_{\alpha}=t_{\alpha}e^{i\alpha}\in r_{\alpha}$, with
$t_{\alpha}=\frac{s^{2}+\lambda_{\alpha}^{2}}{2\lambda_{\alpha}}$. The
quadrilateral $[0,1,q_{\alpha},w_{\alpha}]$ belongs to
$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1})$, and the set of possible values
for $z_{4}$ in $r_{\alpha}$ is $\\{te^{i\alpha}\colon t_{\alpha}\leq t\leq
1\\}$. This case occurs for arguments $\alpha_{0}<\alpha\leq\pi-\arcsin(s)$.
Notice that for $\alpha=\alpha_{0}$, $w_{\alpha}=p_{\alpha}$ and
$q_{\alpha}=se^{i\alpha}$, and in the limit angle $\alpha=\pi-\arcsin(s)$,
$q_{\alpha}=-e^{-i\arcsin(s)}=q_{2}$ (see Figure 6.$(b)$).

  3. iii.

If $\alpha>\pi-\arcsin(s)$, then for all $s\leq t\leq 1$, the tangent line to
${\mathcal{C}}_{0}$ through $te^{i\alpha}\in r_{\alpha}$ intersects
$L_{\alpha}$ outside the circle $\\{|z-1|=1\\}$. We conclude that there are no
allowed values of $z_{4}$ in the radii $r_{\alpha}$.

$(a)$$\bullet$$-s$$p_{\alpha}$$\bullet$$p_{\alpha_{0}}$$\bullet$$-ise^{i\alpha_{0}}$$\bullet$$-ise^{i\alpha}$$\bullet$$e^{i\alpha}$$\bullet$$e^{i\alpha_{0}}$$\bullet$$\bullet$$\bullet$$0$$\bullet$$1$$\bullet$$-1$$\bullet$$i$$\bullet$$q_{{}_{2}}$$\bullet$$q_{{}_{1}}$$\bullet$$r_{\alpha}$$r_{\alpha_{0}}$$L_{\alpha}$$L_{\alpha_{0}}$$(b)$$-s$$\bullet$$\bullet$$q_{\alpha}$$\bullet$$w_{\alpha}$$\bullet$$\bullet$$-ise^{i\alpha}$$\bullet$$e^{i\alpha}$$\bullet$$p_{\alpha_{0}}$$\bullet$$-ise^{i\alpha_{0}}$$\bullet$$1$$\bullet$$i$$0$$\bullet$$-1$$\bullet$$q_{{}_{2}}$$\bullet$$q_{{}_{1}}$$\bullet$$\bullet$$e^{i\alpha_{0}}$$\bullet$$r_{\alpha}$$L_{\alpha}$$r_{\alpha_{0}}$$L_{\alpha_{0}}$
Figure 6: The darkest lines represent the set of possible values for the
vertex $z_{4}$ in the radius $r_{\alpha}$ with
$\frac{\pi}{2}\leq\alpha\leq\pi-\arcsin(s)$. The cases $(a)$ and $(b)$
correspond to $\frac{\pi}{2}\leq\alpha\leq\alpha_{0}$ and
$\alpha_{0}\leq\alpha\leq\pi-\arcsin(s)$ respectively. The construction for
the angle $\alpha_{0}$ is represented in light gray in both cases.

We may define $t_{\alpha}=\frac{s}{\sin{\alpha}}$ for
$\arcsin{s}\leq\alpha\leq\frac{\pi}{2}$, and $t_{\alpha}=s$ for
$\frac{\pi}{2}\leq\alpha\leq\alpha_{0}$, so that the set of possible values
for $z_{4}$ is (Figure 7):

$\Omega_{{\mathcal{W}}s}=\\{te^{i\alpha}\in r_{\alpha}\colon t_{\alpha}\leq
t\leq 1,\,\arcsin{s}\leq\alpha\leq\pi-\arcsin(s)\\}.$

Step 2. Fix a point $z_{4}\in\Omega_{{\mathcal{W}}s}$. Let
${\mathcal{C}}_{z_{4}}=\\{|z-z_{4}|=s\\}$ and
$\\{q_{z_{4}}\\}=L_{\arg(z_{4})}\cap{\mathcal{C}}_{z_{4}}$. If
$p_{z_{4}}\in{\mathcal{C}}_{z_{4}}$ is the point such that the line containing
$\overline{1p_{z_{4}}}$ is tangent to ${\mathcal{C}}_{z_{4}}$ and does not
intersect the segment $\overline{z_{4}0}$, then
${\mathcal{A}}_{z_{4}}\subset{\mathcal{C}}_{z_{4}}$ denotes the arc in the
positive sense from $q_{z_{4}}$ to $p_{z_{4}}$ and
${\mathcal{R}}_{z_{4}}=\\{t(p_{z_{4}}-1)+p_{z_{4}}\colon t\geq 0\\}$ denotes
the ray tangent to ${\mathcal{C}}_{z_{4}}$ starting at $p_{z_{4}}$. The curve
$\gamma_{z_{4}}$ of possible values for $z_{3}$ is contained in the union
$\overline{(-ise^{i\arg(z_{4})})(q_{z_{4}})}\cup{\mathcal{A}}_{z_{4}}\cup{\mathcal{R}}_{z_{4}}$.
The boundaries of of $\gamma_{z_{4}}$ are determined different in two cases:

  1. i.

Suppose that $Im(z_{4})\geq s$. If $|z_{4}-1|\geq\sqrt{2}s$, we define

$\gamma_{z_{4}}=\\{z\in\overline{(-ise^{i\arg(z_{4})})(q_{z_{4}})}\cup{\mathcal{A}}_{z_{4}}\cup{\mathcal{R}}_{z_{4}}\colon
d(z,\overline{01})\geq s,~{}\text{and}~{}~{}|z-1|\leq 1\\},$

and if $|z_{4}-1|\leq\sqrt{2}s$, we define

$\gamma_{z_{4}}=\\{z\in\overline{(-ise^{i\arg(z_{4})})(q_{z_{4}})}\cup{\mathcal{A}}_{z_{4}}\cup{\mathcal{R}}_{z_{4}}\colon
d(1,\overline{zz_{4}})\geq s,~{}\text{and}~{}~{}|z-1|\leq 1\\}.$

We observe that the conditions $d(1,\overline{zz_{4}})\geq s$ and
$d(z,\overline{01})\geq s$ coincide when $|z_{4}-1|=\sqrt{2}s$.

  2. ii.

Suppose that $Im(z_{4})\leq s$. If $\omega_{z_{4}}\in L_{\arg(z_{4})}$ is the
point such that the line $\\{t\omega_{z_{4}}+(1-t)z_{4}:t\in\mathbb{R}\\}$ has
positive slope and is tangent ${\mathcal{C}}_{0}$, then

$\gamma_{z_{4}}=\\{z\in\overline{(-ise^{i\arg(z_{4})})(q_{z_{4}})}\cup{\mathcal{A}}_{z_{4}}\cup{\mathcal{R}}_{z_{4}}\colon
Im(z)\geq Im(\omega_{z_{4}}),~{}\text{and}~{}~{}|z-1|\leq 1\\}.$

In the limit case, when $z_{4}\in\Gamma$, then with our previous notation
$z_{4}=q_{\arg(z_{4})}$ and $\omega_{z_{4}}=\omega_{\arg(z_{4})}$, see Figure
6.

As before, the curve $\gamma_{z_{4}}$ is constructed by an arc and two
segments tangent to it, so it is a $C^{1}$-curve whose construction only
depends on the point $z_{4}$ and the parameter $s$.

Figure 7: The shaded region is the set $\Omega_{{\mathcal{W}}s}$. For the
points $z_{4},z^{\prime}_{4},z^{\prime\prime}_{4}\in\Omega_{{\mathcal{W}}s}$,
the curves $\gamma_{z_{4}},\gamma_{z^{\prime}_{4}}$ and
$\gamma_{z^{\prime\prime}_{4}}$ of possible values for the vertex $z_{3}$ are
represented with the darkest lines. The curve $\Gamma=\\{q_{\alpha}\in
r_{\alpha}\colon\alpha_{0}\leq\alpha\leq\pi-\arcsin(s)\\}$ is the boundary
section between the points $q_{2}$ and $se^{i\alpha_{0}}$.

Step 3. For every $z\in\Omega_{{\mathcal{W}}s}$, the curve $\gamma_{z}$ can be
parametrized by a rescaling of the arc-length as a continuous function
$\gamma_{z}\colon[0,1]\rightarrow\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1})$.
We have a projection

$\Omega_{{\mathcal{W}}s}\times[0,1]\rightarrow\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1}),\qquad(z,t)\rightarrow[0,1,\gamma_{z}(t),z],$

where the collapsed fibers correspond to the points in $\Gamma\cup\\{q_{1}\\}$
(Figure 7). As the set $\Gamma\cup\\{q_{1}\\}$ is contained in
$\partial\Omega_{{\mathcal{W}}s}$, a homeomorphism
$\Omega_{{\mathcal{W}}s}\times[0,1]\cong\mathbb{D}^{3}$ induces a
homeomorphism from $\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1})$ to
$\mathbb{D}^{3}$. ∎

######  Remark 5.8.

Three consequences of the proof of Lemma 5.7.

  1. $I$.

For $1\leq j\leq 4$, the sets
$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{j}),\mathrm{cl}(h^{-1}(s)\cap{\mathcal{V}}_{j}),\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{j})$
are $I$-bundles with basis homeomorphic to the disc $\mathbb{D}^{2}$ and with
collapsed fibers at some closed regions on their boundaries.

  2. $II$.

The boundaries of the balls
$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{j}),\mathrm{cl}(h^{-1}(s)\cap{\mathcal{V}}_{j}),\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{j})$
correspond to the loss of uniqueness of the values $\ell$ or $r$ (see
Definitions 5.1 and 5.3). Then for
$A\in\\{{\mathcal{U}}_{j},{\mathcal{W}}_{j},{\mathcal{V}}_{j}:1\leq j\leq
4\\}$, it is satisfies that

$\partial\mathrm{cl}(h^{-1}(s)\cap A)=\bigcup_{B\neq
A}\mathrm{cl}(h^{-1}(s)\cap B)\cap\mathrm{cl}(h^{-1}(s)\cap A),$

where the union is over the set
$\\{{\mathcal{U}}_{j},{\mathcal{W}}_{j},{\mathcal{V}}_{j}:1\leq j\leq 4\\}$.
The decompositions of the boundaries of the balls given by these intersections
are permuted by the action of the dihedral group $\langle\sigma,\tau\rangle$.

  3. $III$.

The points $q_{1}=e^{i\arcsin(s)}$ and $q_{2}=-e^{-i\arcsin(s)}$ (see Figures
5 and 7) are special because the quadrilaterals $Q_{1}=[0,1,q_{1}+1,q_{1}]$
and $Q_{2}=[0,1,q_{2}+1,q_{2}]$ satisfy

$\\{Q_{1},Q_{2}\\}=h^{-1}(s)\cap\bigcap_{j=1}^{4}\mathrm{cl}({\mathcal{U}}_{j})\cap\mathrm{cl}({\mathcal{V}}_{j})\cap\mathrm{cl}({\mathcal{W}}_{j}).$

Moreover, the action of the group $\langle\sigma,\tau\rangle$ in this points
reduces to $\sigma(Q_{1})=\tau(Q_{1})=Q_{2}$,
$\sigma(Q_{2})=\tau(Q_{2})=Q_{1}$ and the fibers of the $I$-bundle structures
on each sets collapse on these quadrilaterals.

In order to give a homeomorphism between $h^{-1}(s)$ and $\mathbb{S}^{3}$, we
describe the intersection pattern of the twelve balls of Lemma 5.7 through
their boundaries. These intersections are codified in a combinatorial
structure that we encounter naturally in $\mathbb{S}^{3}$, in the remainder of
the section we describe this combinatorial pattern.

######  Definition 5.9.

A bigon is a topological closed disc with two marked points on the boundary.
The marked points are the vertices of the bigon and the two arcs in the
boundary bounded by the vertices are the sides of the bigon. A decomposition
of a surface into bigons is a covering of topological bigons such that the
intersection of two such bigons is on the vertices or in one of its sides.

######  Lemma 5.10.

The spheres $\partial\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{j})$,
$\partial\mathrm{cl}(h^{-1}(s)\cap{\mathcal{V}}_{j})$ and
$\partial\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{j})$ admit a decomposition
into four bigons determined by the intersection with their neighboring
spheres. More precisely, the graph $\mathcal{G}$ in Figure 8 represents such a
decomposition as follows:

  1. 1.

each vertex of $\mathcal{G}$ represents a ball in

$\big{\\{}\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{j}),\mathrm{cl}(h^{-1}(s)\cap{\mathcal{V}}_{j}),\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{j}):1\leq
j\leq 4\big{\\}},$

  2. 2.

each edge of ${\mathcal{G}}$ represents a bigon which is the intersection of
the boundaries of the balls corresponding to its vertices.

  3. 3.

each edge of $\mathcal{G}$ is contained in two cycles of the graph of length
three and four respectively, those cycles represent the intersection of the
balls and determine the sides of the bigon.

The quadrilaterals $Q_{1}$ and $Q_{2}$ are the vertices of all the bigons and
the bigon decomposition is preserved by the action of the group
$\langle\sigma,\tau\rangle$, see Remark 5.8.$III$.

$\bullet$$\bullet$$\bullet$$\bullet$${\mathcal{V}}_{3}$${\mathcal{W}}_{4}$${\mathcal{U}}_{4}$${\mathcal{W}}_{3}$$\bullet$$\bullet$$\bullet$$\bullet$${\mathcal{V}}_{1}$${\mathcal{W}}_{1}$${\mathcal{U}}_{2}$${\mathcal{W}}_{2}$$\bullet$${\mathcal{V}}_{2}$$\bullet$${\mathcal{U}}_{3}$$\bullet$${\mathcal{U}}_{1}$$\bullet$${\mathcal{V}}_{4}$

Figure 8: Graph ${\mathcal{G}}$ representing the bigon decomposition of the
boundary spheres of the balls
$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{j}),\mathrm{cl}(h^{-1}(s)\cap{\mathcal{V}}_{j}),$
and $\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{j})$.

###### Proof.

Let $A\in\\{{\mathcal{U}}_{j},{\mathcal{W}}_{j},{\mathcal{V}}_{j}\colon 1\leq
j\leq 4\\}$. The result follows if we prove that the intersections
$\mathrm{cl}(h^{-1}(s)\cap B)\cap\mathrm{cl}(h^{-1}(s)\cap A)$ with
$B\in\\{{\mathcal{U}}_{j},{\mathcal{W}}_{j},{\mathcal{V}}_{j}\colon 1\leq
j\leq 4\\}$ define a bigon decomposition of the boundary sphere
$\partial\mathrm{cl}(h^{-1}(s)\cap A)$. As the bigon decomposition of each set
is defined by pairwise intersections, the Dihedral group
$\langle\sigma,\tau\rangle$ will act preserving such bigon decompositions.
Then, for Remarks 4.5 and 5.4, it is enough to prove the result for the balls
$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})$ and
$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1})$.

Bigon decomposition of $\partial\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})$.
The base space of the $I$-bundle structure in
$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})$ is the topological disc
$\Omega_{{\mathcal{U}}_{s}}=\\{z\in\mathbb{C}\colon|z|\leq 1,\
\textrm{Im}(z)\geq s\\}$ (Remark 5.8), and the fibers are $C^{1}$-curves
$\gamma_{z}$ parametrized by a rescaling of their arc-length (Figure 5). The
induced projection

$\Omega_{{\mathcal{U}}_{s}}\times[0,1]\rightarrow\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1}),\qquad(z,t)\rightarrow[0,1,\gamma_{z}(t),z],$

collapses the fibers at the points
$q_{1}=e^{i\arcsin(s)},q_{2}=-e^{-i\arcsin(s)}\in\partial\Omega_{{\mathcal{U}}_{s}}$.
Let
$\partial\Omega_{{\mathcal{U}}_{s}}={\mathcal{L}}_{s}\cup{\mathcal{A}}_{s}$
with
${\mathcal{L}}_{s}=\\{z\in\Omega_{{\mathcal{U}}_{s}}\colon\textrm{Im}(z)=s\\}$
and ${\mathcal{A}}_{s}=\\{z\in\Omega_{{\mathcal{U}}_{s}}\colon|z|=1\\}$, these
are the segment and the arc sections of the boundary of the shaded region in
Figure 5. Then the projection of the boundary
$\partial(\Omega_{{\mathcal{U}}_{s}}\times[0,1])=\Omega_{{\mathcal{U}}_{s}}\times\\{0,1\\}\cup\partial\Omega_{{\mathcal{U}}_{s}}\times[0,1]$
determines a bigon decomposition in the sphere
$\partial\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})$ with the four bigons:

  * •

${\mathcal{B}}_{1}$ the projection of
$\Omega_{{\mathcal{U}}_{s}}\times\\{0\\}$.

  * •

${\mathcal{B}}_{2}$ the projection of ${\mathcal{L}}_{s}\times[0,1]$

  * •

${\mathcal{B}}_{3}$ the projection of
$\Omega_{{\mathcal{U}}_{s}}\times\\{1\\}$

  * •

${\mathcal{B}}_{4}$ the projection of ${\mathcal{A}}_{s}\times[0,1]$

and with two vertices at $Q_{1},Q_{2}$ (Remark 5.8.$III$). From constructions
in the proof of Lemma 5.7, a quadrilateral
$Z\in\partial\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})$ satisfy that (see
Definition 5.3 and Figure 5):

  * •

if $Z\in{\mathcal{B}}_{1}$, then $r(Z)=d(z_{3},\overline{z_{4}z_{1}})$ and
therefore, $Z\in\partial\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1})$.

  * •

if $Z\in{\mathcal{B}}_{2}$, then $r(Z)=d(z_{4},\overline{z_{1}z_{2}})$ or
$r(Z)=d(z_{1},\overline{z_{3}z_{4}})$ and therefore,
$Z\in\partial\mathrm{cl}(h^{-1}(s)\cap{\mathcal{V}}_{1})$.

  * •

if $Z\in{\mathcal{B}}_{3}$, then $\ell(Z)=|z_{3}-z_{4}|$ and therefore,
$Z\in\partial\mathrm{cl}(h^{-1}(s)\cap{\mathcal{V}}_{3})$.

  * •

if $Z\in{\mathcal{B}}_{4}$, then $\ell(Z)=|z_{1}-z_{4}|$ and therefore,
$Z\in\partial\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{4})$.

The sides of the bigons are (Figure 9.$a)$):

  * •

${\mathcal{B}}_{1}\cap{\mathcal{B}}_{2}=h^{-1}(s)\cap\mathrm{cl}({\mathcal{U}}_{1})\cap\mathrm{cl}({\mathcal{V}}_{1})\cap\mathrm{cl}({\mathcal{W}}_{1})$,
the projection of ${\mathcal{L}}_{s}\times\\{0\\}$.

  * •

${\mathcal{B}}_{2}\cap{\mathcal{B}}_{3}=h^{-1}(s)\cap\mathrm{cl}({\mathcal{U}}_{1})\cap\mathrm{cl}({\mathcal{V}}_{1})\cap\mathrm{cl}({\mathcal{U}}_{3})\cap\mathrm{cl}({\mathcal{V}}_{3})$,
the projection of ${\mathcal{L}}_{s}\times\\{1\\}$.

  * •

${\mathcal{B}}_{3}\cap{\mathcal{B}}_{4}=h^{-1}(s)\cap\mathrm{cl}({\mathcal{U}}_{1})\cap\mathrm{cl}({\mathcal{V}}_{3})\cap\mathrm{cl}({\mathcal{W}}_{4})$,
the projection of ${\mathcal{A}}_{s}\times\\{1\\}$.

  * •

${\mathcal{B}}_{4}\cap{\mathcal{B}}_{1}=h^{-1}(s)\cap\mathrm{cl}({\mathcal{U}}_{1})\cap\mathrm{cl}({\mathcal{V}}_{4})\cap\mathrm{cl}({\mathcal{W}}_{1})\cap\mathrm{cl}({\mathcal{W}}_{4})$,
the projection of ${\mathcal{A}}_{s}\times\\{0\\}$.

$\boldsymbol{a)}$$\bullet$${\mathcal{V}}_{4}$$\bullet$${\mathcal{W}}_{1}$$\bullet$${\mathcal{V}}_{1}$$\bullet$${\mathcal{U}}_{3}$$\bullet$${\mathcal{V}}_{3}$$\bullet$${\mathcal{W}}_{4}$$\bullet$${\mathcal{U}}_{1}$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\boldsymbol{b)}$$\bullet$${\mathcal{W}}_{4}$$\bullet$${\mathcal{U}}_{1}$$\bullet$${\mathcal{V}}_{1}$$\bullet$${\mathcal{W}}_{2}$$\bullet$${\mathcal{U}}_{2}$$\bullet$${\mathcal{V}}_{4}$$\bullet$${\mathcal{W}}_{1}$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$
Figure 9: Quadrilaterals belonging to the sides of the bigon decompositions on
spheres $\partial\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})$ in $a)$ and
$\partial\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1})$ in $b)$. The sides of
the quadrilateral with maximum length $\ell$ are the darkest ones and the
length $r$ is represented with light gray segments.

Bigon decomposition of $\partial\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1})$.
Similarly to te previous case, the base of the $I$-bundle structure in
$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1})$ is the disc
$\Omega_{{\mathcal{W}}_{s}}$ in Figure 7, and the fibers are $C^{1}$-curves
$\gamma_{z}$ parametrized by a rescaling of their arc-length. The projection

$\Omega_{{\mathcal{W}}_{s}}\times[0,1]\to\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1}),\qquad(z,t)\to[0,1,\gamma_{z}(t),z]$

collapses the fibers of points in the set $\Gamma\cup\\{q_{1}\\}$. Let
$\partial\Omega_{{\mathcal{W}}_{s}}={\mathcal{A}}_{s}\cup{\mathcal{L}}_{s}$
with ${\mathcal{A}}_{s}=\\{z\in\Omega_{{\mathcal{U}}_{s}}\colon|z|=1\\}$ and

${\mathcal{L}}_{s}=\Gamma\cup\\{z\in\Omega_{{\mathcal{W}}_{s}}\colon|z|=s\\}\cup\\{z\in\Omega_{{\mathcal{W}}_{s}}\colon\textrm{Im}(z)=s,\textrm{Re}(z)\geq
0\\},$

be a division of the boundary $\partial\Omega_{{\mathcal{W}}_{s}}$ in two
curves joining $q_{1}$ with $q_{2}$, see Figure 7. The four bigons in the
sphere $\partial\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1})$ are:

  * •

${\mathcal{B}}_{1}$ the projection of
$\\{z\in\Omega_{{\mathcal{W}}_{s}}:\textrm{Im}(z)\geq s\\}\times\\{0\\}$

  * •

${\mathcal{B}}_{2}$ the projection of
$\\{z\in\Omega_{{\mathcal{W}}_{s}}:\textrm{Im}(z)\leq
s\\}\times\\{0\\}\cup{\mathcal{L}}_{s}\times[0,1]$

  * •

${\mathcal{B}}_{3}$ the projection of
$\Omega_{{\mathcal{W}}_{s}}\times\\{1\\}$

  * •

${\mathcal{B}}_{4}$ the projection of ${\mathcal{A}}_{s}\times[0,1]$

whose vertices are $Q_{1},Q_{2}$. These bigons satisfy

  * •

${\mathcal{B}}_{1}=h^{-1}(s)\cap\mathrm{cl}({\mathcal{W}}_{1})\cap\mathrm{cl}({\mathcal{U}}_{1})$

  * •

${\mathcal{B}}_{2}=h^{-1}(s)\cap\mathrm{cl}({\mathcal{W}}_{1})\cap\mathrm{cl}({\mathcal{V}}_{1})$

  * •

${\mathcal{B}}_{3}=h^{-1}(s)\cap\mathrm{cl}({\mathcal{W}}_{1})\cap\mathrm{cl}({\mathcal{U}}_{2})$

  * •

${\mathcal{B}}_{4}=h^{-1}(s)\cap\mathrm{cl}({\mathcal{W}}_{1})\cap\mathrm{cl}({\mathcal{V}}_{4})$

and the sides of the bigons are (Figure 9.$b)$):

  * •

${\mathcal{B}}_{1}\cap{\mathcal{B}}_{2}=h^{-1}(s)\cap\mathrm{cl}({\mathcal{U}}_{1})\cap\mathrm{cl}({\mathcal{V}}_{1})\cap\mathrm{cl}({\mathcal{W}}_{1})$,
the projection of
$\\{z\in\Omega_{{\mathcal{W}}_{s}}:\textrm{Im}(z)=s\\}\times\\{0\\}$.

  * •

${\mathcal{B}}_{2}\cap{\mathcal{B}}_{3}=h^{-1}(s)\cap\mathrm{cl}({\mathcal{W}}_{1})\cap\mathrm{cl}({\mathcal{V}}_{1})\cap\mathrm{cl}({\mathcal{U}}_{2})\cap\mathrm{cl}({\mathcal{V}}_{2})$,
the projection of ${\mathcal{L}}_{s}\times\\{1\\}$.

  * •

${\mathcal{B}}_{3}\cap{\mathcal{B}}_{4}=h^{-1}(s)\cap\mathrm{cl}({\mathcal{W}}_{1})\cap\mathrm{cl}({\mathcal{U}}_{2})\cap\mathrm{cl}({\mathcal{V}}_{4})$,
the projection of ${\mathcal{A}}_{s}\times\\{1\\}$.

  * •

${\mathcal{B}}_{4}\cap{\mathcal{B}}_{1}=h^{-1}(s)\cap\mathrm{cl}({\mathcal{W}}_{1})\cap\mathrm{cl}({\mathcal{V}}_{4})\cap\mathrm{cl}({\mathcal{U}}_{1})\cap\mathrm{cl}({\mathcal{W}}_{4})$
is the projection of ${\mathcal{A}}_{s}\times\\{0\\}$.

∎

######  Remark 5.11.

The sides of the bigons in the decomposition of Lemma 5.10 correspond to
either repetition of three extreme values $\ell$ or $r$, obtained by
intersecting three balls, or to repeat two times $\ell$ and two times $r$,
obtained by intersecting four balls. This is captured in the diagram of Figure
9.

######  Theorem 5.12.

If $s\in(0,1)$, then the level set $h^{-1}(s)\subset S(4)$ is homeomorphic to
the sphere $\mathbb{S}^{3}$.

###### Proof.

Let $\mathcal{C}_{0}\subset\mathbb{R}^{3}$ be the cube centered at $(0,0,0)$
and with vertices at $\\{(\pm 1,\pm 1,\pm 1)\\}$. The graph $\mathcal{G}_{0}$
obtained by joining the middle points of adjacent edges at each square face of
${\mathcal{C}}_{0}$ is a regular cuboctahedron with vertices in the set

$V=\\{(\pm 1,\pm 1,0),(\pm 1,0,\pm 1),(0,\pm 1,\pm 1)\\},$

see Figure 10.$a)$. For every $v\in V$, $P_{v}=\\{w\in\mathbb{R}^{3}:\langle
v,w\rangle=0\\}$ denotes the orthogonal plane to $v$. These planes determine
twelve closed discs $D_{j}$ in the sphere
$\mathbb{S}^{2}\subset\mathbb{R}^{3}$, such that, for all $1\leq j\leq 12$,
the infinite cone region

$C_{j}=\\{tw\colon w\in D_{j},\ t\geq 0\\}\subset\mathbb{R}^{3},$

contains one and only one edge of the cube $\mathcal{C}_{0}$, see Figure
10.$b)$. Then, the one point compactification $\mathbb{R}^{3}\cup\\{\infty\\}$
have a decomposition into twelve closed balls
$C_{j}^{*}:=C_{j}\cup\\{\infty\\}\cong\mathbb{D}^{3}$ that are the one point
compactifications of the cones $C_{j}$. This decomposition satisfies:

  1. 1.

There is exactly one ball $C_{j}^{*}$ for each vertex of the cuboctahedron
$\mathcal{G}_{0}$.

  2. 2.

The intersection $C_{j}^{*}\cap C_{k}^{*}$ is a bigon if and only if the
corresponding vertices of the graph $\mathcal{G}_{0}$ share an edge. This
bigon is contained in the compactification of a plane $P_{v}$ and its vertices
are $\\{0,\infty\\}$.

  3. 3.

Let $E=\\{(\pm 1,\pm 1,\pm 1)\\}\cup\\{(\pm 1,0,0)\\}\cup\\{(0,\pm
1,0)\\}\cup\\{(0,0,\pm 1)\\}\subset{\mathcal{C}}_{0}$. The rays $\\{te\colon
t\geq 0\\}$ with $e\in E$, are the sides of the bigons $\\{C_{j}^{*}\cap
C_{k}^{*}\\}$. Moreover, in one hand, the vertices of the cube $\\{(\pm 1,\pm
1,\pm 1)\\}$, correspond to the triangular faces of $\mathcal{G}_{0}$ and thus
parameterize triple intersections $C^{*}_{j}\cap C^{*}_{k}\cap C^{*}_{l}$, and
in the other hand, the centers of the faces in the cube $\\{(\pm
1,0,0)\\}\cup\\{(0,\pm 1,0)\\}\cup\\{(0,0,\pm 1)\\}$, correspond to square
faces of $\mathcal{G}_{0}$ and parameterize quadruple intersections
$C^{*}_{i}\cap C^{*}_{j}\cap C^{*}_{k}\cap C^{*}_{l}$.

  4. 4.

The intersection of all the balls is

$\bigcap_{j=1}^{12}C_{j}^{*}=\\{0,\infty\\}.$

Summarizing: For all $1\leq j\leq 12$, the boundary of the closed ball
$C_{j}^{*}\subset\mathbb{R}^{3}\cup\\{\infty\\}$ is decomposed into four
bigons determined by the intersection with its neighboring balls, the
intersection pattern is given by the cuboctahedron graph $\mathcal{G}_{0}$.
Since ${\mathcal{G}}_{0}$ is isomorphic to the graph ${\mathcal{G}}$ in Figure
8, then the decomposition of $\mathbb{R}^{3}\cup\\{\infty\\}$ into closed
balls is combinatorically isomorphic to the decomposition of the level sets
$h^{-1}(s)$ given in Lemma 5.10, for all $0<s<1$. As the union of two discs
through a disc at their boundaries is again a disc (Theorem A.2), we conclude
that $h^{-1}(s)$ is homeomorphic to the sphere
$\mathbb{S}^{3}\cong\mathbb{R}^{3}\cup\\{\infty\\}$. ∎

$a)$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$b)$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$
Figure 10: Cuboctahedron graph ${\mathcal{G}}_{0}$, constructed from the
middle points of the regular cube. The right image is one of the cones,
constructed using the dual structure of the graph which gives a bigon on the
sphere $\mathbb{S}^{3}$ as a one-point compactification of $\mathbb{R}^{3}$.

######  Remark 5.13.

It is possible to refine the decomposition of $h^{-1}(s)$, into the cones
$\\{\textrm{cl}({\mathcal{U}}_{j}),\textrm{cl}({\mathcal{V}}_{j}),\textrm{cl}({\mathcal{W}}_{j})\\}$,
to a triangulation. One way to make this refinement explicit is by adding the
faces of the cube in Figure 10 and the planes defined by the diagonals so that
the resulting pieces are tetrahedrons. For this reason, we take the liberty to
call such balls, with the combinatorial structure given by Lemma 5.10,
polyhedrons (even though the faces are bigons, not triangles).

######  Remark 5.14.

Conjugating the action of $\Sigma$ and $T$ on $\mathbb{C}^{2}$ (see Section 3)
with the involution $(z_{1},z_{2})\mapsto(z_{1},\overline{z_{2}}),$ we obtain
$\hat{\Sigma}(z_{1},z_{2})=(-z_{1},iz_{2})$ and
$\hat{T}(z_{1},z_{2})=(\overline{z_{1}},i\overline{z_{2}})$. Let
$\varphi\colon\mathbb{S}^{3}\to\mathbb{R}^{3}$ denote the stereographic
projection $(a+ib,c+id)\mapsto\frac{1}{1-b}(a,c,d)$. The compositions
$\varphi\circ\hat{\Sigma}\circ\varphi^{-1},\varphi\circ\hat{T}\circ\varphi^{-1}\colon\mathbb{R}^{3}\to\mathbb{R}^{3}$
are given by

$(x,y,z)\mapsto\frac{1}{x^{2}+y^{2}+z^{2}}(-x,-z,y)\quad\text{and}\quad(x,y,z)\mapsto\frac{1}{x^{2}+y^{2}+z^{2}}(x,z,y),$

respectively. The Dihedral group
$\langle\varphi\circ\hat{\Sigma}\circ\varphi^{-1},\varphi\circ\hat{T}\circ\varphi^{-1}\rangle$
acts in $\mathbb{R}^{3}$, permutes the planes $\\{P_{v}\colon v\in V\\}$, and
preserve the decomposition $\mathbb{R}^{3}=\bigcup_{i=1}^{12}C_{i}$ given in
the proof of Theorem 5.12. Then, the identification
$h^{-1}(s)\cong\mathbb{S}^{3}$ given by Theorem 5.12 is equivariant under the
corresponding actions of the Dihedral groups in the cones.

##  6 Cone structure on $S(4)$ and ${\mathcal{M}}(4)$

In this section we prove that $S(4)$ is homeomorphic to $\mathbb{R}^{4}$ by
showing that the level sets of the height function $h\colon S(4)\to\mathbb{R}$
provide in $S(4)$ a structure of open cone over the sphere $\mathbb{S}^{3}$,
we also describe the moduli space ${\mathcal{M}}(4)$. It is worth mentioning
that in [3] it is already proved that $S(4)\cong\mathbb{R}^{4}$, however, the
result provided in this section is stronger because the cone structure in
$S(4)$ naturally descends to an open cone structure of
${\mathcal{M}}(4)=S(4)/\langle\sigma,\tau\rangle$.

######  Lemma 6.1.

If $1\leq j\leq 4$, then the sets
$\textrm{cl}({\mathcal{U}}_{j}),\textrm{cl}({\mathcal{V}}_{j}),\textrm{cl}({\mathcal{W}}_{j})$
are homeomorphic to the open cone over the closed ball $C(\mathbb{D}^{3})$.
Moreover, under such homeomorphism, the height function $h$ corresponds to the
function on $C(\mathbb{D}^{3})\rightarrow(0,1]$ induced from the projection

$\mathbb{D}^{3}\times(0,1]\rightarrow(0,1].$

###### Proof.

In order to prove the result for a set
$\textrm{cl}({\mathcal{U}})\in\\{\textrm{cl}({\mathcal{U}}_{j}),\textrm{cl}({\mathcal{V}}_{j}),\textrm{cl}({\mathcal{W}}_{j})\\}$,
we construct a family of homeomorphisms $f_{s}\colon
h^{-1}(s)\cap\textrm{cl}({\mathcal{U}})\rightarrow
h^{-1}(1/2)\cap\textrm{cl}({\mathcal{U}})$ that vary continuously on the
parameter $s$ and a curve
$\mu\colon(0,1)\rightarrow\mathrm{cl}({\mathcal{U}})$ such that $h(\mu(t))$ is
strictly decreasing (in fact, after a reparametrization of $\mu$, we can make
$h(\mu(t))=t$). As the curve $\mu$ is transversal to the level sets of the
height function $h$, we can conclude that
$\mathrm{cl}({\mathcal{U}})\smallsetminus\\{\mathfrak{R}_{4}\\}$ has a
structure of a fiber bundle with base $(0,1)$ and fiber
$h^{-1}(1/2)\cap\textrm{cl}({\mathcal{U}})$, homeomorphic to $\mathbb{D}^{3}$
by Lemma 5.7; this fiber bundle is trivial because it has a contractible base
space, see [6, Theorem 9.9]). We can write down the trivialization as

$h^{-1}\left(\frac{1}{2}\right)\cap\mathrm{cl}({\mathcal{U}})\times(0,1)\rightarrow\mathrm{cl}({\mathcal{U}})\smallsetminus\\{\mathfrak{R}_{4}\\},\qquad(x,t)\mapsto
f^{-1}_{h(\mu(t))}(x)=f^{-1}_{t}(x),$

with continuous inverse $Z\mapsto(f_{h(Z)}(Z),h(Z))$. As the fibers
$h^{-1}(s)\cap\textrm{cl}({\mathcal{U}})$ collapse to
$\\{\mathfrak{R}_{4}\\}$, when $s\rightarrow 1$, the product structure on
$\mathrm{cl}({\mathcal{U}})\smallsetminus\\{\mathfrak{R}_{4}\\}$ induces the
structure of an open cone on $\mathrm{cl}({\mathcal{U}})\subset S(4)$, where
the height function $h$ corresponds to the projection onto the second factor.

Recall from Lemma 5.7 that for all $s\in(0,1)$, the level set
$h^{-1}(s)\cap\textrm{cl}({\mathcal{U}})$ have the structure of a fiber bundle
over a topological disc with some collapsed fibers at the boundary. Using this
structure, we can construct the function
$f_{s}:h^{-1}(s)\cap\textrm{cl}({\mathcal{U}})\rightarrow
h^{-1}(1/2)\cap\textrm{cl}({\mathcal{U}})$ by giving first the homeomorphism
between the base spaces and then extending such homeomorphism to the fibers as
linear maps on the arc-length parameter defining them. If the homeomorphisms
on the base spaces depend continuously on the parameter $s$, then the
extensions on the fibers depend continuously on the parameter $s$ as well.

If ${\mathcal{U}}$ is either ${\mathcal{U}}_{1}$ or ${\mathcal{W}}_{1}$, then
the base space of the fiber bundle structure of
$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}})$, is a topological disc contained in
$\mathbb{D}^{2}$, which we can denote temporarily here by $\Omega_{s}$. We
decompose the domain $\Omega_{s}$ into vertical intervals as follows: observe
that for every $z\in\Omega_{s}$, the real part satisfies
$-\cos(\arcsin(s))\leq\textrm{Re}(z)\leq\cos(\arcsin(s))$, and every $x$ in
the interval $[-\cos(\arcsin(s)),\cos(\arcsin(s))]$, defines the vertical
interval $I_{x}=\\{y\in\mathbb{R}:x+iy\in\Omega_{s}\\}$. Using this
decomposition of $\Omega_{s}$, we can give the homeomorphism
$f_{s}\colon\Omega_{s}\to\Omega_{1/2}$ as $x+iy\mapsto
x^{\prime}(x,s)+iy^{\prime}(x,y,s)$, where

$x^{\prime}(x,s)=\left(\frac{\cos(\arcsin(\frac{1}{2}))}{\cos(\arcsin(s))}\right)x,$

and $y\mapsto y^{\prime}(x,y,s)$ is the linear map between the intervals
$I_{x}\subset\Omega_{s}$ and $I_{x^{\prime}}\subset\Omega_{1/2}$. The
expression for the imaginary part can be made explicit for ${\mathcal{U}}_{1}$
as

$y^{\prime}(x,y,s)=\left(\frac{\sqrt{1-(x^{\prime})^{2}~{}}-s}{\sqrt{1-x^{2}~{}}-s}\right)(x-s)+s.$

The corresponding function for ${\mathcal{W}}_{1}$ has a much more involved
expression, due to the complicated behaviour of the curve $\Gamma$ in the
lower part of the boundary (determined by the points $q_{\alpha}$ in the proof
of Lemma 5.7, see Figure 7). This homeomorphism depends continuously on $s$
and thus, as discussed above, can be extended to the homeomorphism
$f_{s}:h^{-1}(s)\cap\textrm{cl}({\mathcal{U}})\rightarrow
h^{-1}(1/2)\cap\textrm{cl}({\mathcal{U}})$ which still depends continuously on
the parameter $s$. The curves
$\mu_{1}\colon[0,1)\to\mathrm{cl}({\mathcal{U}}_{1})$ and
$\mu_{2}\colon[0,1)\to\mathrm{cl}({\mathcal{W}}_{1})$ given by

$\mu_{1}(t)=[0,1,1-\frac{t}{2}+i(1-t),i(1-\frac{t}{2})]~{}~{}\text{and}~{}~{}~{}\mu_{2}(t)=[0,1,(1-\frac{t}{2})(1+i),\frac{t}{2}+i(1-\frac{t}{2})],$

satisfy that $\mu_{j}(0)=\mathfrak{R}_{4}$, $h(\mu_{j}(t))=1-t$, and they are
transversal to the level sets of the height function $h$. By the previous
discussion, these curves give us the missing part for the construction of the
homeomorphisms $f\colon\mathrm{cl}({\mathcal{U}}_{1})\to C(\mathbb{D}^{3})$
and $g\colon\mathrm{cl}({\mathcal{W}}_{1})\to C(\mathbb{D}^{3})$.

Finally, from Remark 5.4, it follows that for all $j\in\\{1,2,3,4\\}$, the
homeomorphisms sought can be constructed of the form
$f\circ\sigma^{-j}\colon\mathrm{cl}({\mathcal{U}}_{j})\to C(\mathbb{D}^{3})$,
$f\circ\sigma^{-j}\circ\tau\colon\mathrm{cl}({\mathcal{V}}_{j})\to
C(\mathbb{D}^{3}),$ and
$g\circ\sigma^{-j}\colon\mathrm{cl}({\mathcal{W}}_{j})\to C(\mathbb{D}^{3})$.
∎

The strategy in the proof of Lemma 6.1, can be used to prove the cone
structure of $S(4)$ over $\mathbb{S}^{3}$. For this, we should be able to
construct a family of functions $f_{s}\colon h^{-1}(s)\rightarrow
h^{-1}\left(\frac{1}{2}\right)$ which depend continuously on the $s$
parameter. Such type of functions were constructed in Lemma 6.1 for each piece
$\mathrm{cl}({\mathcal{U}}_{j}),\mathrm{cl}({\mathcal{V}}_{j}),\mathrm{cl}({\mathcal{W}}_{j})$
in the decomposition of the level set $h^{-1}(s)$, however they do not
coincide in their intersections, and remains unsolved to construct the family
$f_{s}\colon h^{-1}(s)\rightarrow h^{-1}\left(\frac{1}{2}\right)$. We overcome
this difficulty by using the combinatorics of the well-understood
decomposition of $S(4)$ into the cones $\mathrm{cl}({\mathcal{U}})$.

######  Theorem 6.2.

There is a homeomorphism $S(4)\cong C(\mathbb{S}^{3})\cong\mathbb{R}^{4}$,
such that the height function $h$ corresponds to the function
$C(\mathbb{S}^{3})\rightarrow(0,1]$, induced from the projection
$\mathbb{S}^{3}\times(0,1]\rightarrow(0,1]$.

###### Proof.

We will proceed in a similar way as in the proof Theorem 5.12, by showing two
combinatorially equivalent decompositions.

By taking the decomposition $\mathbb{S}^{3}=\cup_{j=1}^{12}C_{j}^{*}$ given in
the proof of Theorem 5.12 over each level sphere, we define decompositions of
$\mathbb{R}^{4}\cong C(\mathbb{S}^{3})$ and
$\mathbb{S}^{4}\cong\Sigma(\mathbb{S}^{3})$. In both cases, the combinatorics
of the decomposition is codified by the graph $\mathcal{G}_{0}$, Figure 10.

The one point compactification $S(4)^{*}=S(4)\cup\\{\infty\\}$ satisfies that

$S(4)^{*}=\bigcup_{i=1}^{4}\mathrm{cl}({\mathcal{U}}_{i})^{*}\cup\mathrm{cl}({\mathcal{V}}_{i})^{*}\cup\mathrm{cl}({\mathcal{W}}_{i})^{*},$

where for every
${\mathcal{U}}\in\\{{\mathcal{U}}_{i},{\mathcal{V}}_{i},{\mathcal{W}}_{i}\\}$,
$\mathrm{cl}({\mathcal{U}})^{*}=\mathrm{cl}({\mathcal{U}})\cup\\{\infty\\}$ is
the one point compactification of $\mathrm{cl}({\mathcal{U}})$. From Lemma
6.1, it follows that $\mathrm{cl}({\mathcal{U}})^{*}$ is homeomorphic to the
suspension
$\Sigma(h^{-1}(1/2)\cap\mathrm{cl}({\mathcal{U}}))\cong\Sigma(\mathbb{D}^{3})\cong\mathbb{D}^{4}$.
Since the graph ${\mathcal{G}}$ (Figure 8) codifies the combinatorics of the
decomposition of $h^{-1}(s)$ into the sets
$\\{h^{-1}(s)\cap\mathrm{cl}({\mathcal{U}}_{i}),h^{-1}(s)\cap\mathrm{cl}({\mathcal{V}}_{i}),h^{-1}(s)\cap\mathrm{cl}({\mathcal{W}}_{i})\\}$
(Lemma 5.10), then ${\mathcal{G}}$ also codifies the decomposition of
$S(4)^{*}$ determined by taking the suspension structure of each set
$\mathrm{cl}({\mathcal{U}})^{*}$.

We conclude that these decompositions of $S(4)^{*}$ and $\mathbb{S}^{4}$ into
twelve copies of the closed disc $\mathbb{D}^{4}$ are combinatorically
isomorphic, so that, $S(4)^{*}\cong\mathbb{S}^{4}$ and $S(4)$ is homeomorphic
to $\mathbb{R}^{4}$. Moreover, each transversal section in $S(4)\cong
C(\mathbb{S}^{3})$ corresponds to a level set of the height function $h\colon
S(4)\to\mathbb{R}$ and by Lemma 6.1, $h$ corresponds to the function on the
cone $C(\mathbb{S}^{3})$ corresponding to the projection
$\mathbb{S}^{3}\times(0,1]\rightarrow(0,1]$ under such identifications, thus
the result follows. ∎

In the rest of this section we use the cone structure in $S(4)$ just proved,
to describe the topology of the moduli space ${\mathcal{M}}(4)$. By a
combination of Theorem 3.2 and Example 3.4, we can see that the quotients
$h^{-1}(s)/\langle\sigma,\tau\rangle$ are homeomorphic to the sphere
$\mathbb{S}^{3}$, for $s$ close enough to $1$. We prove this fact for every
$s\in(0,1)$, because such construction will pay off when we study the boundary
$h^{-1}(0)=\partial S(4)$.

######  Lemma 6.3.

For every $s\in(0,1)$, $h^{-1}(s)/\langle\sigma,\tau\rangle$ is homeomorphic
to $\mathbb{S}^{3}$.

###### Proof.

A fundamental domain in the level set $h^{-1}(s)$ for the action of the
subgroup $\langle\sigma\rangle<\langle\sigma,\tau\rangle$ is the set

$\big{[}\mathrm{cl}({\mathcal{U}}_{1})\cup\mathrm{cl}({\mathcal{V}}_{1})\cup\mathrm{cl}({\mathcal{W}}_{1})\big{]}\cap
h^{-1}(s).$

The only reflection that leaves setwise fixed this subset is $\tau$, the
action of which permutes $\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})$ with
$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{V}}_{1})$ and leaves setwise fixed
$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1})$. Thus, the set

$B_{s}=\big{\\{}[0,1,z_{3},z_{4}]\in\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1}):\textrm{Im}(z_{3})\leq\textrm{Im}(z_{4})\big{\\}},$

is a closed ball (Lemma 5.7), and the union
$B_{s}\cup\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})$ is a fundamental domain
for the action of the dihedral group $\langle\sigma,\tau\rangle$.

If we take the identifications on the sphere
$\partial\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})$ determined by the
actions of elements of the dihedral group $\langle\sigma,\tau\rangle$, we
obtain a closed ball with boundary sphere splitted into two discs obtained
from the intersections
$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})\cap\mathrm{cl}({\mathcal{W}}_{1})$
and
$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})\cap\mathrm{cl}({\mathcal{W}}_{4})$.
If we do the analogous identifications on $\partial B_{s}$, we obtain a closed
ball with boundary sphere splitted into two discs obtained from the
intersections
$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1})\cap\mathrm{cl}({\mathcal{U}}_{1})$
and
$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1})\cap\mathrm{cl}({\mathcal{V}}_{4})$.
As
$\sigma^{3}\tau(\textrm{cl}({\mathcal{W}}_{1})\cap\textrm{cl}({\mathcal{V}}_{4}))=\textrm{cl}({\mathcal{W}}_{4})\cap\textrm{cl}({\mathcal{U}}_{1})$,
we conclude that after performing the identifications by elements of the
dihedral group $\langle\sigma,\tau\rangle$ in the fundamental domain, we
obtain two closed balls glued together through their boundary spheres. The
space obtained in this way is homeomorphic to $\mathbb{S}^{3}$, finishing the
proof.

As the statements on the identifications on
$\partial\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})$ and $\partial B_{s}$ by
elements of $\langle\sigma,\tau\rangle$ are not obvious, we make a detailed
account of them separately.

Identifications on $\partial\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})$.
Recall from Lemma 5.10 that $\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})$ is
homeomorphic to a closed ball with boundary sphere divided the bigons,

  1. 1.

$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})\cap\mathrm{cl}({\mathcal{V}}_{1})$,
consisting of trapezoids with height $s$.

  2. 2.

$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})\cap\mathrm{cl}({\mathcal{W}}_{4})$,
whose elements have two adjacent sides of maximal length.

  3. 3.

$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})\cap\mathrm{cl}({\mathcal{V}}_{3})$,
whose elements have two oposite sides of maximal length

  4. 4.

$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})\cap\mathrm{cl}({\mathcal{W}}_{1})$,
whose elements satisfy that $z_{3}$ lies in the bisector of the angle
$\measuredangle z_{4}z_{1}z_{2}$ and it is at distance $s$ of both sides
$\overline{z_{1}z_{2}}$ and $\overline{z_{1}z_{4}}$.

As the geometric properties of the quadrilaterals at each intersection are
mutually exclusive, we see that there are no identifications among different
bigons under the action of the dihedral group $\langle\sigma,\tau\rangle$. We
have two maps leaving fixed setwise two faces:

$\tau(\textrm{cl}({\mathcal{U}}_{1})\cap\textrm{cl}({\mathcal{V}}_{1}))=\textrm{cl}({\mathcal{V}}_{1})\cap\textrm{cl}({\mathcal{U}}_{1})\quad\textrm{and}\quad\sigma^{2}\tau(\textrm{cl}({\mathcal{U}}_{1})\cap\textrm{cl}({\mathcal{V}}_{3}))=\textrm{cl}({\mathcal{V}}_{3})\cap\textrm{cl}({\mathcal{U}}_{1}).$

The map $\tau$ folds the face
$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})\cap\textrm{cl}({\mathcal{V}}_{1})$
in half along the curve of symmetric trapezoids

$\big{\\{}[0,1,1/2+t+is,1/2-t+is]\colon s\leq 2t\leq 1\big{\\}}.$

Analogously, the map $\sigma^{2}\tau$ is a reflection in
$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})\cap\textrm{cl}({\mathcal{V}}_{3})$
that swaps the angles $\measuredangle z_{3}z_{4}0$ with $\measuredangle
z_{4}01$ and $\measuredangle z_{4}z_{3}1$ with $\measuredangle 01z_{3}$, so it
folds this face along the one-dimensional curve

$\left\\{[0,1,z_{3},z_{4}]:\arccos\left(\frac{1-s}{2}\right)\leq\measuredangle
z_{3}z_{4}0=\measuredangle z_{4}01\leq\pi/2\right\\},$

corresponding to quadrilaterals that are truncated isosceles triangles. The
two curves, along which the maps $\tau$ and $\sigma^{2}\tau$ fold the
corresponding faces, are connected in the rectangle $[0,1,1+is,is]$, so that,
the two identifications can be combined to be seen as folding in half the
topological disc

$\Big{[}\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})\cap\textrm{cl}({\mathcal{V}}_{1})\Big{]}\bigcup\Big{[}\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})\cap\textrm{cl}({\mathcal{V}}_{3})\Big{]}.$

After doing such identifications, we are left with a space homeomorphic to a
closed $3$-ball, whose boundary sphere is decomposed as the union of two
topological discs, coming from the intersections
$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})\cap\textrm{cl}({\mathcal{W}}_{1})$
and
$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})\cap\textrm{cl}({\mathcal{W}}_{4})$.

Identifications on $B_{s}$. Because of the condition
$\textrm{Im}(z_{3})\leq\textrm{Im}(z_{4})$, there are no longer elements with
$\textrm{Im}(z_{4})<s$ in the closed ball
$B_{s}\subset\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1})$ (see Figure 7).
Thus, just as in the case of $\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})$, an
element $[0,1,z_{3},z_{4}]\in B_{s}$ satisfies that

$z_{4}\in\Omega_{{\mathcal{U}}_{s}}=\\{z\in\mathbb{D}:\textrm{Im}(z)\geq
s\\}.$

There is a curve $\Lambda\subset\Omega_{{\mathcal{U}}_{s}}$, which is
determined by the property: $z_{4}\in\Lambda$, if and only if, there is a
$z_{3}\in\\{z\in\mathbb{C}\colon|z-1|=1\\}$ such that

$[0,1,z_{3},z_{4}]\in\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1}),\quad\text{and}\quad\textrm{Im}(z_{3})=\textrm{Im}(z_{4}),$

see Figure 12. The curve $\Lambda$ splits the set $\Omega_{{\mathcal{U}}_{s}}$
into two connected components
$\Omega_{{\mathcal{U}}_{s}}=\Omega_{\ell}\cup\Omega_{T}$, so that, for
$z_{4}\in\Omega_{\ell}$, the fiber of the $I$-bundle of
$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1})$ is completely contained in
$B_{s}$ and for $z_{4}\in\Omega_{T}$, the fiber is cut at the point $z_{3}$,
such that $[0,1,z_{3},z_{4}]$ is a trapezoid, see Figure 11 and Figure 12. We
have thus, a splitting of the boundary of $B_{s}$ into the following boundary
components, determined by the domination of a constraint

  1. 1.

The intersection $\\{[0,1,r+s+it,r+it]:0\leq r,t\leq
1\\}\cap\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1})$, consisting of trapezoids
with parallel sides of lengths $\ell=1$ and $s$.

  2. 2.

$[0,1,z_{3},z_{4}]\in\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1})\cap\mathrm{cl}({\mathcal{V}}_{4})$,
such that $\textrm{Im}(z_{4})\geq\textrm{Im}(z_{3})$ (or equivalently
$z_{4}\in\Omega_{T}$) whose elements have two adjacent sides of maximal length
$\ell=1$.

  3. 3.

$[0,1,z_{3},z_{4}]\in\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1})\cap\mathrm{cl}({\mathcal{U}}_{2})$,
such that $z_{4}\in\Omega_{\ell}$, whose elements have two adjacent sides of
maximal length.

  4. 4.

$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1})\cap\mathrm{cl}({\mathcal{U}}_{1})$.

We can observe that for $Z=[0,1,z_{3},z_{4}]$ belonging to the face described
in the point 2, it is satisfied that
$\tau(Z)=[0,1,z_{3}^{\prime},z_{4}^{\prime}]\in\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1})\cap\mathrm{cl}({\mathcal{U}}_{2})$
with $z_{4}^{\prime}\in\Omega_{T}$. On the other side, for
$Z=[0,1,z_{3},z_{4}]$ in the face described in point 3, it is satisfied that
$\tau(Z)=[0,1,z_{3}^{\prime},z_{4}^{\prime}]\in\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1})\cap\mathrm{cl}({\mathcal{V}}_{4})$
with $\textrm{Im}(z_{4}^{\prime})\leq\textrm{Im}(z_{3}^{\prime})$. Then these
two faces of $B_{s}$ complement each other (under the action of $\tau$), to
obtain a face equivalent to
$\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1})\cap\mathrm{cl}({\mathcal{V}}_{4})$
in $\mathrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1})$.

It is immediate from the previous descriptions that the reflection $\tau$,
acts as a folding map on the face of $B_{s}$ described in point 1, defined by
trapezoids. After doing this folding in $B_{s}$, we obtain a space
homeomorphic to a closed $3$-ball, whose boundary sphere is decomposed as the
union of two topological discs, one coming from the intersection
$\textrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1})\cap\mathrm{cl}({\mathcal{U}}_{1})$
and the other one, as a combination of faces described in points 2 and 3,
which is a face equivalent to
$\textrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1})\cap\mathrm{cl}({\mathcal{V}}_{4})$.

Finally, as we have that
$\sigma^{3}\tau({\mathcal{W}}_{1}\cap{\mathcal{V}}_{4})={\mathcal{U}}_{1}\cap{\mathcal{W}}_{4}$,
then we conclude that the quotient $h^{-1}(s)/\langle\sigma,\tau\rangle$ is
obtained by gluing together two copies of a closed $3$-ball by a homeomorphism
between their boundaries

$\big{(}\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})/\\{\tau,\sigma^{2}\tau\\}\big{)}\cup\big{(}B_{s}/\\{\tau,\sigma\\}\big{)}\cong(\mathbb{D}^{3}\cup\mathbb{D}^{3})/\sim.$

This space is homeomorphic to the sphere $\mathbb{S}^{3}$. ∎

######  Remark 6.4.

The fundamental domain of the action of the dihedral group
$\langle\sigma,\tau\rangle$ in $h^{-1}(s)$, constructed in Lemma 6.3, is
characterized by two properties, on their elements as follows:
$Z=[0,1,z_{3},z_{4}]$ belongs to the fundamental domain if and only if

  * •

$\textrm{Im}(z_{4})\geq s$ and $|z_{4}|\leq 1$, and

  * •

$z_{3}$ belongs to a piecewise differentiable curve $\gamma_{z_{4}}$, which
deppends continuously on the parameter $z_{4}$.

This is shown by two generic elements in Figure 11, and endows naturally with
the structure of an $I$-bundle on the fundamental domain, over the topological
disc $\Omega_{{\mathcal{U}}_{s}}=\\{z\in\mathbb{D}\colon\textrm{Im}(z)\geq
s\\}$.

$s$$\bullet$$z_{4}$$\bullet$$\bullet$$\bullet$$\gamma_{z_{4}}$$\bullet$$\bullet$$z^{\prime}_{4}$$s$$\gamma_{z^{\prime}_{4}}$$\bullet$$\bullet$$-1$$\bullet$$i$$\bullet$$0$$\bullet$$1$$\bullet$$2$$\bullet$$\Omega_{\mathcal{U}s}$
Figure 11: Curves of possible values for the $z_{3}$ coordinate of a
quadrialteral, with a fixed $z_{4}$ coordinate. These curves are cut from
above with either the condition $\textrm{Im}(z_{4})\geq\textrm{Im}(z_{3})$, or
$\ell_{3}=|z_{3}-1|\leq 1$. Figure 12: A vertex $z_{4}\in\Lambda$, satisfies
that $[0,1,z_{3},z_{4}]\in\textrm{cl}(h^{-1}\cap{\mathcal{W}}_{1})$ is a
trapezoid and $|1-z_{3}|=1$. For $z_{4}\in\Omega_{T}$, we cut the fibers of
the $I$-bundle at trapezoids, and for $z_{4}\in\Omega_{\ell}$, the fibers of
the $I$-bundle are not cut by the new condition, so they are still cut by the
condition $\ell_{3}=1=\ell_{2}$.

######  Theorem 6.5.

There is a homeomorphism ${\mathcal{M}}(4)\cong
C(\mathbb{S}^{3})\cong\mathbb{R}^{4}$.

###### Proof.

For every $0<s<1$, the fundamental domain
$B_{s}\cup\mathrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})$ for the action of
$\langle\sigma,\tau\rangle$ on the level set $h^{-1}(s)$, constructed in the
proof of Lemma 6.3, is an $I$-bundle over the topological disc
$\Omega_{{\mathcal{U}}_{s}}=\\{z\in\mathbb{D}\colon\textrm{Im}(z)\geq s\\}$,
see Remark 6.4 and Figure 11. By using the strategy used in Lemma 6.1, we can
conclude that these fundamental domains can be uniformized with a continuous
function as follows,

$\big{(}\mathbb{D}^{2}\times[0,1]\big{)}\times(0,1)\rightarrow
S(4),\qquad(w,t,s)\mapsto[0,1,\gamma_{\varphi_{s}(w)}(t),\varphi_{s}(w)].$

From Proposition 4.3, it follows that this function can be extended by the
constant with value $\mathfrak{R}_{4}\in S(4)$ in
$\big{(}\mathbb{D}^{3}\times[0,1]\big{)}\times\\{1\\}$.

Finally, it is possible to define an action of $\langle\sigma,\tau\rangle$
into $\mathbb{S}^{3}$ so that the uniformizing function
$C(\mathbb{S}^{3})\rightarrow S(4)$ is equivariant. By Lemma 6.3, we have
$h^{-1}(s)/\langle\sigma,\tau\rangle\cong\mathbb{S}^{3}$ and so, the result
follows. ∎

##  7 Asymptotic description of $\overline{S(4)}$ and
$\overline{{\mathcal{M}}(4)}$

As seen in the previous section, we have the cone structures $S(4)\cong
C(\mathbb{S}^{3})\cong\mathbb{R}^{4}$, but despite such well behaved
structure, the closure $\overline{S(4)}$ is not even a manifold. In this
section we describe precisely the region in $\partial S(4)$ which captures
this bad asymptotic behaviour and prove that outside of such set, we have a
very well behaved structure. As this section is harder to follow than the
previous ones, we break down the ideas into four subsections.

Recall that for a subset $A\subset S(4)\subset\mathbb{C}\mathbb{P}^{2}$, we
are considering two types of closures, as a subset of
$\mathbb{C}\mathbb{P}^{2}$ denoted as $\overline{A}$, and as a subset of
$S(4)$, denoted as $\textrm{cl}(A)=\overline{A}\cap S(4)$.

###  7.1 Bad asymptotic behaviour at the boundary

Here we describe the different shapes a quadrilateral can have in the boundary
and show a bad asymptotic behaviour as the level sets $h^{-1}(s)$ approaches
to $h^{-1}(0)=\partial S(4)$.

######  Remark 7.1.

A quadrilateral $Z=[z_{1},z_{2},z_{3},z_{4}]\in\partial S(4)$ satisfies that
there exists $j\in\\{1,2,3,4\\}$ such that
$z_{j}\in{\mathcal{L}}_{j}=\overline{z_{j+1}z_{j+2}}\cup\overline{z_{j+2}z_{j+3}}$
(Proposition 4.4). There are four types of quadrilaterals with this property,

  1. a)

Flags: The vertex $z_{j}$ belongs to
${\mathcal{L}}_{j}\smallsetminus\\{z_{j+1},z_{j+2},z_{j+3}\\}$.

  2. b)

Triangles: It is satisfied that $z_{j}=z_{j+1}$ or $z_{j}=z_{j+3}$.

  3. c)

Wedges: It is satisfied that $z_{j}=z_{j+2}$. We denote the set of wedges as
$\mathcal{K}$.

  4. d)

4-segments. ${\mathcal{L}}_{j}$ degenerates to a segment and therefore, the
four vertices are collinear.

Notice that in general, flags and wedges are limits of sequences of
necessarily non-convex quadrilaterals.

######  Definition 7.2.

Let $Z\in\partial S(4)$ be a 4-segment.

  1. (a)

We say that $Z$ is convex if it can be obtained as the limit of convex
quadrilaterals.

  2. (b)

We say that $Z$ is non-convex if it cannot be obtained as limit of convex
quadrilaterals. We denote as ${\mathcal{N}}\subset\partial S(4)$ the closure
of the space of non-convex $4$-segments.

We can observe that in a non-convex $4$-segment, the vertices must be in a
zig-zag configuration, for example, the quadrilateral
$\left[0,\frac{3}{4},\frac{1}{4},1\right]$.

######  Proposition 7.3.

The set of wedges ${\mathcal{K}}$ is homeomorphic to a wedge of two spheres
$\mathbb{S}^{2}\vee\mathbb{S}^{2}$. The space of non-convex $4$-segments
${\mathcal{N}}$ is homeomorphic to a topological square and each sphere of
$\mathcal{K}$ intersects $\mathcal{N}$ into one of the diagonals of the
square. The intersection of the two diagonals of ${\mathcal{N}}$, which is
also the intersection of the two spheres of ${\mathcal{K}}$, is the “zig-zag”
${\mathcal{Z}}=[0,1,0,1]$, Figure 13.

###### Proof.

The set of wedges can be written as the union

${\mathcal{K}}=\\{[0,1,0,w]\in\partial S(4)\colon
w\in\mathbb{CP}^{1}\\}\cup\\{[0,1,z,1]\in\partial
S(4):z\in\mathbb{CP}^{1}\\},$

where both sets in the union are homeomorphic to $\mathbb{S}^{2}$ and
intersect at $\\{{\mathcal{Z}}\\}$, giving us the homeomorphism
$\mathcal{K}\cong\mathbb{S}^{2}\vee\mathbb{S}^{2}$. The set of non-convex
$4$-segments can be triangulated as
${\mathcal{N}}=\bigcup_{j=0}^{3}\sigma^{j}T$, with the simplex
$T=\\{[0,1,t,x]\colon 0\leq t\leq x\leq 1\\}.$ As there are four triangles in
such description, we get a topological square, with diagonals

$\\{[0,1,0,t]:0\leq t\leq 1\\}\cup\\{[0,t,0,1]:0\leq t\leq 1\\},\text{~{}
and}$ $\\{[0,1,t,1]:0\leq t\leq 1\\}\cup\\{[0,1-t,-t,1-t]:0\leq t\leq 1\\},$

which correspond to the intersection with each sphere of ${\mathcal{K}}$,
Figure 13. ∎

$\bullet$$[0,1,0,0]$$\bullet$${\mathcal{Z}}$$\bullet$$[0,0,0,1]$$\bullet$$[1,0,0,0]$$\bullet$$[0,1,0,0]$$\bullet$$[0,0,1,0]$$\bullet$$[0,0,0,1]$$\bullet$${\mathcal{Z}}$$\bullet$$[1,0,0,0]$$\bullet$$[0,0,1,0]$${\mathcal{Z}}$$\bullet$
Figure 13: ${\mathcal{D}}_{0}$ as a wedge of two spheres, corresponding to the
set of wedges $\mathcal{K}$, union with a square in the middle, corresponding
to the set of non-convex 4-segments ${\mathcal{N}}$. The union is through the
diagonal segments of ${\mathcal{N}}$ and the two spheres intersect at the
center of the square that correspond to the zig-zag ${\mathcal{Z}}=[0,1,0,1]$.

We introduce a decomposition of the boundary into subsets analogous to the
ones given in Definition 5.3 for a level set $h^{-1}(s)$, with $0<s<1$.

######  Definition 7.4.

Let $j\in\\{1,2,3,4\\}$, then the sets
$\widehat{{\mathcal{U}}}_{j},\widehat{{\mathcal{V}}}_{j},\widehat{{\mathcal{W}}}_{j}\subset\partial
S(4)$ consist of the points which are limit of quadrilaterals in
$\mathrm{cl}({\mathcal{U}}_{j})$, $\mathrm{cl}({\mathcal{V}}_{j})$, and
$\mathrm{cl}({\mathcal{W}}_{j})$, respectively.

######  Remark 7.5.

From Remark 7.1 it follows that every quadrilateral in $\partial S(4)$ belongs
to one of the sets $\widehat{{\mathcal{U}}}_{j}$,
$\widehat{{\mathcal{V}}}_{j},$ or $\widehat{{\mathcal{W}}}_{j}$, with
$j\in\\{1,2,3,4\\}$. This gives us the identity

$h^{-1}(0)\cap\overline{S(4)}=\partial
S(4)=\bigcup_{j=1}^{4}\left(\widehat{{\mathcal{U}}}_{j}\cup\widehat{{\mathcal{V}}}_{j}\cup\widehat{{\mathcal{W}}}_{j}\right).$

The dihedral group $\langle\sigma,\tau\rangle$ permutes the sets
$\widehat{{\mathcal{U}}}_{j}$, $\widehat{{\mathcal{V}}}_{j},$ and
$\widehat{{\mathcal{W}}}_{j}$ just like permutes the sets ${\mathcal{U}}_{j}$,
${\mathcal{V}}_{j},$ and ${\mathcal{W}}_{j}$ in the level sets $h^{-1}(s)\cap
S(4)$, with $0<s<1$, see Remark 5.4.

In order to understand the sets $\widehat{{\mathcal{U}}}_{j}$,
$\widehat{{\mathcal{V}}}_{j},$ and $\widehat{{\mathcal{W}}}_{j}$, for
$j\in\\{1,2,3,4\\}$, we proceed to describe $\widehat{{\mathcal{U}}}_{1}$ and
$\widehat{{\mathcal{W}}}_{1}$.

######  Construction 7.6.

We construct $\widehat{{\mathcal{U}}}_{1}$ and $\widehat{{\mathcal{W}}}_{1}$,
the other sets can be obtained by using the action of the dihedral group
$\langle\sigma,\tau\rangle$.

$a)$ Construction of $\widehat{{\mathcal{U}}}_{1}$. From Lemma 5.7 we know
that a quadrilateral
$Z(s)=[0,1,z_{3}(s),z_{4}(s)]\in\textrm{cl}(h^{-1}(s)\cap{\mathcal{U}}_{1})$,
satisfies that $z_{4}(s)$ lies in the set $\Omega_{{\mathcal{U}}s}$, and
$z_{3}(s)$ lies in the $C^{1}$-curve $\gamma_{z_{4}(s)}$ constructed as a
concatenation of an arc of a circle and two segments tangent to it, as shown
in Figure 5. These conditions become as $s\rightarrow 0$, to the fact that a
quadrilateral $Z=[0,1,z_{3},z_{4}]\in\widehat{{\mathcal{U}}}_{1}$ must satisfy
that $z_{4}$ lies in the closed set
$\Omega_{{\mathcal{U}}_{0}}=\\{z\in\mathbb{D}^{2}\colon\textrm{Im}(z)\geq
0\\}$, and the set of possible values for $z_{3}$ is the limit $\lim_{s\to
0}\gamma_{z_{4}(s)}$. For
$z_{4}\in\Omega_{{\mathcal{U}}_{0}}\smallsetminus\\{1\\}$, the limit
$\lim_{s\to 0}\gamma_{z_{4}(s)}$ is a curve $\gamma_{z_{4}}$ obtained by
cutting the union $\overline{01}\cup\\{(1-z_{4})t+1\colon t\geq 0\\}$ with the
constraint $|z_{3}-z_{4}|\leq 1$, Figure 15.$a)$. In the remaining case the
quadrilaterals are wedges with $z_{4}=z_{2}=1$, in this case the limit
$\lim_{s\to 0}\gamma_{z_{4}(s)}$ behaves somewhat pathologically, because it
is the two-dimensional set

$\overline{01}\cup\big{\\{}z+1\in\mathbb{C}\colon|z|\leq 1,Re(z)\geq
0,\textrm{Im}(z)\leq 0\big{\\}}.$

$b)$ Construction of $\widehat{{\mathcal{W}}}_{1}$. A quadrilateral
$Z(s)=[0,1,z_{3}(s),z_{4}(s)]\in\textrm{cl}(h^{-1}(s)\cap{\mathcal{W}}_{1})$
is determined by $z_{4}(s)\in\Omega_{{\mathcal{W}}s}$ and $z_{3}(s)$ belonging
to a $C^{1}$-curve $\gamma_{z_{4}(s)}$ which is again the union of an arc and
two segments tangent to it, see Figure 7. As $s\rightarrow 0$, these
conditions translate to the fact that a quadrilateral
$Z=[0,1,z_{3},z_{4}]\in\widehat{{\mathcal{W}}}_{1}$, must satisfy that $z_{4}$
lies in the set
$\Omega_{{\mathcal{W}}_{0}}=\\{z\in\mathbb{D}^{2}:\textrm{Im}(z)\geq 0\\}$
(observe $\Omega_{{\mathcal{W}}_{0}}=\Omega_{{\mathcal{U}}_{0}}$ and it is
better behaved than $\Omega_{{\mathcal{W}}s}$), and $z_{3}$ lies in the limit
$\lim_{s\to 0}\gamma_{z_{4}(s)}$. For every
$z_{4}\in\Omega_{{\mathcal{W}}_{0}}\smallsetminus\\{1\\}$, the set of possible
values for $z_{3}$ so that
$Z=[0,1,z_{3},z_{4}]\in\widehat{{\mathcal{W}}}_{1}$, is the curve
$\gamma_{z_{4}}$ contained in the union
$\overline{0z_{4}}\cup\\{z_{4}+t(z_{4}-1)\colon t\geq 0\\}$ and cut with the
constraint $|z_{3}-1|\leq 1$. The case $z_{4}=z_{2}=1$, which corresponds to
wedges, is again pathological because the set of possible values for $z_{3}$
is

$\overline{01}\cup\\{z+1\in\mathbb{C}\colon|z|\leq 1,Re(z)\geq 0,Im(z)\geq
0\\}.$

######  Lemma 7.7.

The set of non-convex $4$-segments can be written as the union

$\mathcal{N}=\bigcup_{j=1}^{4}\left(\widehat{{\mathcal{W}}}_{j}\cap\widehat{{\mathcal{U}}}_{j}\cap\widehat{{\mathcal{V}}}_{j}\right),$

and the set of wedges $\mathcal{K}$ is the closure of

$\bigcup_{j=1}^{4}\left[\left(\widehat{{\mathcal{W}}}_{j}\cap\widehat{{\mathcal{U}}}_{j}\right)\cup\left(\widehat{{\mathcal{W}}}_{j}\cap\widehat{{\mathcal{V}}}_{j}\right)\right]\smallsetminus\left(\widehat{{\mathcal{W}}}_{j}\cap\widehat{{\mathcal{U}}}_{j}\cap\widehat{{\mathcal{V}}}_{j}\right).$

###### Proof.

We proceed by using the same strategy as in previous lemmas, i.e. we describe
the sets of $4$-segments and wedges contained in
$\widehat{{\mathcal{U}}}_{1}$, $\widehat{{\mathcal{V}}}_{1}$ and
$\widehat{{\mathcal{W}}}_{1}$, and then using the dihedral group
$\langle\sigma,\tau\rangle$, we obtain the description of the whole spaces
${\mathcal{N}}$ and ${\mathcal{K}}$.

Notice from Construction 7.6$.a)$ that a quadrilateral
$Z\in\widehat{{\mathcal{U}}}_{1}$ is a $4$-segment if and only if
$Z=[0,1,s,t]$, with $-1\leq t\leq 1$ and $0\leq s\leq 2$. If $Z$ is non-
convex, then the inequality $s\leq t$ is satisfied. Also from Construction
7.6$.b)$, we can see that the same conditions must hold for the non-convex
$4$-segments in $\widehat{{\mathcal{W}}}_{1}$, giving us
$\widehat{{\mathcal{U}}}_{1}\cap{\mathcal{N}}=\widehat{{\mathcal{W}}}_{1}\cap{\mathcal{N}}$.
Now, the set $\widehat{{\mathcal{U}}}_{1}\cap{\mathcal{N}}=\\{[0,1,s,t]\colon
0\leq s\leq t\leq 1\\}$ is a $2$-simplex, which is invariant under the
homeomorphism $\tau([0,1,s,t])=[0,1,1-t,1-s]$, and as we have
$\tau(\widehat{{\mathcal{U}}}_{1}\cap{\mathcal{N}})=\widehat{{\mathcal{V}}}_{1}\cap{\mathcal{N}}$,
then
$\widehat{{\mathcal{U}}}_{1}\cap{\mathcal{N}}=\widehat{{\mathcal{V}}}_{1}\cap{\mathcal{N}}=\widehat{{\mathcal{W}}}_{1}\cap{\mathcal{N}}=\widehat{{\mathcal{W}}}_{1}\cap\widehat{{\mathcal{U}}}_{1}\cap\widehat{{\mathcal{V}}}_{1}$.
If we act with the homeomorphism $\sigma$, we get the equivalent conditions
for all $j\in\\{1,2,3,4\\}$

$\widehat{{\mathcal{U}}}_{j}\cap{\mathcal{N}}=\widehat{{\mathcal{V}}}_{j}\cap{\mathcal{N}}=\widehat{{\mathcal{W}}}_{j}\cap{\mathcal{N}}=\widehat{{\mathcal{W}}}_{j}\cap\widehat{{\mathcal{U}}}_{j}\cap\widehat{{\mathcal{V}}}_{j},$

so in particular
${\mathcal{N}}=\bigcup_{j=1}^{4}\left(\widehat{{\mathcal{W}}}_{j}\cap\widehat{{\mathcal{U}}}_{j}\cap\widehat{{\mathcal{V}}}_{j}\right)$.
As the intersection
$\widehat{{\mathcal{W}}}_{j}\cap\widehat{{\mathcal{U}}}_{j}\cap\widehat{{\mathcal{V}}}_{j}$
is a $2$-simplex, then the union of such intersections defines a triangulation
of ${\mathcal{N}}$.

From Construction 7.6$.a)$ we can see that the wedges in
$\widehat{{\mathcal{U}}}_{1}$ are of the form $[0,1,0,z_{4}]$ and the wedges
in $\widehat{{\mathcal{V}}}_{1}$ are of the form $[0,1,z_{3},1]$, which are
obtained by applying the refection $\tau$ to the wedges in
$\widehat{{\mathcal{U}}}_{1}$. In both cases, the wedges belong to the set
$\widehat{{\mathcal{W}}}_{1}$ as well. Moreover, from the Construction
7.6$.b)$ and using the previous case of non-convex 4-segments, we can see that
the intersection $\mathcal{K}\cap\widehat{{\mathcal{W}}}_{1}$ is the closure
of

$\left[\left(\widehat{{\mathcal{W}}}_{1}\cap\widehat{{\mathcal{U}}}_{1}\right)\cup\left(\widehat{{\mathcal{W}}}_{1}\cap\widehat{{\mathcal{V}}}_{1}\right)\right]\setminus\left(\widehat{{\mathcal{W}}}_{1}\cap\widehat{{\mathcal{U}}}_{1}\cap\widehat{{\mathcal{V}}}_{1}\right).$

If we act with the homeomorphism $\sigma$, we obtain the description of
$\mathcal{K}\cap\widehat{{\mathcal{W}}}_{j}$, for every $j\in\\{1,2,3,4\\}$,
in terms of double and triple intersections as in the case $j=1$, and the
result follows. ∎

The previous results motivates the following definition.

######  Definition 7.8.

We define the set

$\mathcal{D}:=\bigcup_{j=1}^{4}\Big{[}\big{(}\overline{{\mathcal{W}}_{j}}\cap\overline{{\mathcal{U}}_{j}}\big{)}\cup\big{(}\overline{{\mathcal{W}}_{j}}\cap\overline{{\mathcal{V}}_{j}}\big{)}\Big{]}\subset\overline{S(4)},$

and for every $s\in[0,1]$, ${\mathcal{D}}_{s}:={\mathcal{D}}\cap h^{-1}(s)$.

Observe that
${\mathcal{D}}_{0}=\bigcup_{j=1}^{4}\big{[}\big{(}\widehat{{\mathcal{W}}}_{j}\cap\widehat{{\mathcal{U}}}_{j}\big{)}\cup\big{(}\widehat{{\mathcal{W}}}_{j}\cap\widehat{{\mathcal{V}}}_{j}\big{)}\big{]}={\mathcal{K}}\cup{\mathcal{N}}$
and ${\mathcal{D}}_{1}=\\{\mathfrak{R}_{4}\\}$. We proceed to describe the
topology of the sets ${\mathcal{D}}_{s}$.

######  Proposition 7.9.

For every $0<s<1$, it is satisfied that
${\mathcal{D}}_{s}\cong\mathbb{S}^{2}$, and

$\lim_{s\rightarrow
1}{\mathcal{D}}_{s}={\mathcal{D}}_{1}=\\{\mathfrak{R}_{4}\\},\qquad\lim_{s\rightarrow
0}{\mathcal{D}}_{s}={\mathcal{D}}_{0}\cong(\mathbb{S}^{2}\vee\mathbb{S}^{2})\cup\mathbb{D}^{2}.$

Thus, ${\mathcal{D}}$ is homeomorphic to an open cone
$C(\mathbb{S}^{2})\cong\mathbb{R}^{3}$ compactified with ${\mathcal{D}}_{0}$.

###### Proof.

Quadrilaterals in ${\mathcal{D}}_{s}$ have a vertex $z_{j}$ such that
$d(z_{j},\overline{z_{j+1}z_{j+2}})=s=d(z_{j},\overline{z_{j+2}z_{j+3}})$, and
therefore, $z_{j}$ belongs to the bisector of the angle $\measuredangle
z_{j+1}z_{j+2}z_{j+3}$. This fact is used to prove that $\lim_{s\rightarrow
0}\mathcal{D}_{s}=\mathcal{D}_{0}={\mathcal{K}}\cup{\mathcal{N}}$ as follows:

  * •

$\mathcal{D}_{0}\subset\lim_{s\rightarrow 0}\mathcal{D}_{s}.$ A non-convex
$4$-segment
$Z=[0,1,x,y]\in\widehat{{\mathcal{U}}}_{1}\cap\widehat{{\mathcal{W}}}_{1}$
with $0\leq x\leq y\leq 1$, can be approximated by the quadrilaterals

$Z_{\theta}=\left[0,1,\frac{x}{\cos(\theta/2)}e^{i\theta/2},\frac{y}{\cos(\theta)}e^{i\theta}\right]\in\Big{(}\textrm{cl}({\mathcal{U}}_{1})\cap\textrm{cl}({\mathcal{W}}_{1})\Big{)},$

satisfying $h(Z_{\theta})=x\tan(\theta/2)\rightarrow 0$ as $\theta\rightarrow
0$. Thus, $Z\in\lim_{s\rightarrow 0}\mathcal{D}_{s}$. Analogously, a wedge
$Z=[0,1,0,re^{i\beta}]\in\widehat{{\mathcal{U}}}_{1}\cap\widehat{{\mathcal{W}}}_{1}$
belongs to $\lim_{s\rightarrow 0}\mathcal{D}_{s}$ because it can be
approximated by the sequence of quadrilaterals
$Z_{t}=[0,1,te^{i\beta/2},re^{i\beta}]\in\textrm{cl}({\mathcal{U}}_{1})\cap\textrm{cl}({\mathcal{W}}_{1})$,
satisfying $h(Z_{t})=t\sin(\beta/2)\rightarrow 0$, as $t\rightarrow 0$. The
statement for every other element in ${\mathcal{D}}_{0}$ can be proved by
using the action of the dihedral group $\langle\sigma,\tau\rangle$.

  * •

$\lim_{s\rightarrow 0}\mathcal{D}_{s}\subset\mathcal{D}_{0}.$ Observe that if
$Z=[z_{1},z_{2},z_{3},z_{4}]\in\partial S(4)$ with $z_{j}\in{\mathcal{L}}_{j}$
is a flag, a triangle or a convex $4$-segment (i.e. $Z$ is not in
${\mathcal{D}}_{0}$), see Remark 7.1 and Definition 7.2, and
$\\{Z_{n}=[z_{1}(n),z_{2}(n),z_{3}(n),z_{4}(n)]\\}_{n\in\mathbb{N}}$ is a
sequence of simple quadrilaterals which converge to $Z$, then for a
sufficiently large subscript $n$, the quadrilateral $Z_{n}$ have the property
that the vertex $z_{j}(n)$ is far from the bisector of the internal angle
$\measuredangle z_{j+1}(n)z_{j+2}(n)z_{j+3}(n)$. Thus,
$Z\notin\lim_{s\rightarrow 0}{\mathcal{D}}_{s}.$

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$${\mathcal{U}}_{2}$$\bullet$${\mathcal{W}}_{2}$$\bullet$${\mathcal{W}}_{1}$$\bullet$${\mathcal{V}}_{1}$$\bullet$${\mathcal{V}}_{4}$$\bullet$${\mathcal{V}}_{2}$$\bullet$${\mathcal{U}}_{3}$$\bullet$${\mathcal{U}}_{1}$$\bullet$${\mathcal{U}}_{4}$$\bullet$${\mathcal{W}}_{4}$$\bullet$${\mathcal{W}}_{3}$$\bullet$${\mathcal{V}}_{3}$$\bullet$$\bullet$$\bullet$$\bullet$
Figure 14: For $0<s<1$, the set $\mathcal{D}_{s}\subset h^{-1}(s)$ is
homeomorphic to the suspension over the darkest closed curve that separates
the cube. Here, we are removing the edges of the cuboctahedron corresponding
to intersections
$\overline{{\mathcal{W}}_{j}}\cap\overline{{\mathcal{U}}_{j}}$ and
$\overline{{\mathcal{W}}_{j}}\cap\overline{{\mathcal{V}}_{j}}$ in $h^{-1}(s)$.

From the decomposition of $h^{-1}(s)$ into the sets
$\\{\textrm{cl}({\mathcal{U}}_{j}),\textrm{cl}({\mathcal{V}}_{j}),\textrm{cl}({\mathcal{W}}_{j})\\}$
given in Theorem 5.12 and represented in Figure 10, we can see that the set
$\mathcal{D}_{s}$ is the suspension over a closed curve in the regular cube
shown in Figure 14, and so, ${\mathcal{D}}_{s}$ is homeomorphic to
$\mathbb{S}^{2}$. Thus, the homemomorphism ${\mathcal{D}}\cap S(4)\cong
C(\mathbb{S}^{2})$ follows from the cone structure on $S(4)$ given by Theorem
6.2. ∎

Proposition 7.9 justifies the assertion that the quadrilaterals in
${\mathcal{D}}_{0}$ generate bad asymptotic behaviour.

###  7.2 Good behaviour at the boundary

As observed in the previous section, there is bad asymptotic behaviour at the
set ${\mathcal{D}}_{0}={\mathcal{N}}\cup\mathcal{K}$, as
${\mathcal{D}}_{s}\rightarrow{\mathcal{D}}_{0}$ and
${\mathcal{D}}_{s}\ncong{\mathcal{D}}_{0}$. In this section, we prove that the
set ${\mathcal{D}}_{0}$ captures all the bad asymptotic behaviour by proving
that the topology of $h^{-1}(s)\smallsetminus{\mathcal{D}}_{s}$ doesn’t change
as $s\rightarrow 0$.

To prove this, we procede as in the proof of Lemma 5.7, describing first the
topology of the sets
$\widehat{{\mathcal{U}}}_{j},\widehat{{\mathcal{V}}}_{j},\widehat{{\mathcal{W}}}_{j}\subset\partial
S(4)$, by identifying a structure of an $I$-bundle on them. We note that this
structure exists only outside of the set of wedges and non-convex $4$-segments
${\mathcal{D}}_{0}$.

######  Lemma 7.10.

The sets $\widehat{{\mathcal{U}}}_{j}\smallsetminus{\mathcal{D}}_{0}$,
$\widehat{{\mathcal{V}}}_{j}\smallsetminus{\mathcal{D}}_{0}$, and
$\widehat{{\mathcal{W}}}_{j}\smallsetminus{\mathcal{D}}_{0}$, are homeomorphic
to a closed 3-ball with a closed 2-disc removed from their boundary.

###### Proof.

We describe the $I$-bundle structure in the sets
$\widehat{{\mathcal{U}}}_{j}\smallsetminus{\mathcal{D}}_{0}$,
$\widehat{{\mathcal{V}}}_{j}\smallsetminus{\mathcal{D}}_{0}$,
$\widehat{{\mathcal{W}}}_{j}\smallsetminus{\mathcal{D}}_{0}$ as the limit of
the $I$-bundle structures in $\textrm{cl}({\mathcal{U}}_{j})$,
$\textrm{cl}({\mathcal{V}}_{j})$, $\textrm{cl}({\mathcal{W}}_{j}),$ given in
Lemma 5.7. From Remark 7.5, it follows that we only need to describe the
$I$-bundle structure for the sets
$\widehat{{\mathcal{U}}}_{1}\smallsetminus{\mathcal{D}}_{0}$ and
$\widehat{{\mathcal{W}}}_{1}\smallsetminus{\mathcal{D}}_{0}$.

Construction of $\widehat{{\mathcal{U}}}_{1}$. Recall from Construction 7.6,
that a quadrilateral $Z=[0,1,z_{3},z_{4}]\in\widehat{{\mathcal{U}}}_{1}$
satisfies that
$z_{4}\in\Omega_{{\mathcal{U}}_{0}}=\\{z\in\mathbb{D}^{2}\colon\textrm{Im}(z)\geq
0\\}$ and when $z_{4}\neq 1$, the set of possible values for $z_{3}$ is the
curve $\gamma_{z_{4}}$ obtained by cutting the union
$\overline{01}\cup\\{(1-z_{4})t+1\colon t\geq 0\\}$ with the constraint
$|z_{3}-z_{4}|\leq 1$, Figure 15.$a)$ (the case $z_{4}=1$ is pathological but
consists of wedges, so it belongs to ${\mathcal{D}}_{0}$). Since for
$z_{4}\in\Omega_{{\mathcal{U}}_{0}}\smallsetminus\\{1\\}$ the curves
$\gamma_{z_{4}}$ are piecewise linear, then they can be parametrized by
rescaling of the piecewise arc-length. So that, the function

$(\Omega_{{\mathcal{U}}_{0}}\smallsetminus\\{1\\})\times[0,1]\rightarrow\widehat{{\mathcal{U}}}_{1},\quad(z,t)\mapsto[0,1,\gamma_{z}(t),z],$

is a projection map onto its image which we denote by
$X_{\widehat{{\mathcal{U}}}}\subset\widehat{{\mathcal{U}}}_{1}$. This
projection defines a structure of an $I$-bundle, with collapsed fibers when
both conditions, $|z_{4}|=1$ and $Re(z_{4})\leq 0$ hold. Thus,
$X_{\widehat{{\mathcal{U}}}}$ is homeomorphic to a tetrahedron with an edge
removed, corresponding to the fiber at $z_{4}=1$, see Figure 15.$b)$.

$a)$$\Omega_{{\mathcal{U}}_{0}}$$\bullet$$\bullet$$\bullet$$0$$1$$-1$$i$$\bullet$$z_{4}$$\gamma_{z_{4}}$$b)$$X_{\widehat{{\mathcal{U}}}}$$\Omega_{{\mathcal{U}}_{0}}$$[0,1,0,0]$$[0,1,1,1]$$\bullet$$\bullet$$\bullet$$[0,1,0,-1]$$[0,1,0,i]$${\mathcal{Z}}=[0,1,0,1]$collapsed
fibersremoved point Figure 15: $a)$ The half of disc
$\Omega_{{\mathcal{U}}_{0}}$ of possible values for $z_{4}$ in
$\widehat{{\mathcal{U}}}_{1}$ and an example of a fiber $\gamma_{z_{4}}$. For
points $z_{4}$ in the dashed arc, the fibers $\gamma_{z_{4}}$ collapse to a
point. $b)$ The topological tetrahedron with and edge removed, corresponding
to the “fiber” at the zig-zag ${\mathcal{Z}}$. The shaded regions represent
the intersection ${\mathcal{D}}_{0}\cap X_{\widehat{{\mathcal{U}}}}$.

We may observe that the remaining wedges on the set
$\widehat{{\mathcal{U}}}_{1}$ are quadrilaterals of the form $[0,1,0,z_{4}]$,
in the $I$-bundle structure they correspond to the bottom part of the fibers,
i.e. when $z_{3}=\gamma_{z_{4}}(0)$. Then removing such wedges is equivalent
to remove the bottom face of the topological tetrahedron
$X_{\widehat{{\mathcal{U}}}}$.

The quadrilaterals in the front face of the tetrahedron
$X_{\widehat{{\mathcal{U}}}}$ are $4$-segments of the form $[0,1,s,t]$, with
$-1\leq t<1$, $0<s\leq 2$ and $|s-t|\leq 1$. In this face, the closure of the
set of non-convex 4-segments is the triangle $\\{[0,1,s,t]\colon 0\leq s\leq
t\leq 1\\}$, and it is adjacent to the bottom face of wedges previously
mentioned.

We conclude that the set
$\widehat{{\mathcal{U}}}_{1}\smallsetminus{\mathcal{D}}_{0}$ is equal to the
tetrahedron $X_{\widehat{{\mathcal{U}}}}$ with two adjacent triangles removed
from its boundary, in particular, it is homeomorphic to a closed 3-ball with a
disc removed from its boundary, Figure 15.$b)$.

Construction of $\widehat{{\mathcal{W}}}_{1}$. As in the previous case, from
Construction 7.6, a quadrilateral
$Z=[0,1,z_{3},z_{4}]\in\widehat{{\mathcal{W}}}_{1}$ satisfies that
$z_{4}\in\Omega_{{\mathcal{W}}_{0}}=\\{z\in\mathbb{D}^{2}:\textrm{Im}(z)\geq
0\\}$ and when $z_{4}\neq 1$, the set of possible values for $z_{3}$ is a
curve $\gamma_{z_{4}}$ contained in the union
$\overline{0z_{4}}\cup\\{z_{4}+t(z_{4}-1)\colon t\geq 0\\}$ and cutted with
the constraint $|z_{3}-1|\leq 1$ (here, the case $z_{4}=1$ is again
pathological and corresponds to wedges of the form $[0,1,z_{3},1]$). Notice
that the curves $\gamma_{z_{4}}$ are degenerated as point curves when
$\textrm{Re}(z_{4})\leq 0$, corresponding to the wedges $[0,1,0,z_{4}]$. So we
may take the region
$\Omega^{\prime}_{{\mathcal{W}}_{0}}=\\{z\in\Omega_{{\mathcal{W}}_{0}}:\textrm{Re}(z)\geq
0,\ z\neq 1\\}$ and parameterize the curves $\gamma_{z_{4}}$ by a rescaling of
the piecewise arc-length in order to have a projection map onto its image

$\Omega^{\prime}_{{\mathcal{W}}_{0}}\times[0,1]\rightarrow\widehat{{\mathcal{W}}}_{1},\quad(z,t)\mapsto[0,1,\gamma_{z}(t),z].$

This projection defines a structure of an $I$-bundle, with collapsed fibers at
the segment $\textrm{Re}(z_{4})=0$. As before, if we denote by
$X_{\widehat{{\mathcal{W}}}}$ the image of the projection map, then it is
homeomorphic to a tetrahedron with an edge removed (the fiber that would
correspond to $z_{4}=1$). We may observe that the remaining wedges on the set
$\widehat{{\mathcal{W}}}_{1}$, lying in the $I$-bundle structure are the
quadrilaterals in the bottom part of the fibers, i.e. when
$z_{3}=\gamma_{z_{4}}(0)$ and having the form $[0,1,0,z_{4}]$, and thus, it is
the bottom face of the tetrahedron $X_{\widehat{{\mathcal{W}}}}$. The
$4$-segments on the $I$-bundle structure are of the form $[0,1,s,t]$ with
$0\leq s\leq t$, they are all non-convex and correspond to the front face of
the tetrahedron $X_{\widehat{{\mathcal{W}}}}$. In particular,
$\widehat{{\mathcal{W}}}_{1}\smallsetminus{\mathcal{D}}_{0}$ is homeomorphic
to a tetrahedron with two adjacent faces removed from its boundary, and it is
thus homeomorphic to a closed ball with a disc removed from its boundary. ∎

We want to understand the combinatorics of the intersections of the balls in
Lemma 7.10. To do this, we first observe that it is possible to give to these
balls the structure of a polyhedron, in the sense of Remark 5.13, and codify
the combinatorics of the intersections between neighboring balls into a graph,
Figure 16. We describe this structure in the following Lemma.

######  Lemma 7.11.

The boundaries of the balls
$\widehat{{\mathcal{U}}}_{j}\smallsetminus{\mathcal{D}}_{0}$,
$\widehat{{\mathcal{V}}}_{j}\smallsetminus{\mathcal{D}}_{0}$, and
$\widehat{{\mathcal{W}}}_{j}\smallsetminus{\mathcal{D}}_{0}$ are splitted into
bigons, corresponding to the intersection of pairs of such balls, but with
some sides of the bigons removed. This combinatorial structure is captured in
the graph of Figure 16 as follows:

  * •

each vertex of the graph represents a ball,

  * •

each black edge of the graph represents a bigon, such bigon is the
intersection of the boundaries of the balls corresponding to the vertices of
the graph,

  * •

each grey edge of the graph represents a disc which is removed to the
boundaries of both balls,

  * •

each triangular or quadrialteral cycle of the graph whose edges are all black,
represents the intersection of the corresponding balls and determine a side of
the bigon in each ball,

  * •

each triangular or quadrialteral cycle of the graph which contains a grey
edge, represents a removed side of the bigon.

$\widehat{{\mathcal{V}}}_{3}$$\widehat{{\mathcal{W}}}_{4}$$\widehat{{\mathcal{U}}}_{4}$$\widehat{{\mathcal{W}}}_{3}$$\widehat{{\mathcal{V}}}_{1}$$\widehat{{\mathcal{W}}}_{1}$$\widehat{{\mathcal{U}}}_{2}$$\widehat{{\mathcal{W}}}_{2}$$\widehat{{\mathcal{V}}}_{2}$$\widehat{{\mathcal{U}}}_{3}$$\widehat{{\mathcal{U}}}_{1}$$\widehat{{\mathcal{V}}}_{4}$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$

Figure 16: Graph describing the combinatorics of the intersections of the sets
$\widehat{{\mathcal{U}}}_{j}\smallsetminus{\mathcal{D}}_{0}$,
$\widehat{{\mathcal{V}}}_{j}\smallsetminus{\mathcal{D}}_{0}$,
$\widehat{{\mathcal{W}}}_{j}\smallsetminus{\mathcal{D}}_{0}$; and the bigon
decompositions they induce.

######  Example 7.12.

To illustrate the combinatorial description in Lemma 7.11 we give two
examples: Figure 16 tells us that the boundary of