The smoothness of the real projective deformation spaces of orderable Coxeter 3-polytopes

Suhyoung Choi Department of Mathematical Sciences, KAIST, 291 Daehak-ro Yuseong-gu Daejeon, South Korea schoi@math.kaist.ac.kr and Seungyeol Park Department of Mathematical Sciences, KAIST, 291 Daehak-ro Yuseong-gu Daejeon, South Korea sypark14@kaist.ac.kr

Abstract.

A Coxeter polytope is a convex polytope in a real projective space equipped with linear reflections in its facets, such that the orbits of the polytope under the action of the group generated by the linear reflections tessellate a convex domain in the real projective space. Vinberg proved that the group generated by these reflections acts properly discontinuously on the interior of the convex domain, thus inducing a natural orbifold structure on the polytope.

In this paper, we consider labeled combinatorial polytopes $\mathcal{G}$ associated to such orbifolds, and study the deformation space $\mathcal{C}(\mathcal{G})$ of Coxeter polytopes realizing $\mathcal{G}$ . We prove that if $\mathcal{G}$ is orderable and of normal type then the deformation space $\mathcal{C}(\mathcal{G})$ of real projective Coxeter 3-polytopes realizing $\mathcal{G}$ is a smooth manifold. This result is achieved by analyzing a natural map of $\mathcal{C}(\mathcal{G})$ into a smooth manifold called the realization space.

Key words and phrases:

Real projective structure, orbifold, Coxeter group, Moduli space

2020 Mathematics Subject Classification:

Primary 57M50; Secondary 53A20, 53C15

The first and second authors were supported by NRF Grant 2022R1A2C300316213.

1. Introduction

Let $\mathbb{S}^{d}=(\mathbb{R}^{d+1}\setminus\{0\})/\sim$ be the real projective $d$ -sphere consisting of the open rays of $\mathbb{R}^{d+1}$ from the origin. The group $\text{SL}_{\pm}(d+1,\mathbb{R})$ can then be viewed as the group of real projective automorphisms of $\mathbb{S}^{d}$ . A real projective structure on a smooth $d$ -orbifold $\mathcal{O}$ is an atlas of coordinate charts on $\mathcal{O}$ valued in $\mathbb{S}^{d}$ such that the changes of coordinates locally lie in $\text{SL}_{\pm}(d+1,\mathbb{R})$ . The deformation space of real projective structures on $\mathcal{O}$ is the space of real projective structures modulo the isotopy equivalence.

Many real projective $d$ -orbifolds (i.e., $d$ -dimensional orbifolds with real projective structures) arise from proper actions of discrete subgroups $\Gamma\subset\text{SL}_{\pm}(d+1,\mathbb{R})$ on convex open domains $\Omega\subset\mathbb{S}^{d}$ . Those orbifolds whose underlying spaces are of the form $\Omega/\Gamma$ are called convex real projective orbifolds. Finding discrete subgroups $\Gamma\subset\text{SL}_{\pm}(d+1,\mathbb{R})$ that act properly on convex open domains is directly related to constructing examples of convex real projective manifolds and orbifolds.

A large class of such discrete subgroups $\Gamma$ are the linear reflection groups (see Section 2.1). A linear reflection group is generated by linear reflections in the codimension-one faces (called facets) of a convex polytope in $\mathbb{S}^{d}$ . The linear reflection groups have been studied and characterized by Vinberg [15], enabling the study of representations of abstract Coxeter groups as linear reflection groups acting on real projective spaces. For instance, those results are used in the study of convex cocompact representations of Coxeter groups [5] in $\mathbb{RP}^{n}$ .

These contexts motivate the study of orbifolds whose fundamental groups are Coxeter groups and the study of the real projective structures on such orbifolds, with holonomy representations given by linear reflection groups. More precisely, those real projective orbifolds arise from the following observation. If $P\subset\mathbb{S}^{d}$ is a convex polyhedron and $\Gamma\subset\text{SL}_{\pm}(d+1,\mathbb{R})$ is the linear reflection group generated by reflections in the facets of $P$ , then the union $\bigcup_{\gamma\in\Gamma}\gamma(P)$ and its interior $\Omega$ are convex (see Theorem 3.2). Then the quotient space $\Omega/\Gamma$ admits a natural orbifold structure where the stabilizer subgroups of the singular points are finite subgroups of $\Gamma$ . The underlying space of the real projective orbifold $\Omega/\Gamma$ is homeomorphic to $P$ minus some of its faces, where a face of $P$ is removed if and only if its stabilizer subgroup in $\Gamma$ is infinite.

Following this observation, we define a class of orbifolds known as reflection orbifolds or Coxeter orbifolds. A Coxeter orbifold is an orbifold $\widehat{P}$ whose underlying space is a convex polyhedron $P$ in $\mathbb{S}^{d}$ with certain faces of codimension $\geq 3$ removed. The singular locus of the orbifold is the union of the non-removed faces of codimension $\geq 1$ , i.e. the boundary of $P$ minus the removed faces.

The motivation of this paper is to study the deformation space of real projective structures on 3-dimensional Coxeter orbifolds, whence the removed faces are some vertices of the underlying polytope. However, to effectively analyze the topological structure of the deformation space, we avoid the direct use of orbifold terminologies. Instead, we associate each Coxeter orbifold with a combinatorial object called a labeled combinatorial polytope $\mathcal{G}$ , which encodes the orbifold structure. Consequently, the study of the deformation space of real projective structures is reduced to considering the space $\mathcal{C}(\mathcal{G})$ of isomorphism classes of Coxeter polytopes realizing $\mathcal{G}$ (see Sections 2.1 and 2.2 for precise definitions). We will also call the space $\mathcal{C}(\mathcal{G})$ the deformation space of Coxeter polytopes realizing $\mathcal{G}$ . By making such replacements, the space $\mathcal{C}(\mathcal{G})$ can be identified with the solution space of certain polynomial equalities and inequalities using results of Vinberg [15].

A labeled combinatorial polytope $\mathcal{G}$ is a CW-complex whose underlying space is a convex polyhedron $P$ . The cells of the CW-complex are the faces of $P$ across all dimensions, where the unique top-dimensional cell is $P$ itself, the facets of $P$ are the $(n-1)$ -dimensional cells, and the ridges are the $(n-2)$ -dimensional cells, and so on. We index the facets of $P$ by $P_{1},\dots,P_{f}$ , and assign to each ridge $P_{i}\cap P_{j}$ an integer label $m_{i,j}\geq 2$ .

A Coxeter polytope realizing $\mathcal{G}$ is a convex polyhedron $Q\subset\mathbb{S}^{d}$ equipped with linear reflections $r_{1},\cdots,r_{f}$ in its $f$ facets, such that $Q$ is combinatorially equivalent to $\mathcal{G}$ and the linear reflections $r_{1},\cdots,r_{f}$ generate a discrete subgroup $\Gamma\subset\text{SL}_{\pm}(d+1,\mathbb{R})$ such that the translates $\gamma\cdot Q$ for $\gamma\in\Gamma$ tessellate a convex domain of $\mathbb{S}^{d}$ (see Definition 2.2 for the precise definition). Each Coxeter polytope realizing $\mathcal{G}$ gives rise to a convex real projective structure on the Coxeter orbifold associated to $\mathcal{G}$ . In this way, the deformation space $\mathcal{C}(\mathcal{G})$ is embedded in the deformation space of the real projective structures on the Coxeter orbifold associated to $\mathcal{G}$ .

Figure 1. A truncation of a convex polytope

P

by a hyperplane

\Pi\subset\mathbb{S}^{3}

As a part of this field, Marquis [8] studied labeled combinatorial polytopes whose underlying spaces are truncation polytopes, which are the polytopes obtained from a 3-simplex (i.e. a tetrahedron) by successively truncating vertices (Figure 1 illustrates a truncation process.). Marquis [8] proved that the deformation space $\mathcal{C}(\mathcal{G})$ is a union of $\mathbb{R}^{k}$ for some $k\geq 0$ . See [4] for the analogous result for higher-dimensional truncation polytopes. These motivate the following generalized question:

Question 1.1.

What can be determined about the local and global structures of the deformation space of Coxeter 3-orbifolds under suitable combinatorial conditions?

There have been several results answering the above question. For example, Choi-Hodgson-Lee [2] and Choi-Lee [3] found some classes of Coxeter 3-orbifolds admitting a (unique) hyperbolic structure, such that the point in the deformation space corresponding to the hyperbolic structure admits a neighborhood homeomorphic to an open cell.

In contrast, as explained by Choi-Lee ([3], Section 5.3), there is a Coxeter 4-orbifold whose deformation space is the union of two lines intersecting at a single point. Hence, some real projective structures on certain Coxeter orbifolds may not even admit open neighborhoods homeomorphic to cells, so in particular their corresponding deformation spaces are not a topological manifold.

In this paper, we provide a sufficient condition on a labeled combinatorial 3-polytope $\mathcal{G}$ for the space $\mathcal{C}(\mathcal{G})$ to be a smooth manifold, ensuring that each real projective structure on the corresponding orbifold admits deformations of a fixed dimension. The main condition we assume is the notion of orderability, whose precise definition is given in Definition 4.1. To determine an element of $\mathcal{C}(\mathcal{G})$ , we have to consider a convex polytope $Q\subset\mathbb{S}^{3}$ together with linear reflections $r_{1},\dots,r_{f}\in\text{SL}_{\pm}(4,\mathbb{R})$ across the facets $Q_{1},\dots,Q_{f}$ of $Q$ . The orderability condition ensures that there exists an indexing $Q_{1},\dots,Q_{f}$ of the $f$ facets of $Q$ such that the linear reflections $r_{1},\dots,r_{f}$ can be constructed successively by solving certain linear equalities and inequalities. Under the orderability assumption on $\mathcal{G}$ , the space $\mathcal{C}(\mathcal{G})$ fibers over the “space of polytopes” that are combinatorially equivalent to $\mathcal{G}$ , with the fiber over each polytope determined by solving linear equalities and inequalities.

In addition to assuming the orderability condition, we further exclude certain labeled combinatorial polytopes from our discussion. These polytopes form a small and exceptional class whose deformation spaces can be computed individually in elementary ways. Namely, we do not consider the labeled combinatorial polytopes $\mathcal{G}$ such that one Coxeter polytope (hence all Coxeter polytopes) realizing $\mathcal{G}$ tessellate the sphere $\mathbb{S}^{3}$ or an affine chart $\mathbb{A}^{3}\subset\mathbb{S}^{3}$ under the action of its linear reflection group. The former case occurs if and only if the associated Coxeter group $\Gamma_{\mathcal{G}}$ is finite (see (2.2) for the definition of $\Gamma_{\mathcal{G}}$ ). Additionally, we exclude Coxeter polytopes that are either cones over polygons or products of polygons with closed intervals. The remaining Coxeter polytopes, which do not fall into any of these excluded categories, are referred to as being of normal type (see Definition 4.2). This distinction allows us to focus on the cases where our main results apply.

Choi [1] proved that if a Coxeter 3-orbifold is orderable and of normal type, then the subspace of $\mathcal{C}(\mathcal{G})$ consisting of real projective structures sharing a common fundamental domain is a smooth manifold. We will state this result precisely in Section 4 and use it to prove the following result about the smoothness of the global space $\mathcal{C}(\mathcal{G})$ .

Theorem 1.2.

Let $\mathcal{G}$ be a labeled combinatorial 3-polytope. Let $f$ and $e_{2}$ be the numbers of facets and the edges of order 2 of $\mathcal{G}$ . Suppose that $\mathcal{G}$ is orderable and is of normal type. Suppose further that the stabilizer subgroup of $\textup{SL}_{\pm}(4,\mathbb{R})$ fixing any polyhedron $P\subset\mathbb{S}^{3}$ of combinatorial type $\mathcal{G}$ is trivial. Then $\mathcal{C}(\mathcal{G})$ is a smooth manifold of dimension $3f-e_{2}-9$ .

The proof of Theorem 1.2 follows the idea suggested by Hodgson, which is formulated in Corollary 2 of [1]. His idea presents another way to parametrize the deformation space of real projective structures on Coxeter orbifolds. The basic idea is that the deformation space $\mathcal{C}(\mathcal{G})$ fibers over a smooth manifold whose elements are polytopes representing the class $\mathcal{G}$ . The base space is called the realization space, and it is a smooth manifold if $\mathcal{G}$ is 3-dimensional (see Section 5, or [6]). On the other hand, the non-empty fibers of the projection are called the restricted deformation spaces. It is proved by Choi [1] that under the hypothesis of Theorem 1.2, the restricted deformation spaces are smooth manifolds of a common dimension. We combine these results to construct smooth charts around each element of $\mathcal{C}(\mathcal{G})$ . We cannot apply this argument for labeled combinatorial polytopes $\mathcal{G}$ of general dimensions. One reason is that the realization spaces for some polytopes of dimension $\neq 3$ may not even be a topological manifold [11].

The orderability condition in Theorem 1.2 has a connection with Marquis’ result [8]. Namely, if $\mathcal{G}$ is simple (i.e. each vertex of $\mathcal{G}$ is joined to exactly three edges) and orderable, then $\mathcal{G}$ is a truncation polytope. If the assumptions of Theorem 1.2 are satisfied and $\mathcal{G}$ is a truncation polytope, then the space $\mathcal{C}(\mathcal{G})$ is homeomorphic to a disjoint union of some copies of $\mathbb{R}^{e_{+}-3}$ by Marquis (Theorem 3.16, [8]). Then we see that Theorem 1.2 verifies Marquis’ result, since a truncation polytope always satisfy $e=3(f-2)$ , and we have

e_{+}-3=(e-e_{2})-3=3f-e_{2}-9.

Therefore, Theorem 1.2 extends to the labeled combinatorial polytopes $\mathcal{G}$ which is orderable but not necessarily simple.

The study of deformations of Coxeter 3-orbifolds provides a way to construct new examples of real projective orbifolds via truncation and gluing. This method was introduced in [4], and we briefly mention the process below.

Consider two Coxeter 3-orbifolds arising from hyperbolic polyhedra with ideal vertices. If each orbifold can be deformed so that one of its ideal vertices moves outside of the closure of the hyperbolic 3-space, then it is possible to truncate the orbifolds at these ‘hyperideal’ vertices, yielding new real projective orbifolds. By gluing the resulting orbifolds along the triangular faces produced by truncation, we obtain another non-hyperbolic real projective orbifold.

Understanding when a given ideal vertex admits a deformation of this type is therefore useful. The dimension of the deformation space near the hyperbolic structure of a Coxeter 3-orbifold provides information on the extent to which such deformations can occur.

Table 1 shows the counts of orderable labeled combinatorial polytopes. In the counting process, one notes that the only factor determining the orderability of a labeled combinatorial polytope is whether the integer assigned to each edge is 2 or not. For example, when counting the number of all labeled polytopes with 5 facets, we note that there are exactly two combinatorial polytopes: the cone over a quadrilateral and the triangular prism. The cone over a quadrilateral has 8 edges, and the triangular prism has 9 edges. Hence, we have $2^{8}+2^{9}=768$ labeled combinatorial polytopes in total. Among them, we have 654 labeled combinatorial polytopes which are orderable. The other cases follow the same procedure. Due to computational difficulty we cannot go beyond 7 faces yet. The Mathematica code for the counts can be found at the webpage [10].

# of facets	Orderable labeled polytopes	All labeled polytopes	Ratio
4	64	64	1
5	654	768	0.851563
6	7130	14848	0.480199
7	157334	421888	0.372928

Table 1. Portions of orderable Coxeter 3-orbifolds

1.1. Outline of the Paper

Sections 2 to 5 present the necessary preliminaries for stating and proving Theorem 1.2.

In Section 2, we define labeled combinatorial polytopes $\mathcal{G}$ and introduce related concepts, including the deformation spaces $\mathcal{C}(\mathcal{G})$ and the Coxeter group $\Gamma_{\mathcal{G}}$ . To facilitate the proof of our main theorem, we identify $\mathcal{C}(\mathcal{G})$ with a subspace of the quotient

((\mathbb{R}^{n+1})^{*})^{f}\times(\mathbb{R}^{n+1})^{f}/\textup{SL}_{\pm}(n+1,\mathbb{R})\times\mathbb{R}_{+}^{f}.

In Section 3, we review some results of Vinberg on linear reflection groups. Specifically, we recall the necessary and sufficient conditions for a subgroup of $\text{SL}_{\pm}(n+1,\mathbb{R})$ , generated by linear reflections in the facets of a polyhedral cone, to generate a linear Coxeter group. These results are closely related to the Coxeter polytopes considered in Section 2 and allow us to describe the elements of

((\mathbb{R}^{n+1})^{*})^{f}\times(\mathbb{R}^{n+1})^{f}/\textup{SL}_{\pm}(n+1,\mathbb{R})\times\mathbb{R}_{+}^{f}

that are identified with elements of $\mathcal{C}(\mathcal{G})$ (see Remark 3.4).

In Section 4, we recall Choi’s result [1] on restricted deformation spaces of $\mathcal{G}$ . We recall the result on the smoothness of the restricted deformation spaces (Theorem 4.5), which is necessary in our proof. We also introduce concepts such as labeled combinatorial polytopes of normal type and describe the restricted deformation spaces in our terminology.

In Section 5, we introduce the notion of realization spaces. We review the results of Steinitz [13] and Filpo Molina [6], which state the smoothness results on the realization spaces of 3-dimensional polytopes necessary for proving our main theorem.

In Section 6, we prove our main theorem by constructing a map $\overline{\rho}$ from the deformation space $\mathcal{C}(\mathcal{G})$ into the associated realization space. We use the smoothness results mentioned in previous sections to complete the proof.

In Section 7, we provide two examples of $\mathcal{G}$ and compute their deformation spaces $\mathcal{C}(\mathcal{G})$ . The first example illustrates in detail the process of computing restricted deformation spaces and the deformation space itself using the orderability conditions. This explicit computation verifies Theorem 1.2 and describes the global structure of $\mathcal{C}(\mathcal{G})$ . The second example follows a similar computation but proves that the map $\overline{\rho}$ is not necessarily a fiber bundle.

Acknowledgement This work is supported by the National Research Foundation of Korea under Grant number 2022R1A2C300316213.

We are grateful to Gye-Seon Lee and Jean-Paul Filpo Molina for their valuable information and discussions. We also thank Ludovic Marquis for his helpful information and lectures.

2. Deformation spaces of Coxeter polytopes

In Section 2.1, we review the notion of labeled combinatorial 3-polytopes $\mathcal{G}$ , which can be associated to Coxeter 3-orbifolds in the manner described in the previous section. In Section 2.2, we recall the notion of the deformation space of Coxeter 3-polytopes $\mathcal{C}(\mathcal{G})$ realizing a labeled combinatorial polytope $\mathcal{G}$ .

2.1. Labeled combinatorial polytopes

We recall the definition of labeled combinatorial polytopes and related notions such as their associated Coxeter groups and Coxeter graphs.

Let $\mathbb{S}^{n}$ denote the projective $n$ -sphere, defined as the quotient space $(\mathbb{R}^{n+1}\setminus\{0\})/\sim$ , where two nonzero vectors $v$ and $w$ of $\mathbb{R}^{n+1}$ are equivalent if and only if $v=tw$ for some $t>0$ . For each subset $A\subset\mathbb{R}^{n+1}$ , we let $\mathbb{S}(A)$ denote the image of $A\setminus\{0\}$ under the projection $\mathbb{R}^{n+1}\setminus\{0\}\to\mathbb{S}^{n}$ . An affine chart of $\mathbb{S}^{n}$ is a subset of the form $\{[v]\in\mathbb{S}^{n}\ |\ \alpha(v)>0\}$ for some nonzero linear functional $\alpha$ on $\mathbb{R}^{n+1}$ . Any affine chart $\Omega$ of $\mathbb{S}^{n}$ can be identified with $\mathbb{R}^{n}$ in such a way (though not uniquely) that for each nontrivial linear subspace $W\subset\mathbb{R}^{n+1}$ , the intersection $\mathbb{S}(W)\cap\Omega$ is either empty or forms an affine subspace of $\Omega=\mathbb{R}^{n}$ . A subset $C\subset\mathbb{S}^{n}$ is said to be convex if it is contained in an affine chart of $\mathbb{S}^{n}$ and is convex in the affine chart. Furthermore, the set $C$ is said to be properly convex if its closure $\overline{C}$ is a convex and bounded subset of an affine chart of $\mathbb{S}^{n}$ .

A convex $n$ -polytope is a properly convex subset of $\mathbb{S}^{n}$ , given by

(2.1)

\displaystyle P=\{[v]\in\mathbb{S}^{n}\mid\alpha_{1}(v)\geq 0,\dots,\alpha_{f}(v)\geq 0\}

for some nonzero linear functionals $\alpha_{1},\dots,\alpha_{f}$ on $\mathbb{R}^{n+1}$ . Throughout this paper, we will assume that none of these defining inequalities are redundant, meaning none of the inequalities $\alpha_{j}\geq 0$ is implied by the others. The boundary $\partial P$ of $P$ in $\mathbb{S}^{n}$ is the union of the faces of $P$ , each of which is a convex $k$ -polytope for some $k<n$ in a $k$ -dimensional projective subspace of $\mathbb{S}^{n}$ . We call the 0-dimensional (resp. 1-dimensional) faces of $P$ the vertices (resp. edges). We call the codimension-1 (resp. codimension-2) faces of $P$ the facets (resp. ridges). Since the inequalities $\alpha_{1}\geq 0,\cdots,\alpha_{f}\geq 0$ are assumed to be non-redundant, the convex $n$ -polytope $P$ has exactly $f$ facets.

Let $P\subset\mathbb{S}^{n}$ be a convex $n$ -polytope. The face lattice $\text{FL}(P)$ of $P$ is the set of faces of $P$ partially ordered by the set inclusion. Two convex $n$ -polytopes $P$ and $P^{\prime}$ are combinatorially equivalent if there is a bijection $\phi:\text{FL}(P)\to\text{FL}(P^{\prime})$ preserving the inclusion relations: $f_{1}\subset f_{2}$ if and only if $\phi(f_{1})\subset\phi(f_{2})$ . A combinatorial $n$ -polytope $\mathcal{G}$ is a combinatorial equivalence class of convex $n$ -polytopes.

We provide an alternative description of combinatorial polytopes, which will be essential for our subsequent discussion. The labeled combinatorial polytope $\mathcal{G}$ can be understood as a homeomorphism class of CW-complexes, represented by a CW-complex whose underlying space is $P$ , and whose $i$ -dimensional cells correspond to the $i$ -dimensional faces of $P$ . The notions of faces, vertices, edges, facets, ridges, and so on in $\mathcal{G}$ are naturally derived from the corresponding notions in any of its representatives. For example, if the facets of $P\subset\mathbb{S}^{n}$ are denoted by $P_{1},\ldots,P_{f}$ , we define the facets $\mathcal{G}_{1},\ldots,\mathcal{G}_{f}$ as the codimension-1 subcomplexes represented by $P_{1},\ldots,P_{f}$ , respectively.

For each pair of adjacent facets, $\mathcal{G}_{i}$ and $\mathcal{G}_{j}$ , we assign an integer $m_{i,j}\geq 2$ to the ridge $\mathcal{G}_{i}\cap\mathcal{G}_{j}$ . The combinatorial polytope $\mathcal{G}$ , together with the assigned integers, is called a labeled combinatorial polytope.

For each labeled combinatorial polytope $\mathcal{G}$ with facets $\mathcal{G}_{1},\cdots,\mathcal{G}_{f}$ and an assignment of integers $m_{i,j}\geq 2$ , we can associate a Coxeter group, an abstract group with the group presentation

(2.2)

\displaystyle\Gamma_{\mathcal{G}}:=\left\langle\gamma_{1},\cdots,\gamma_{f}\ |\ \gamma_{i}^{2}=1,\ (\gamma_{i}\gamma_{j})^{m_{i,j}}=1\right\rangle.

We set $m_{i,j}=\infty$ if $\mathcal{G}_{i}$ and $\mathcal{G}_{j}$ are not adjacent.

Definition 2.1.

For a Coxeter group $\Gamma=\Gamma_{\mathcal{G}}$ in the form (2.2), we can associate a Coxeter graph $G_{\Gamma}$ . It is a weighted graph such that

(i)

the vertices (or the nodes) are the generators $\gamma_{1},\cdots,\gamma_{f}$ ;
(ii)

two vertices $\gamma_{i}$ , $\gamma_{j}$ are joined by a single edge if and only if $m_{i,j}\in\{3,4,\cdots,\infty\}$ , and in that case the edge is labeled by $m_{i,j}$ .

The labels $m_{i,j}=3$ are conventionally omitted in the representations of Coxeter graphs; however, this convention will not be relevant in this paper. We say that the Coxeter group $\Gamma$ is irreducible if its associated Coxeter graph $G_{\Gamma}$ is a connected graph.

2.2. The deformation space $\mathcal{C}(\mathcal{G})$ of Coxeter polytopes

In this section, we define the notion of Coxeter polytopes and their deformation spaces. We will also explain the topology assigned to the deformation spaces.

Definition 2.2.

Let $\mathcal{G}$ be a labeled combinatorial polytope with $f$ facets $\mathcal{G}_{1},\ldots,\mathcal{G}_{f}$ . A Coxeter n-polytope realizing $\mathcal{G}$ is a convex $n$ -polytope $P\subset\mathbb{S}^{n}$ together with linear reflections $r_{1},\ldots,r_{f}\in\text{SL}_{\pm}(n+1,\mathbb{R})$ such that:

(i)

$P$ is a convex $n$ -polytope combinatorially equivalent to $\mathcal{G}$ , with facets $P_{1},\ldots,P_{f}\subset P$ corresponding to the facets $\mathcal{G}_{1},\ldots,\mathcal{G}_{f}$ ;
(ii)

each $r_{j}$ is a reflection in the hyperplane of $\mathbb{S}^{n}$ supporting the facet $P_{j}$ ;
(iii)

if the facets $P_{i},P_{j}$ are adjacent and the integer assigned to the edge $P_{i}\cap P_{j}$ is $m_{i,j}$ , then the product $r_{i}r_{j}$ has order $m_{i,j}$ in the group $\text{SL}_{\pm}(n+1,\mathbb{R})$ ; and
(iv)

the subgroup $\Gamma\subset\text{SL}_{\pm}(n+1,\mathbb{R})$ generated by the linear reflections $r_{1},\ldots,r_{f}$ satisfies the condition

$\text{Int}(P)\cap\gamma\cdot\text{Int}(P)=\varnothing\quad\text{for}\ \gamma\in\Gamma\setminus\{1\}.$

Note that the group $\text{SL}_{\pm}(n+1,\mathbb{R})$ acts (on the left) on the Coxeter $n$ -polytopes realizing $\mathcal{G}$ by

g\cdot(P,r_{1},\ldots,r_{f}):=(g(P),gr_{1}g^{-1},\ldots,gr_{f}g^{-1}),\quad g\in\text{SL}_{\pm}(n+1,\mathbb{R}).

Definition 2.3.

Let $\mathcal{G}$ be a labeled combinatorial polytope with $f$ facets $\mathcal{G}_{1},\ldots,\mathcal{G}_{f}$ . The deformation space of Coxeter n-polytopes realizing $\mathcal{G}$ is the space

\mathcal{C}(\mathcal{G}):=\{\text{Coxeter}\ n\text{-polytopes}\ (P,r_{1},\ldots,r_{f})\ \text{realizing}\ \mathcal{G}\}/\text{SL}_{\pm}(n+1,\mathbb{R}).

Let $\mathbb{R}_{+}$ be the multiplicative group of positive real numbers. The product Lie group $\text{SL}_{\pm}(n+1,\mathbb{R})\times\mathbb{R}_{+}^{f}$ acts on the vector space $\left((\mathbb{R}^{n+1})^{*}\right)^{f}\times(\mathbb{R}^{n+1})^{f}$ by

		$\displaystyle(A,c_{1},\ldots,c_{f})\cdot(\alpha_{1},\ldots,\alpha_{f},v_{1},\ldots,v_{f})$
(2.3)			$\displaystyle\quad:=(c_{1}^{-1}\alpha_{1}\circ A^{-1},\ldots,c_{f}^{-1}\alpha_{f}\circ A^{-1},c_{1}Av_{1},\ldots,c_{f}Av_{f}).$

To topologize $\mathcal{C}(\mathcal{G})$ , we identify the space $\mathcal{C}(\mathcal{G})$ as a subset of the quotient space

(2.4)

\displaystyle\left((\mathbb{R}^{n+1})^{*}\right)^{f}\times(\mathbb{R}^{n+1})^{f}/(\text{SL}_{\pm}(n+1,\mathbb{R})\times\mathbb{R}_{+}^{f})

in the following manner.

Let $(P,r_{1},\ldots,r_{f})$ be a tuple representing an element of $\mathcal{C}(\mathcal{G})$ , where $r_{1},\ldots,r_{f}\in\text{SL}_{\pm}(n+1,\mathbb{R})$ are linear reflections in the facets $P_{1},\ldots,P_{f}$ of the convex $n$ -polytope $P$ , generating a linear reflection group. For each $j\in\{1,\ldots,f\}$ , we have an expression $r_{j}=\text{Id}-\alpha_{j}\otimes v_{j}$ for some $\alpha_{j}\in(\mathbb{R}^{n+1})^{*}$ and $v_{j}\in\mathbb{R}^{n+1}$ with $\alpha_{j}(v_{j})=2$ and $\alpha_{j}(x)\geq 0$ for $x\in P$ . Then $P$ can be expressed as (2.1).

For each element $[(P,r_{1},\cdots,r_{f})]\in\mathcal{C}(\mathcal{G})$ , the tuple $(r_{1},\ldots,r_{f})$ of linear reflections is uniquely determined by the action of the group $\text{SL}_{\pm}(n+1,\mathbb{R})$ where the action is given by

g\cdot(r_{1},\ldots,r_{f}):=(gr_{1}g^{-1},\ldots,gr_{f}g^{-1}).

Moreover, for each reflection $r_{j}=\text{Id}-\alpha_{j}\otimes v_{j}$ with $\alpha_{j}(v_{j})=2$ , the tuple $(\alpha_{j},v_{j})$ is determined by the action of the multiplicative group $\mathbb{R}_{+}$ given by

c\cdot(\alpha_{j},v_{j}):=(c^{-1}\alpha_{j},cv_{j}).

Hence, the element $[(P,r_{1},\ldots,r_{f})]\in\mathcal{C}(\mathcal{G})$ determines a unique element

\Phi([(P,r_{1},\ldots,r_{f})]):=[(\alpha_{1},\ldots,\alpha_{f},v_{1},\ldots,v_{f})]

of the orbit space in (2.4).

Thus, we have obtained a map

\Phi:\mathcal{C}(\mathcal{G})\to\left((\mathbb{R}^{n+1})^{*}\right)^{f}\times(\mathbb{R}^{n+1})^{f}/(\text{SL}_{\pm}(n+1,\mathbb{R})\times\mathbb{R}_{+}^{f}).

By the following lemma, the map $\Phi$ is injective. Throughout this paper, we give $\mathcal{C}(\mathcal{G})$ the topology induced by the injective map $\Phi$ . This identification will be useful in the proof of our main theorem, particularly in relating the space $\mathcal{C}(\mathcal{G})$ to the realization space, which will be discussed in Section 5.

Lemma 2.4.

The map $\Phi$ is injective. In particular, there is a unique topology on $\mathcal{C}(\mathcal{G})$ such that the map $\Phi$ is a topological embedding.

Proof.

The map $\Phi$ is defined by associating to each equivalence class $[(P,r_{1},\ldots,r_{f})]\in\mathcal{C}(\mathcal{G})$ the orbit $[(\alpha_{1},\ldots,\alpha_{f},v_{1},\ldots,v_{f})]$ of $(\alpha_{1},\ldots,\alpha_{f},v_{1},\ldots,v_{f})$ with $r_{j}=\text{Id}-\alpha_{j}\otimes v_{j}$ under the action of $\text{SL}_{\pm}(n+1,\mathbb{R})\times\mathbb{R}_{+}^{f}$ as described in (2.2). Since each orbit $[(\alpha_{j},v_{j})]$ uniquely determines the linear reflection $r_{j}=\text{Id}-\alpha_{j}\otimes v_{j}$ up to projective equivalence, and since the convex polytope $P$ is determined (up to projective equivalence) by the collection of inequalities $\alpha_{j}(x)\geq 0$ , $\Phi$ is injective. ∎

3. The theory of Vinberg

Vinberg [15] characterized the discrete subgroups $\Gamma$ of $\text{SL}_{\pm}(n+1,\mathbb{R})$ generated by reflections in the facets of convex polyhedra in $\mathbb{S}^{n}$ , in terms of the linear functionals and vectors defining these reflections. It is shown that these linear functionals and vectors must satisfy specific real polynomial equalities and inequalities.

Moreover, Vinberg’s results provide a method to construct the universal covering manifold of a given Coxeter $n$ -orbifold. Using this method, we can identify the deformation space $\mathcal{C}(\mathcal{G})$ as a subspace of the deformation space of real projective structures on the Coxeter orbifold associated with $\mathcal{G}$ . This process will be briefly mentioned in Theorem 3.2 and Remark 3.3 but will not be discussed in detail, as it does not appear in the subsequent discussions.

In this section, we present some of Vinberg’s results, the first of which will be frequently used in our discussions. Additionally, we strengthen Lemma 2.4 by describing the image of the embedding $\Phi$ (Remark 3.4).

A linear transformation $A\in\text{GL}(n+1,\mathbb{R})$ is called a linear reflection if $A^{2}=I$ (hence diagonalizable) and it has $-1$ as its eigenvalue with algebraic multiplicity $1$ . Then $A$ can be written in the form $A=\text{Id}-\alpha\otimes v$ for some linear functional $\alpha\in(\mathbb{R}^{d+1})^{*}\setminus\{0\}$ and vector $v\in\mathbb{R}^{d+1}\setminus\{0\}$ with $\alpha(v)=2$ . The vector $v$ is an eigenvector of $A$ corresponding to the simple eigenvalue $-1$ , and the kernel of $\alpha$ is fixed pointwise by the linear reflection $A$ .

Let $K\subset\mathbb{R}^{n+1}$ be a convex polyhedral cone given by

K:=\{x\in\mathbb{R}^{n+1}\ |\ \alpha_{j}(x)\geq 0,\ j=1,\cdots,f\},

for some linear functionals $\alpha_{1},\cdots,\alpha_{f}\in(\mathbb{R}^{n+1})^{*}\setminus\{0\}$ . Let $v_{1},\cdots,v_{f}\in\mathbb{R}^{n+1}$ be vectors with $\alpha_{j}(v_{j})=2$ and let $R_{j}:=I-\alpha_{j}\otimes v_{j}$ be the reflection determined by $\alpha_{j},v_{j}$ . We further assume that $K$ has nonempty interior and each inequality $\alpha_{j}\geq 0$ is not implied by the other $(f-1)$ inequalities.

The subgroup $\Gamma$ of $\text{GL}(n+1,\mathbb{R})$ generated by the $f$ reflections $R_{1},\cdots,R_{f}$ is called a linear reflection group generated by the reflections $R_{1},\cdots,R_{f}$ if we further have

\gamma\cdot K\cap K=\varnothing\quad\text{for}\ \gamma\in\Gamma\setminus\{1\}.

Linear reflection groups are also called linear Coxeter groups or discrete linear groups [15].

In this setting, we have the following theorem.

Theorem 3.1.

([15], Theorem 1) Let $K$ , $\alpha_{j}$ , $v_{j}$ and $R_{j}$ , $j=1,\cdots,f$ be as above. Then the subgroup of $\textup{GL}(n+1,\mathbb{R})$ generated by $R_{1},\cdots,R_{f}$ is a linear reflection group if and only if $\alpha_{1},\cdots,\alpha_{f}$ , $v_{1},\cdots,v_{f}$ satisfy the following conditions:

(i)

$\alpha_{i}(v_{j})\leq 0$ if $i\neq j$ ;
(ii)

$\alpha_{i}(v_{j})=0$ if and only if $\alpha_{j}(v_{i})=0$ ;
(iii)

$\alpha_{i}(v_{j})\alpha_{j}(v_{i})\geq 4$ or $\alpha_{i}(v_{j})\alpha_{j}(v_{i})=4\cos^{2}(\frac{\pi}{n_{i,j}})$ for some integer $n_{i,j}\geq 2$ .

In this case, the polyhedral cone $K$ is a fundamental domain of the discrete subgroup, and the subgroup is isomorphic to the abstract group

(3.1)

\displaystyle\left\langle\gamma_{1},\cdots,\gamma_{f}\ |\ \gamma_{1}^{2}=\cdots=\gamma_{f}^{2}=1,\ (\gamma_{i}\gamma_{j})^{n_{i,j}}=1\right\rangle,

via the isomorphism $R_{j}\mapsto r_{j}$ . Here, we have $n_{i,j}=1$ if $i=j$ . If $i\neq j$ and $\alpha_{i}(v_{j})\alpha_{j}(v_{i})\geq 4$ , then the relation $(\gamma_{i}\gamma_{j})^{n_{i,j}}=1$ is omitted from the group presentation (3.1).

The following theorem and its subsequent Remark 3.3 enable us to relate the deformation space $\mathcal{C}(\mathcal{G})$ of Coxeter polytopes realizing the labeled combinatorial polytope $\mathcal{G}$ with the deformation space of real projective structures on the Coxeter orbifold associated to $\mathcal{G}$ (see Section 1).

Theorem 3.2.

([15], Theorem 2) Let $\Gamma\subset\textup{SL}_{\pm}(n+1,\mathbb{R})$ be a linear reflection group generated by reflections $R_{1},\cdots,R_{f}\in\textup{SL}_{\pm}(n+1,\mathbb{R})$ in the facets of a convex polyhedral cone $K$ . For each $x\in K$ , let $\Gamma_{x}$ denote the subgroup of $\Gamma$ generated by reflections in those facets of $K$ which contain $x$ . Define

K^{f}:=\{x\in K\ |\ \Gamma_{x}\ \textup{is finite}\}.

Then the following assertions are true.

(i)

$\bigcup_{\gamma\in\Gamma}\gamma K$ is a convex cone.
(ii)

$\Gamma$ acts discretely on the interior $\widetilde{\Omega}:=\textup{Int}\left(\bigcup_{\gamma\in\Gamma}\gamma K\right)$ of the cone $\bigcup_{\gamma\in\Gamma}\gamma K$ .
(iii)

$\widetilde{\Omega}\cap K=K^{f}$ .
(iv)

The canonical map $K^{f}\to\widetilde{\Omega}/\Gamma$ is a homeomorphism.
(v)

For every $x\in K$ , $\Gamma_{x}\subset\Gamma$ is the stabilizer of $x$ in $\Gamma$ .
(vi)

For every pair $K_{i},K_{j}$ of $K$ supported by the functionals $\alpha_{i},\alpha_{j}$ , let $n_{ij}$ denote the order of $R_{i}R_{j}$ ( $n_{ij}$ may be infinite). Then

$R_{i}^{2}=1,\quad(R_{i}R_{j})^{n_{ij}}=1$

is a system of defining relations for $\Gamma$ .

Remark 3.3.

Let $K^{f},\widetilde{\Omega}$ be as in the theorem. Let $\pi:\mathbb{R}^{n+1}\setminus\{0\}\to\mathbb{S}^{n}$ be the natural projection and let $\Omega:=\pi(\widetilde{\Omega})$ , $\widehat{P}:=\pi(K^{f})$ . Since $\widetilde{\Omega}$ is convex by Theorem 3.2, its projectivization $\Omega\subset\mathbb{S}^{n}$ is also convex. Therefore, the quotient $\Omega/\Gamma$ gives a convex real projective structure on the orbifold $\widehat{P}$ . In fact, Theorem 2 of [1] states that every real projective structure on a Coxeter orbifold is convex when the orbifold is 3-dimensional.

Remark 3.4.

Recall from Lemma 2.4 that the space $\mathcal{C}(\mathcal{G})$ can be identified as a subspace of the quotient space (2.4). We can improve the lemma by describing the image of the embedding $\Phi$ as follows. It follows from Theorem 3.1 and the definition of $\mathcal{C}(\mathcal{G})$ that

[(\alpha,v)]=[(\alpha_{1},\cdots,\alpha_{f},v_{1},\cdots,v_{f})]\in((\mathbb{R}^{n+1})^{*})^{f}\times(\mathbb{R}^{n+1})^{f}/\textup{SL}_{\pm}(n+1,\mathbb{R})\times\mathbb{R}_{+}^{f}

belongs to (the image of) $\mathcal{C}(\mathcal{G})$ if and only if

(i)

$\alpha_{i}(v_{j})\leq 0$ if $i\neq j$ ;
(ii)

$\alpha_{i}(v_{j})=0$ if and only if $\alpha_{j}(v_{i})=0$ ;
(iii)

$\alpha_{i}(v_{j})\alpha_{j}(v_{i})\geq 4$ or $\alpha_{i}(v_{j})\alpha_{j}(v_{i})=4\cos^{2}(\frac{\pi}{n_{i,j}})$ for some integer $n_{i,j}\geq 2$ ;
(iv)

the projectivization of the polyhedral cone $\{x\in\mathbb{R}^{n+1}\ |\ \alpha_{j}\geq 0\ \text{for each}\ j\}$ in $\mathbb{S}^{n}$ is a convex $n$ -polytope representing the combinatorial polytope $\mathcal{G}$ , so that the facet supported by $\alpha_{j}$ represents $\mathcal{G}_{j}$ .

4. Restricted deformation spaces

Given a discrete subgroup $\Gamma\subset\text{SL}_{\pm}(n+1,\mathbb{R})$ preserving a convex open domain $\Omega\subset\mathbb{S}^{n}$ , it admits fundamental domains $D\subset\Omega$ such that $\bigcup_{\gamma\in\Gamma}\gamma\cdot D=\Omega$ . Choi [1] proved that if a Coxeter 3-orbifold is orderable and satisfies certain generic conditions, then the space consisting of real projective structures whose holonomy representations share a common fixed fundamental domain is a smooth manifold. Such spaces are called the restricted deformation spaces.

In this section, we recall the precise notion of restricted deformation spaces and the sufficient conditions under which Coxeter 3-orbifolds admit smooth restricted deformation spaces.

Let $P,Q\subset\mathbb{S}^{3}$ be two convex 3-polytopes. We say that they are projectively equivalent if there is some automorphism $A\in\text{SL}_{\pm}(4,\mathbb{R})$ of $\mathbb{S}^{3}$ such that $A(P)=Q$ . Let $\mathcal{G}$ be a labeled combinatorial polytope and let $\mathcal{C}(\mathcal{G})$ be the deformation space of Coxeter 3-polytopes. We consider an equivalence relation $\sim$ on $\mathcal{C}(\mathcal{G})$ by saying that $[(P,r_{1},\cdots,r_{f})]\sim[(P^{\prime},r_{1}^{\prime},\cdots,r_{f}^{\prime})]$ if and only if $P$ and $P^{\prime}$ are projectively equivalent. We denote each equivalence class represented by $[(P,r_{1},\cdots,r_{f})]$ by $\mathcal{C}_{P}(\mathcal{G})$ and call it the restricted deformation space with fundamental polytope $P$ .

The following two definitions involve our main assumptions on the labeled combinatorial polytopes $\mathcal{G}$ .

Definition 4.1.

A labeled combinatorial polytope $\mathcal{G}$ is orderable if the $f$ facets of $\mathcal{G}$ can be totally ordered as $\mathcal{G}_{1},\cdots,\mathcal{G}_{f}$ so that each facet $\mathcal{G}_{i}$ has at most three edges such that each of them is either (i) of order 2 or (ii) the common edge of the facets $\mathcal{G}_{i}$ and $\mathcal{G}_{j}$ with $j<i$ .

Definition 4.2.

We say that $\mathcal{G}$ is of normal type if $\mathcal{G}$ does not satisfy any of the following conditions:

(i)

$\mathcal{G}$ is a cone over a polygon and the integers assigned to the edges of the base polygon are all 2;
(ii)

$\mathcal{G}$ equals a polygon times a closed interval and the edge orders assigned to the edges of the two base polygons are all 2;
(iii)

the associated Coxeter group $\Gamma_{\mathcal{G}}$ (see (2.2)) is finite.
(iv)

$\mathcal{G}$ admits an affine Coxeter group representation, i.e. there is a Coxeter polytope $(P,r_{1},\cdots,r_{f})$ realizing $\mathcal{G}$ such that the interior of $\bigcup_{\gamma\in\Gamma_{\mathcal{G}}}\gamma\cdot P$ is contained in an affine chart of $\mathbb{S}^{3}$ which is invariant under $r_{1},\cdots,r_{f}$ .

Remark 4.3.

Let $\Gamma_{\mathcal{G}}$ be the Coxeter group associated with $\mathcal{G}$ . If $\Gamma_{\mathcal{G}}$ is irreducible (see Definition 2.1), then by Margulis-Vinberg [7], the group is either:

(i)

spherical (i.e. finite);
(ii)

affine (i.e. infinite and virtually abelian);
(iii)

large (i.e. there exists a finite-index subgroup of $\Gamma_{\mathcal{G}}$ admitting a surjective homomorphism onto a free group of rank $\geq 2$ ).

If $\Gamma_{\mathcal{G}}$ is both irreducible and large, then $\mathcal{G}$ is of normal type. In particular, many Coxeter orbifolds including hyperbolic Coxeter orbifolds of finite volume correspond to irreducible and large Coxeter groups and hence they are of normal type.

Proposition 4.4.

Let $\mathcal{G}$ be a labeled combinatorial 3-polytope and let $\Gamma_{\mathcal{G}}$ be its associated Coxeter group. If the Coxeter group $\Gamma_{\mathcal{G}}$ is irreducible and large, then $\mathcal{G}$ is of normal type.

Proof.

Since the Coxeter group $\Gamma_{\mathcal{G}}$ is irreducible but not spherical, none of the conditions (i), (ii), and (iii) of Definition 4.2 is satisfied.

Let $(P,r_{1},\dots,r_{f})$ be a Coxeter 3-polytope realizing $\mathcal{G}$ . It remains to show that the reflections $r_{1},\dots,r_{f}$ do not simultaneously preserve an affine chart of $\mathbb{S}^{3}$ containing the interior of $\bigcup_{\gamma\in\Gamma_{\mathcal{G}}}\gamma\cdot P$ .

The linear reflections $r_{1},\dots,r_{f}\in\mathrm{SL}_{\pm}(4,\mathbb{R})$ can be expressed as

r_{i}(x)=x-\alpha_{i}(x)v_{i},

for some $\alpha_{i}\in(\mathbb{R}^{4})^{*}$ and $v_{i}\in\mathbb{R}^{4}$ such that $\alpha_{i}(v_{i})=2$ . We consider the corresponding Cartan matrix $A:=[\alpha_{i}(v_{j})]_{i,j=1}^{f}$ (see [15], [5], or [9] for the definition of Cartan matrices).

Since $\Gamma_{\mathcal{G}}$ is irreducible and large, the Cartan matrix $A$ is of negative type, i.e., its lowest real eigenvalue is negative (see Fact 3.17 of [5]). Moreover, the rank of $A$ is either 3 or 4 by Proposition 15 in [15].

Case 1: $\text{rank}(A)=4$ .

If $\text{rank}(A)=4$ , then the reflections $r_{1},\dots,r_{f}$ do not simultaneously preserve a nontrivial linear subspace of $\mathbb{R}^{4}$ by Proposition 19 in [15] and its corollary. Consequently, they do not preserve any affine chart of $\mathbb{S}^{3}$ .

Case 2: $\text{rank}(A)=3$ .

If $\text{rank}(A)=3$ , then the vectors $v_{1},\dots,v_{f}$ span a 3-dimensional subspace of $\mathbb{R}^{4}$ . Suppose, for contradiction, that $r_{1},\dots,r_{f}$ simultaneously preserve an affine chart $\Omega$ containing the interior of $\bigcup_{\gamma\in\Gamma_{\mathcal{G}}}\gamma\cdot P$ . Then the boundary $\partial\Omega\subset\mathbb{S}^{3}$ is invariant under $r_{1},\dots,r_{f}$ and must equal the span of $v_{1},\dots,v_{f}$ by Proposition 19 of [15].

On the other hand, since $A$ is of negative type, there exists a vector $x=(x_{1},\dots,x_{f})\in\mathbb{R}^{f}$ with $x_{i}>0$ for all $i$ such that all the entries of $Ax\in\mathbb{R}^{f}$ are negative (see Theorem 3 of [15]). Define the vector

v:=-\sum_{j=1}^{f}x_{j}v_{j}.

The vector $v$ projects into the interior of the polytope $P\subset\mathbb{S}^{3}$ , implying that $[v]\in\Omega$ . However, this contradicts the fact that $[v]\in\mathbb{S}(\mathrm{Span}\{v_{1},\dots,v_{f}\})=\partial\Omega$ .

In both cases, we conclude that the reflections $r_{1},\dots,r_{f}$ do not simultaneously preserve an affine chart of $\mathbb{S}^{3}$ . Thus, $\mathcal{G}$ is of normal type. ∎

We present a rephrazed version of Choi’s result [1] on the smoothness of the restricted deformation spaces.

Theorem 4.5.

([1], Theorem 4) Let $\mathcal{G}$ be a labeled combinatorial 3-polytope. Let $f,e$ be the number of facets, edges of $\mathcal{G}$ and let $e_{2}$ be the number of edges of $\mathcal{G}$ of order 2. Suppose that $\mathcal{G}$ is orderable and is of normal type. Let $k(\mathcal{G})$ be the dimension of the stabilizer subgroup of $\text{SL}_{\pm}(4,\mathbb{R})$ fixing a convex 3-polytope in $\mathbb{S}^{3}$ combinatorially equivalent to $\mathcal{G}$ . Then the restricted deformation space $\mathcal{C}_{P}(\mathcal{G})$ is a smooth manifold of dimension $3f-e-e_{2}-k(\mathcal{G})$ if it is not empty.

Remark 4.6.

The number $k(\mathcal{G})$ does not depend on the choice of the convex 3-polytopes combinatorially equivalent to $\mathcal{G}$ . It can be easily checked that $k(\mathcal{G})=3$ if $\mathcal{G}$ is a tetrahedron, $k(\mathcal{G})=1$ if $\mathcal{G}$ is a cone over a polygon other than a triangle, and $k(\mathcal{G})=0$ otherwise.

Remark 4.7.

There was an error in the proof of Proposition 2 of [1]. We need to exclude Coxeter orbifolds admitting affine structures, which is necessary for the proof of Theorem 4.5. We will try to mend the proof here.

Let $P$ be a properly convex fundamental polytope. Let $F_{i}$ be the sides of $P$ . Let $R_{i}$ be the reflections on the sides $F_{i}$ of $P$ .

First, note that if the sphere of fixed points of a reflection contains an antipodal fixed point of another reflection, then those two reflections must commute, and their associated sides must meet in an edge, and their edge order is $2$ .

We need to show that there is no holonomy-invariant disjoint union of one or two $1$ -dimensional subspaces or holonomy-invariant $2$ -dimensional subspaces.

Suppose that $l$ is a holonomy-invariant disjoint union of one or two $1$ -dimensional subspaces. The case of the two $1$ -dimensional subspaces reduces to the first one because a reflection must act on each $1$ -dimensional subspace if it acts on a disjoint union of two $1$ -dimensional subspaces.

Let $l$ be a holonomy-invariant $1$ -dimensional subspace. If a face is contained in a $2$ -dimensional subspace containing $l$ , we call it a parallel face. The associated reflection is also called parallel. If not, it is called a transverse face. The associated reflection is called transverse. In this case, the antipodal fixed point must lie on $l$ , and the fixed point subspace must intersect $l$ transversely.

Suppose that $\Omega\cap l=\varnothing$ . Then $P$ must have at most two parallel sides $F_{i}$ and $F_{j}$ . If there are exactly two parallel sides $F_{i}$ and $F_{j}$ , then $R_{i}$ and $R_{j}$ commute with all other reflections. Hence, we violated the normality. Otherwise, we cannot have a compact $P$ : If there is one parallel face, then the group is just an extension of the $2$ -dimensional Coxeter group by a reflection of the face. This follows from the first paragraph above. If there is no parallel face, then each sphere containing $l$ is invariant by the nature of transversal reflections, and again we have a $2$ -dimensional Coxeter group. These do not have properly convex fundamental polytopes with some vertices removed.

Suppose that $\Omega\cap l\neq\varnothing$ . If there is a pair of adjacent transverse faces $F_{i}$ and $F_{j}$ , then $R_{i}$ and $R_{j}$ generate a finite group, and by their action and convexity, we must have $\Omega\supset l$ . This implies $\Omega=\mathbb{S}^{3}$ by convexity, and we must have a finite Coxeter group, contradicting the normality. Also, there can be at most two parallel faces since $P$ is a properly convex polytope. From these two facts, it follows that there must be at most one transverse face and at most two parallel faces. We cannot construct a properly convex $P$ in this situation.

Suppose that $S$ is a holonomy-invariant $2$ -dimensional subspace. If $S\cap\Omega=\varnothing$ , then there is an invariant affine subspace containing $\Omega$ , and our orbifold admits an affine structure.

The old proof correctly rules out $S\cap\Omega\neq\varnothing$ . We explain a bit more. In the old proof, we chose the fundamental polytope $P$ so that $P\cap S\neq\varnothing$ . Also, to deduce the infinite edge orders, we tacitly used the fact that our holonomy group acts as a $2$ -dimensional Coxeter group on $S\cap\Omega$ . ∎

For two non-projectively-equivalent convex 3-polytopes $P,P^{\prime}\subset\mathbb{S}^{3}$ representing a common combinatorial polytope $\mathcal{G}$ , the associated restricted deformation spaces $\mathcal{C}_{P}(\mathcal{G})$ and $\mathcal{C}_{P^{\prime}}(\mathcal{G})$ may not be homeomorphic to each other, even if $\mathcal{G}$ satisfies the hypothesis of Theorem 4.1. It may happen that one restricted deformation space $\mathcal{C}_{P}(\mathcal{G})$ is empty while another restricted deformation space $\mathcal{C}_{P^{\prime}}(\mathcal{G})$ is not. Moreover, even if both $\mathcal{C}_{P}(\mathcal{G})$ and $\mathcal{C}_{P^{\prime}}(\mathcal{G})$ are non-empty, those spaces may not be homeomorphic to each other in general. We will see an example (Example 7.2) in which the above phenomena occur simultaneously.

Let $\mathcal{G}$ be a labeled combinatorial polytope. Each element $[(P,r_{1},\cdots,r_{f})]\in\mathcal{C}(\mathcal{G})$ determines a projective equivalence class $[P]$ of a convex 3-polytope $P\subset\mathbb{S}^{3}$ , where $P$ represents the combinatorial polytope $\mathcal{G}$ . Thus it is natural to consider the space of projective equivalence classes $[P]$ of convex polytopes $P\subset\mathbb{S}^{3}$ representing $\mathcal{G}$ , and consider the projection given by $[(P,r_{1},\cdots,r_{f})]\mapsto[P]$ of $\mathcal{C}(\mathcal{G})$ into that space. The nonempty fibers of this projection are precisely the restricted deformation spaces of $\mathcal{G}$ by definition. In the next section, we study this space of polytopes representing $\mathcal{G}$ .

5. Realization spaces

In this section, we recall the notion of realization spaces for convex 3-polytopes. Essentially, the realization space of a fixed combinatorial 3-polytope is the set of all convex 3-polytopes in $\mathbb{S}^{3}$ that share the same combinatorial type, modulo projective equivalence. Steinitz [13] studied a related realization space consisting of 3-dimensional polytopes in the affine 3-space $\mathbb{A}^{3}$ modulo affine equivalence, proving that the realization space of each affine 3-polytope is a cell. See also [12], which introduces the realization spaces of affine 3-polytopes and includes a proof of Steinitz’s result.

For the proof of Theorem 1.2, we require an analogous result for the realization spaces of projective 3-polytopes. At the end of this section, we will recall a theorem regarding the smoothness of the realization spaces of projective 3-polytopes.

For each subset $A\subset\mathbb{S}^{3}$ , let $\text{conv}(A)\subset\mathbb{S}^{3}$ denote the convex hull of $A$ in $\mathbb{S}^{3}$ .

Definition 5.1.

Let $\mathcal{G}$ be a combinatorial 3-polytope, whose edges need not be labeled by integers. Let $\mathcal{V}$ be the set of vertices of $\mathcal{G}$ . A realization of $\mathcal{G}$ is a tuple $(p_{v})_{v\in\mathcal{V}}\in(\mathbb{S}^{3})^{\mathcal{V}}$ such that

(i)

$P:=\text{conv}\{p_{v}\ |\ v\in\mathcal{V}\}\subset\mathbb{S}^{3}$ is a convex 3-polytope combinatorially equivalent to $\mathcal{G}$ ;
(ii)

for each subset $\mathcal{F}\subset\mathcal{V}$ , the subset $\text{conv}(\{p_{v}\ |\ v\in\mathcal{F}\})\subset P$ is a face of $P$ (in an arbitrary dimension) if and only if $\mathcal{F}$ is the set of vertices of a face of $\mathcal{G}$ of same dimension.

Definition 5.2.

The pre-realization space $\overline{\mathcal{RS}}(\mathcal{G})$ is the set of all the realizations of $\mathcal{G}$ .

Note that the pre-realization space $\overline{\mathcal{RS}}(\mathcal{G})$ is a subset of the product $(\mathbb{S}^{3})^{\mathcal{V}}$ . We endow $\overline{\mathcal{RS}}(\mathcal{G})$ the subspace topology.

Note that the group $\text{SL}_{\pm}(4,\mathbb{R})$ of projective automorphisms of $\mathbb{S}^{3}$ acts on the space $\overline{\mathcal{RS}}(\mathcal{G})$ by

(5.1)

\displaystyle A\cdot(p_{v})_{v\in\mathcal{V}}:=(A\cdot p_{v})_{v\in\mathcal{V}},

where we identify $A\in\text{SL}_{\pm}(4,\mathbb{R})$ as a projective automorphism $A:\mathbb{S}^{3}\to\mathbb{S}^{3}$ .

Definition 5.3.

The (projective) realization space of $\mathcal{G}$ is the quotient space $\mathcal{RS}(\mathcal{G}):=\overline{\mathcal{RS}}(\mathcal{G})/\text{SL}_{\pm}(4,\mathbb{R})$ .

We can define the affine realization spaces of $\mathcal{G}$ analogously by requiring its realizations lie in an affine space $\mathbb{A}^{3}$ and by replacing the group $\text{SL}_{\pm}(4,\mathbb{R})$ by the group $\text{Aff}(3,\mathbb{R})$ of affine automorphisms. A classical result of Steinitz [13] states that the affine realization space of a 3-polytope having $e$ edges is homeomorphic to $\mathbb{R}^{e-6}$ . Filpo Molina [6] extended the result to the projective geometry. We state the result in our terminology as follows.

Theorem 5.4.

([6], Theorem B) Let $\mathcal{G}$ be a combinatorial 3-polytope with $e$ -edges that is not combinatorially equivalent to a cone over a polygonal base. Then the projective realization space $\mathcal{RS}(\mathcal{G})$ is a smooth manifold of dimension $e-9$ .

Note that if $P\subset\mathbb{S}^{3}$ is a convex 3-polytope that is not combinatorially equivalent to a cone over a polygonal base, then $e\geq 9$ . This can be verified as follows. Let $e_{j}$ denote the number of edges of the $j$ -th facet of $P$ . Since $e_{j}\geq 3$ for each $j$ , we have

e=\frac{1}{2}\sum_{j=1}^{f}e_{j}\geq\frac{3f}{2}.

This implies that $e\geq 9$ if $f\geq 6$ . If $f=5$ , then $P$ is either a triangular prism or a cone over a quadrilateral. Each triangular prism has exactly 9 edges, and cones over quadrilaterals are excluded by our assumption. If $f=4$ , then $P$ is a tetrahedron, which is always a cone over a triangle.

Remark 5.5.

For dimensions $d>3$ , the realization spaces of convex $d$ -polytopes can be defined analogously. However, the realization spaces of some polytopes may not be topological manifolds in general. (See [11].) For the purpose of proving the smoothness of the deformation space $\mathcal{C}(\mathcal{G})$ , we thus restrict our attention to the 3-dimensional polytopes.

6. The proof of the main theorem

In this section, we prove Theorem 1.2. The starting point is to consider a map $\mathcal{C}(\mathcal{G})\to\mathcal{RS}(\mathcal{G})$ that “forgets” the reflection data (explained in Section 6.2). The smoothness of $\mathcal{C}(\mathcal{G})$ will essentially follow from the smoothness of the realization space $\mathcal{RS}(\mathcal{G})$ and the nonempty fibers of the map $\mathcal{C}(\mathcal{G})\to\mathcal{RS}(\mathcal{G})$ .

The realization space $\mathcal{RS}(\mathcal{G})$ , as described in Section 5, is expressed in terms of the vertices of the polytopes. It will be more convenient, however, to describe the realization space in terms of the facets of the polytopes. We achieve this in Section 6.1 by embedding $\mathcal{RS}(\mathcal{G})$ in a quotient space of $(V^{*})^{f}$ .

In Section 6.2, we consider an open map $\mathcal{C}(\mathcal{G})\to\mathcal{RS}(\mathcal{G})$ whose nonempty fibers correspond to the restricted deformation spaces discussed in Section 4. Section 6.3 uses this map to construct a smooth manifold $\widetilde{\mathcal{D}}(\mathcal{G})$ . Finally, in Section 6.4, we show that the quotient of $\widetilde{\mathcal{D}}(\mathcal{G})$ by the Lie group $\text{SL}_{\pm}(4,\mathbb{R})\times\mathbb{R}_{+}^{f}$ is homeomorphic to $\mathcal{C}(\mathcal{G})$ , thereby inducing a smooth structure on $\mathcal{C}(\mathcal{G})$ .

Throughout this section, we adopt the following settings. Let $\mathcal{G}$ be an orderable, labeled combinatorial 3-polytope satisfying the conditions of Theorem 1.2. Let $\mathcal{V}$ be the set of vertices of $\mathcal{G}$ . We label the $f$ facets of $\mathcal{G}$ as $\mathcal{G}_{1},\ldots,\mathcal{G}_{f}$ , and denote by $m_{i,j}$ the integer $\geq 2$ assigned to the edge (= ridge) $\mathcal{G}_{i}\cap\mathcal{G}_{j}$ , whenever it is defined. Let $V=\mathbb{R}^{4}$ be a 4-dimensional real vector space, and let $\mathbb{S}^{3}:=\mathbb{S}(V)$ denote the real projective 3-sphere obtained by projectivizing $V$ . The group of linear automorphisms of $V$ with determinant $\pm 1$ is denoted by $\text{SL}_{\pm}(V)=\text{SL}_{\pm}(4,\mathbb{R})$ , which can also be viewed as the group of projective automorphisms of $\mathbb{S}^{3}$ . Lastly, let $\mathbb{R}_{+}$ be the one-dimensional group of positive real numbers, and define $\mathbb{G}:=\text{SL}_{\pm}(V)\times\mathbb{R}_{+}^{f}$ , which is a product Lie group of dimension $15+f$ .

6.1. Reparametrizing the realization space

In Section 5, we considered the realization spaces of convex 3-polytopes. In the settings of Section 5, each polytope (realization) is described in terms of its vertices, as the vertices determine the polytope by taking the convex hull. However, it will be more beneficial for us to describe the polytopes in terms of their facets.

More specifically, for each convex 3-polytope $P$ with $f$ facets $P_{1},\cdots,P_{f}\subset P$ , there exist $f$ linear functionals $\alpha_{1},\cdots,\alpha_{f}\in V^{*}$ such that $P$ is defined by the linear inequalities $\alpha_{i}\geq 0$ , with each facet $P_{i}$ supported by the hyperplane of $\mathbb{S}^{3}$ determined by $\alpha_{i}$ . These functionals $\alpha_{i}$ are not uniquely defined by $P$ , as any positive scalar multiple of $\alpha_{i}$ also supports the facet $P_{i}$ .

In this way, $P$ determines an element of the quotient $(V^{*})^{f}/\mathbb{R}_{+}^{f}$ . On the other hand, each element of the realization space is a projective equivalence class of a convex 3-polytope, so the corresponding element of $(V^{*})^{f}/\mathbb{R}_{+}^{f}$ is uniquely determined up to the action of $\text{SL}_{\pm}(V)$ . Thus, each realization of a polytope determines a unique element of

(V^{*})^{f}/(\mathbb{R}_{+}^{f}\times\text{SL}_{\pm}(V))=(V^{*})^{f}/\mathbb{G}.

We describe the elements of $(V^{*})^{f}/\mathbb{G}$ that arise as the image of a realization. For computational purposes, it will be convenient to consider the lift of these elements in $(V^{*})^{f}$ under the projection $(V^{*})^{f}\to(V^{*})^{f}/\mathbb{G}$ . In this subsection, we consider these constructions in detail.

Recall from Lemma 2.4 that we have an embedding $\mathcal{C}(\mathcal{G})\to((V^{*})^{f}\times V^{f})/\mathbb{G}$ . Similarly, for the realization spaces, we will consider an embedding of the realization space $\mathcal{RS}(\mathcal{G})$ into $(V^{*})^{f}/\mathbb{G}$ .

Now we make the above considerations precise. Let $\overline{\mathcal{RS}}(\mathcal{G})$ be the pre-realization space of $\mathcal{G}$ (see Definition 5.2). Then, $\mathcal{RS}(\mathcal{G})$ is the quotient space $\overline{\mathcal{RS}}(\mathcal{G})/\text{SL}_{\pm}(V)$ . We first construct an embedding $\overline{\mathcal{RS}}(\mathcal{G})\to(V^{*})^{f}/\mathbb{R}_{+}^{f}$ , where $(V^{*})^{f}/\mathbb{R}_{+}^{f}$ is the quotient space obtained by the action of the group $\mathbb{R}_{+}^{f}$ on $(V^{*})^{f}$ given by

(c_{1},\ldots,c_{f})\cdot(\alpha_{1},\ldots,\alpha_{f}):=(c_{1}^{-1}\alpha_{1},\ldots,c_{f}^{-1}\alpha_{f}).

Let $(p_{v})_{v\in\mathcal{V}}\in\overline{\mathcal{RS}}(\mathcal{G})$ be a realization of $\mathcal{G}$ (see Definition 5.1). The convex hull $P:=\text{conv}\{p_{v}\ |\ v\in\mathcal{V}\}\subset\mathbb{S}^{3}$ is a convex 3-polytope combinatorially equivalent to $\mathcal{G}$ , so in particular $P$ has $f$ facets. We label the $f$ facets by $P_{1},\cdots,P_{f}$ in alignment with the labeling $\mathcal{G}_{1},\cdots,\mathcal{G}_{f}$ of the facets of $\mathcal{G}$ , i.e., in such a way that for each vertex $v$ of $\mathcal{G}$ , $v\in\mathcal{G}_{i}$ if and only if $p_{v}\in P_{i}$ .

We choose elements $\alpha_{1},\ldots,\alpha_{f}\in V^{*}$ such that for each $i$ the hyperplane $\ker\alpha_{i}\subset V$ is projected onto the hypersphere of $\mathbb{S}^{3}$ containing $P_{i}$ . Additionally, we impose the condition $\alpha_{i}(x)\geq 0$ for each $x\in V$ lifting a point of $P\subset\mathbb{S}^{3}$ . The tuple $(\alpha_{1},\ldots,\alpha_{f})\in(V^{*})^{f}$ is uniquely determined up to multiplication by positive real numbers. In this way, we obtain a map $\overline{\iota}:\overline{\mathcal{RS}}(\mathcal{G})\to(V^{*})^{f}/\mathbb{R}_{+}^{f}$ .

Lemma 6.1.

The map $\overline{\iota}:\overline{\mathcal{RS}}(\mathcal{G})\to(V^{*})^{f}/\mathbb{R}_{+}^{f}$ is a topological embedding.

Proof.

We first prove that $\overline{\iota}$ is injective. Let $(p_{v})_{v\in\mathcal{V}}$ , $(p_{v}^{\prime})_{v\in\mathcal{V}}\in\overline{\mathcal{RS}}(\mathcal{G})$ be two realizations of $\mathcal{G}$ , and suppose that

\overline{\iota}((p_{v})_{v\in\mathcal{V}})=\overline{\iota}((p_{v}^{\prime})_{v\in\mathcal{V}})=[(\alpha_{1},\ldots,\alpha_{f})].

We need to show that $p_{v}=p_{v}^{\prime}$ for each $v\in\mathcal{V}$ .

Let $P:=\text{conv}\{p_{v}\ |\ v\in\mathcal{V}\}$ , $P^{\prime}:=\text{conv}\{p_{v}^{\prime}\ |\ v\in\mathcal{V}\}$ be the convex 3-polytopes determined by $(p_{v})_{v\in\mathcal{V}}$ , $(p_{v}^{\prime})_{v\in\mathcal{V}}$ respectively. Label the facets of $P$ and $P^{\prime}$ by $P_{1},\cdots,P_{f}$ and $P_{1}^{\prime},\cdots,P_{f}^{\prime}$ so that they accord with the labeling $\mathcal{G}_{1},\cdots,\mathcal{G}_{f}$ of the facets of $\mathcal{G}$ .

Let $v\in\mathcal{V}$ , and let $\mathcal{G}_{i_{1}},\ldots,\mathcal{G}_{i_{k}}\subset P$ be the facets of $P$ containing the vertex $v$ , so that we have

\{v\}=\mathcal{G}_{i_{1}}\cap\cdots\cap\mathcal{G}_{i_{k}}.

Then $(p_{v})_{v\in\mathcal{V}}$ and $(p_{v}^{\prime})_{v\in\mathcal{V}}$ satisfy analogous relations, i.e., we have

\{p_{v}\}=\mathbb{S}(\ker\alpha_{i_{1}}\cap\cdots\cap\ker\alpha_{i_{k}})=\{p_{v}^{\prime}\}.

This proves that $p_{v}=p_{v}^{\prime}$ for each $v\in\mathcal{V}$ . Hence, $\overline{\iota}$ is injective.

We can check easily that the map $\overline{\iota}$ is continuous. Small perturbations of the vertices of a convex 3-polytope, while maintaining its combinatorial type, result in small perturbations of the supporting hyperplanes of the facets of the resulting convex 3-polytope. Therefore, $\overline{\iota}$ is continuous.

Finally, the inverse map $\overline{\iota}^{-1}:\overline{\iota}(\overline{\mathcal{RS}}(\mathcal{G}))\to\overline{\mathcal{RS}}(\mathcal{G})$ is continuous for a similar reason: small perturbations of the supporting hyperplanes, while maintaining the combinatorial type of the polytope, result in small perturbations of the vertices of the polytope. Hence, $\overline{\iota}^{-1}$ is continuous.

We conclude that $\overline{\iota}$ is a topological embedding. ∎

The group $\text{SL}_{\pm}(V)$ acts on the space $\overline{\mathcal{RS}}(\mathcal{G})$ by the rule (5.1), and also acts on $(V^{*})^{f}/\mathbb{R}_{+}^{f}$ by the rule

A\cdot[(\alpha_{1},\ldots,\alpha_{f})]:=[(\alpha_{1}\circ A^{-1},\ldots,\alpha_{f}\circ A^{-1})].

It can be easily checked that the embedding $\overline{\iota}:\overline{\mathcal{RS}}(\mathcal{G})\to(V^{*})^{f}/\mathbb{R}_{+}^{f}$ in Lemma 6.1 is equivariant with respect to these actions. Therefore, we obtain an embedding

\iota:\mathcal{RS}(\mathcal{G})\to\left((V^{*})^{f}/\mathbb{R}_{+}^{f}\right)/\text{SL}_{\pm}(V)\cong(V^{*})^{f}/\mathbb{G}.

(Recall $\mathbb{G}=\text{SL}_{\pm}(V)\times\mathbb{R}_{+}^{f}$ .)

In the proof of Theorem 1.2, we use the smoothness of the realization space $\mathcal{RS}(\mathcal{G})$ . It turns out to be convenient to use the images of the maps $\overline{\iota}$ and $\iota$ instead of their domains $\overline{\mathcal{RS}}(\mathcal{G})$ and $\mathcal{RS}(\mathcal{G})$ . Hence, we introduce the following notations. We define

\overline{\mathcal{E}}(\mathcal{G}):=\overline{\iota}(\overline{\mathcal{RS}}(\mathcal{G}))\subset(V^{*})^{f}/\mathbb{R}_{+}^{f},\quad\mathcal{E}(\mathcal{G}):=\iota(\mathcal{RS}(\mathcal{G}))\subset(V^{*})^{f}/\mathbb{G}.

Lastly, we define $\widetilde{\mathcal{E}}(\mathcal{G})$ to be the preimage of $\mathcal{E}(\mathcal{G})$ under the projection $(V^{*})^{f}\to(V^{*})^{f}/\mathbb{G}$ . We have the following commuting diagram, where the vertical maps are the quotient maps induced by the corresponding group actions.

We call the face lattice of $\mathcal{G}$ to be the set of faces of $\mathcal{G}$ in all dimensions, partially ordered by the inclusion, and denote it by $\text{FL}(\mathcal{G})$ . By the construction of the map $\overline{\iota}$ , we have the following description of $\widetilde{\mathcal{E}}(\mathcal{G})\subset(V^{*})^{f}$ .

Lemma 6.2.

Let $\alpha_{1},\dots,\alpha_{f}\in V^{*}$ . Then $(\alpha_{1},\dots,\alpha_{f})\in\widetilde{\mathcal{E}}(\mathcal{G})$ if and only if the set

Q:=\{[x]\in\mathbb{S}^{3}\mid\alpha_{i}(x)\geq 0\ \text{for all}\ i\}

is a convex 3-polytope combinatorially equivalent to $\mathcal{G}$ , and there exists a face-lattice isomorphism $\phi:\textup{FL}(\mathcal{G})\to\textup{FL}(Q)$ sending each facet $\mathcal{G}_{i}$ to the facet of $Q$ supported by $\alpha_{i}$ .

Proof.

The “only if” part follows directly from the construction of the map $\overline{\iota}$ .

Conversely, the face-lattice isomorphism $\phi:\textup{FL}(P)\to\textup{FL}(Q)$ restricts to a labeling $q_{v}:=\phi(v)$ , $v\in\mathcal{V}$ of the vertices of $Q$ . The realization $(q_{v})_{v\in\mathcal{V}}$ is then mapped to $[(\alpha_{1},\dots,\alpha_{f})]$ via the map $\overline{\iota}$ . ∎

In the proof of Theorem 1.2, we will construct a smooth manifold and consider a proper free smooth action of the group $\mathbb{G}$ on it, such that $\mathcal{C}(\mathcal{G})$ is the resulting quotient smooth manifold. To achieve this, we need to construct a smooth structure on $\widetilde{\mathcal{E}}(\mathcal{G})$ . For this purpose, we prove the following two lemmas.

Lemma 6.3.

Let $\mathcal{G}$ be a combinatorial 3-polytope (whose facets need not be labeled) with $f$ facets $\mathcal{G}_{1},\cdots,\mathcal{G}_{f}$ . Suppose $\mathcal{G}$ is not a cone over a polygon. Let $\widetilde{\mathcal{E}}(\mathcal{G})\subset(V^{*})^{f}$ be the subspace constructed in the above process. Then the action of $\mathbb{G}$ on $\widetilde{\mathcal{E}}(\mathcal{G})$ is free.

Proof.

Let $\alpha=(\alpha_{1},\cdots,\alpha_{f})\in\widetilde{\mathcal{E}}(\mathcal{G})$ , and let

g=(A,c_{1},\cdots,c_{f})\in\mathbb{G}=\text{SL}_{\pm}(V)\times\mathbb{R}_{+}^{f}.

Suppose that $g\cdot\alpha=\alpha$ . Consider the polytope $Q=\{[x]\in\mathbb{S}^{3}\mid\alpha_{i}(x)\geq 0\ \text{for all}\ i\}$ , which is combinatorially equivalent to $\mathcal{G}$ .

Since $g\cdot\alpha=(c_{1}^{-1}\alpha_{1}\circ A^{-1},\cdots,c_{f}^{-1}\alpha_{f}\circ A^{-1})$ , the assumption $g\cdot\alpha=\alpha$ is equivalent to

(6.1)

\displaystyle c_{i}^{-1}\alpha_{i}\circ A^{-1}=\alpha_{i}\quad\text{for}\ i=1,\cdots,f.

First, we note that the forms $\alpha_{1},\cdots,\alpha_{f}$ span $V^{*}$ . If they did not span $V^{*}$ , then there would be a point $[x]\in\mathbb{S}^{3}$ such that $\alpha_{i}(x)=0$ for all $i$ , which would imply that both $[x]$ and $[-x]$ belong to $Q$ , contradicting the fact that $Q$ is properly convex.

We can assume, by reindexing the facets of $\mathcal{G}$ if necessary, that $\alpha_{1},\cdots,\alpha_{4}$ are linearly independent. Then for each $i\in\{5,\cdots,f\}$ , there are unique coefficients $d_{i,j}\in\mathbb{R}$ , $j=1,2,3,4$ , such that

(6.2)

\displaystyle\alpha_{i}=\sum_{j=1}^{4}d_{i,j}\alpha_{j}.

We make two observations:

1.

For each $i\in\{5,\cdots,f\}$ , at most one of the coefficients $d_{i,j}$ is zero. If two of the coefficients $d_{i,j}$ , say $d_{i,3}=0$ and $d_{i,4}=0$ , this would imply that $\alpha_{i}=d_{i,1}\alpha_{1}+d_{i,2}\alpha_{2}$ . In this case, two of the three inequalities $\alpha_{i}\geq 0$ , $\alpha_{1}\geq 0$ and $\alpha_{2}\geq 0$ would imply the other, contradicting the fact that $Q$ has exactly $f$ facets.
2.

For each $j\in\{1,2,3,4\}$ , there is at least one $i\in\{5,\cdots,f\}$ such that $d_{i,j}\neq 0$ . Without loss of generality, assume $j=1$ . Since $\alpha_{1},\cdots,\alpha_{4}$ are linearly independent, there is a unique point $[x]\in\mathbb{S}^{3}$ such that $\alpha_{2}(x)=\alpha_{3}(x)=\alpha_{4}(x)=0$ and $\alpha_{1}(x)>0$ . If $d_{i,1}=0$ for all $i\in\{5,\cdots,f\}$ , then $\alpha_{i}(x)=0$ for all such $i$ , implying that $Q$ is the cone over the facet $Q_{1}$ with apex $[x]$ , contradicting our assumption.

Now, the equations $c_{j}^{-1}\alpha_{j}\circ A^{-1}=\alpha_{j}$ for $j=1,2,3,4$ imply that the matrix representation of $A$ with respect to the dual basis of $\{\alpha_{1},\cdots,\alpha_{4}\}$ is given by $\text{diag}(c_{1}^{-1},c_{2}^{-1},c_{3}^{-1},c_{4}^{-1})$ . Since $A\in\text{SL}_{\pm}(V)$ , we have $\pm 1=\det(A)=c_{1}^{-1}c_{2}^{-1}c_{3}^{-1}c_{4}^{-1}$ . Since $c_{j}>0$ for all $j$ , we obtain $c_{1}c_{2}c_{3}c_{4}=1$ .

Finally, to conclude the proof, it suffices to show that $c_{1}=c_{2}=c_{3}=c_{4}$ , since this will imply that $A=I$ , and by equation (6.1), $c_{j}=1$ for all $j$ .

Let $i\in\{5,\cdots,f\}$ . Applying (6.1) and (6.2) gives

\alpha_{i}\circ A^{-1}=\sum_{j=1}^{4}d_{i,j}\alpha_{j}\circ A^{-1}=\sum_{j=1}^{4}d_{i,j}c_{j}\alpha_{j}.

On the other hand, we have

\alpha_{i}\circ A^{-1}=c_{i}\alpha_{i}=c_{i}\sum_{j=1}^{4}d_{i,j}\alpha_{j}.

Comparing the coefficients, we obtain $(c_{i}-c_{j})d_{i,j}=0$ for all $i\in\{5,\cdots,f\}$ and $j\in\{1,2,3,4\}$ .

We consider the index $i=5$ . If $d_{5,1},d_{5,2},d_{5,3},d_{5,4}$ are all nonzero, then it follows that $c_{5}=c_{1}=c_{2}=c_{3}=c_{4}$ , so we obtain the desired result. Suppose that one of $d_{5,1},d_{5,2},d_{5,3},d_{5,4}$ is zero. For simplicity, we assume that $d_{5,1}=0$ . Then Observation 1 implies that $d_{5,2},d_{5,3},d_{5,4}$ are nonzero, and this implies that $c_{5}=c_{2}=c_{3}=c_{4}$ . By Observation 2, there is another index $i\in\{6,\cdots,f\}$ such that $d_{i,1}\neq 0$ . Then Observation 1 again implies that there are at least two indices $u,v\in\{2,3,4\}$ such that $d_{i,u},d_{i,v}\neq 0$ , so $c_{i}=c_{1}=c_{u}=c_{v}$ . Hence, we obtain $c_{1}=c_{2}=c_{3}=c_{4}$ .

We conclude that $A=\text{diag}(c_{1}^{-1},c_{2}^{-1},c_{3}^{-1},c_{4}^{-1})=I$ and $g=(I,1,\cdots,1)$ . Therefore, the action of $\mathbb{G}$ on $\widetilde{\mathcal{E}}(\mathcal{G})$ is free. ∎

Lemma 6.4.

Let $\mathcal{G}$ be a combinatorial 3-polytope. Let $f$ and $e$ be the numbers of facets and edges of $\mathcal{G}$ , respectively. Suppose that $\mathcal{G}$ is not a cone over a polygon. Let $\pi:\widetilde{\mathcal{E}}(\mathcal{G})\to\mathcal{E}(\mathcal{G})$ be the quotient map, defined as the restriction of the quotient map $(V^{*})^{f}\to(V^{*})^{f}/\mathbb{G}$ . Then $\pi$ is a locally trivial principal $\mathbb{G}$ -bundle.

In particular, $\widetilde{\mathcal{E}}(\mathcal{G})$ is a topological manifold of dimension

\dim\mathcal{E}(\mathcal{G})+\dim\mathbb{G}=(e-9)+(15+f)=e+f+6,

and admits a smooth structure such that $\pi$ is a smooth submersion.

Proof.

The group $\mathbb{G}$ acts freely on $\widetilde{\mathcal{E}}(\mathcal{G})$ by Lemma 6.3. Let $\alpha=(\alpha_{1},\cdots,\alpha_{f})\in\widetilde{\mathcal{E}}(\mathcal{G})$ . To prove the local triviality of the map $\pi$ , it suffices to construct a neighborhood $U$ of $[\alpha]$ in $\mathcal{E}(\mathcal{G})$ and a continuous section $s:U\to\widetilde{\mathcal{E}}(\mathcal{G})$ of $\pi$ (see Proposition 14.1.5 of [14]).

We reindex the facets of $\mathcal{G}$ if necessary, ensuring that each of the three facets $\mathcal{G}_{2},\mathcal{G}_{3},\mathcal{G}_{4}$ is adjacent to $\mathcal{G}_{1}$ . For each element $\beta=(\beta_{1},\ldots,\beta_{f})\in\widetilde{\mathcal{E}}(\mathcal{G})$ , the set

Q:=\{[x]\in\mathbb{S}^{3}\ |\ \beta_{i}(x)\geq 0\ \text{for all}\ i\}

is a convex 3-polytope that is combinatorially equivalent to $\mathcal{G}$ . Moreover, the four facets $Q_{1},Q_{2},Q_{3},Q_{4}$ do not share a common vertex, since $\mathcal{G}_{1},\mathcal{G}_{2},\mathcal{G}_{3},\mathcal{G}_{4}$ have no common vertex among them. Therefore, for each $\beta=(\beta_{1},\ldots,\beta_{f})\in\widetilde{\mathcal{E}}(\mathcal{G})$ , the first four linear functionals $\beta_{1},\beta_{2},\beta_{3},\beta_{4}\in V^{*}$ are linearly independent. Consequently, for $i\in\{5,\ldots,f\}$ and $j\in\{1,2,3,4\}$ , there are unique continuous maps $t_{i,j}:\widetilde{\mathcal{E}}(\mathcal{G})\to\mathbb{R}$ such that

\beta_{i}=\sum_{j=1}^{4}t_{i,j}(\beta)\beta_{j}.

To choose a local section around the point $[\alpha]$ , we use the observations on the coefficients $t_{i,j}(\alpha)$ from the proof of Lemma 6.3, which are numbered 1 and 2. According to the first observation, there exist three distinct indices $k_{1},k_{2},k_{3}\in\{1,2,3,4\}$ such that $t_{5,k_{j}}(\alpha)\neq 0$ . We define $k_{4}$ as the index in the complement $\{1,2,3,4\}\setminus\{k_{1},k_{2},k_{3}\}$ .

The second observation indicates that there is an index $l\in\{6,\ldots,f\}$ such that $t_{l,k_{4}}(\alpha)\neq 0$ . Moreover, the first observation again implies that $t_{l,k_{j_{0}}}(\alpha)\neq 0$ for some $j_{0}\in\{1,2,3\}$ . Lastly, for $i\in\{7,\ldots,f\}$ , we select arbitrary indices $p_{i}\in\{1,2,3,4\}$ such that $t_{i,k_{i}}(\alpha)\neq 0$ .

For notational simplicity, we assume that $k_{j}=j$ for $j=1,2,3,4$ , $l=6$ , and $j_{0}=1$ . Then the following conditions hold:

(i)

$t_{5,1}(\alpha),t_{5,2}(\alpha),t_{5,3}(\alpha)\neq 0$ ;
(ii)

$t_{6,1}(\alpha),t_{6,4}(\alpha)\neq 0$ ;
(iii)

$t_{i,p_{i}}(\alpha)\neq 0$ for $i=7,\ldots,f$ .

The above choices cannot be made if $f<6$ . In this case, a separate argument is required. If $f=4$ , then $\mathcal{G}$ is a tetrahedron, which means it is a cone over a polygon. If $f=5$ and $\mathcal{G}$ is not a cone over a polygon, then $\mathcal{G}$ is a triangular prism. In this situation, the realization space $\mathcal{RS}(\mathcal{G})\cong\mathcal{E}(\mathcal{G})$ is a singleton, which implies that the projection $\pi$ is a trivial bundle over a point.

By the continuity of $t_{i,j}:\widetilde{\mathcal{E}}(\mathcal{G})\to\mathbb{R}$ , there exists an open neighborhood $\widetilde{U}\subset\widetilde{\mathcal{E}}(\mathcal{G})$ of $\alpha$ such that the signs of

t_{5,1},t_{5,2},t_{5,3},t_{6,1},t_{6,4},t_{i,p_{i}}\quad(\text{for}\ i\geq 7)

remain constant over $\widetilde{U}$ . Let $\sigma_{i,j}\in\{\pm 1\}$ denote the constant signs of $t_{i,j}$ for the specified indices $i,j$ and the neighborhood $\widetilde{U}$ .

We construct a map $\widetilde{s}:\widetilde{U}\to\widetilde{\mathcal{E}}(\mathcal{G})$ , which will descend to the section $s:U\to\widetilde{\mathcal{E}}(\mathcal{G})$ (where $U:=\pi(\widetilde{U})$ ) that we are looking for. Let $\beta=(\beta_{1},\cdots,\beta_{f})\in\widetilde{U}$ . We claim that there is a unique element $\beta^{\prime}=(\beta_{1}^{\prime},\cdots,\beta_{f}^{\prime})$ in the $\mathbb{G}$ -orbit of $\beta$ satisfying the following properties:

-

$\beta_{i}^{\prime}=e_{i}^{*}$ for $i=1,2,3,4$ , where $\{e_{1}^{*},\cdots,e_{4}^{*}\}$ is the standard dual basis of $V^{*}=(\mathbb{R}^{4})^{*}$ ;
-

$t_{5,j}(\beta^{\prime})=\sigma_{5,j}$ for $j=1,2,3$ ;
-

$t_{6,j}(\beta^{\prime})=\sigma_{6,j}$ for $j=1,4$ ;
-

$t_{i,p_{i}}(\beta^{\prime})=\sigma_{i,p_{i}}$ for $i\geq 7$ .

In other words, we need to prove that there exists a unique $g=(A,c_{1},\cdots,c_{f})\in\mathbb{G}$ such that $g\cdot\beta=\beta^{\prime}$ . This equation can be expressed as:

(c_{1}^{-1}\beta_{1}\circ A^{-1},\cdots,c_{f}^{-1}\beta_{f}\circ A^{-1})=(\beta_{1}^{\prime},\cdots,\beta_{f}^{\prime}).

The first condition, $c_{i}^{-1}\beta_{i}\circ A^{-1}=\beta_{i}^{\prime}=e_{i}^{*}$ for $i=1,2,3,4$ , is equivalent to

(6.3)

\displaystyle A=\begin{bmatrix}c_{1}^{-1}\beta_{1}\\ c_{2}^{-1}\beta_{2}\\ c_{3}^{-1}\beta_{3}\\ c_{4}^{-1}\beta_{4}\end{bmatrix},

where we interpret the $\beta_{i}$ in each row as the coordinate vector of $\beta_{i}$ with respect to the dual basis $\{e_{1}^{*},\cdots,e_{4}^{*}\}$ . This implies that we are subject to the relation

\pm 1=\det A=c_{1}^{-1}c_{2}^{-1}c_{3}^{-1}c_{4}^{-1}\det(\beta_{1},\cdots,\beta_{4}).

For each $i\geq 5$ , we have

\beta_{i}=\sum_{j=1}^{4}t_{i,j}(\beta)\beta_{j},

thus,

\sum_{j=1}^{4}t_{i,j}(\beta^{\prime})e_{j}^{*}=\beta^{\prime}=c_{i}^{-1}\beta_{i}\circ A^{-1}=c_{i}^{-1}\sum_{j=1}^{4}t_{i,j}(\beta)\beta_{j}\circ A^{-1}=c_{i}^{-1}\sum_{j=1}^{4}t_{i,j}(\beta)c_{j}e_{j}^{*}.

By comparing the coefficients, it can be verified that there are unique $c_{1},\cdots,c_{f}>0$ satisfying $\pm 1=c_{1}^{-1}c_{2}^{-1}c_{3}^{-1}c_{4}^{-1}\det(\beta_{1},\cdots,\beta_{4})$ , and a corresponding matrix $A$ such that the requirements for $\beta^{\prime}$ and equation (6.3) are satisfied.

We further note that the unique elements $c_{1},\cdots,c_{f}$ and $A$ depend continuously on $\beta$ , allowing us to obtain a continuous map $\beta\mapsto\beta^{\prime}=:\widetilde{s}(\beta)$ . Since $\mathbb{G}$ acts continuously on $\widetilde{\mathcal{E}}(\mathcal{G})$ , the map $\pi$ is an open map, which implies that $U:=\pi(\widetilde{U})$ is an open neighborhood of $[a]=\pi(\alpha)$ . Furthermore, since the map $\widetilde{s}:\widetilde{U}\to\widetilde{\mathcal{E}}(\mathcal{G})$ is constant on the intersection of $\widetilde{U}$ with each $\mathbb{G}$ -orbit, the map $\widetilde{s}$ descends to a continuous section $s:U\to\widetilde{\mathcal{E}}(\mathcal{G})$ . This completes the proof. ∎

6.2. The projection $\mathcal{C}(\mathcal{G})\to\mathcal{RS}(\mathcal{G})$ and the restricted deformation spaces

In this section, we consider a natural map $\mathcal{C}(\mathcal{G})\to\mathcal{RS}(\mathcal{G})$ and observe that each nonempty fiber of the map can be identified with a restricted deformation space $\mathcal{C}_{Q}(\mathcal{G})$ . In view of Theorem 4.5 and Theorem 5.4, the space $\mathcal{RS}(\mathcal{G})$ and the nonempty fibers of the map $\mathcal{C}(\mathcal{G})\to\mathcal{RS}(\mathcal{G})$ thus admit smooth structures.

For computational purposes, especially in constructing the smooth structure on $\mathcal{C}(\mathcal{G})$ , we use the embedding of $\mathcal{C}(\mathcal{G})$ onto a subspace $\mathcal{D}(\mathcal{G})\subset((V^{*})^{f}\times V^{f})/\mathbb{G}$ as described in Lemma 2.4. This embedding allows us to transform the map $\mathcal{C}(\mathcal{G})\to\mathcal{RS}(\mathcal{G})$ into a topologically equivalent map $\mathcal{D}(\mathcal{G})\to\mathcal{E}(\mathcal{G})$ given by restricting the projection $((V^{*})^{f}\times V^{f})/\mathbb{G}\to(V^{*})^{f}/\mathbb{G}$ .

In constructing the smooth structure on $\mathcal{C}(\mathcal{G})$ , it will also be convenient to work with the lift $\widetilde{\mathcal{D}}(\mathcal{G})\subset(V^{*})^{f}\times V^{f}$ of $\mathcal{D}(\mathcal{G})$ under the projection $(V^{*})^{f}\times V^{f}\to((V^{*})^{f}\times V^{f})/\mathbb{G}$ , in a manner analogous to our definition of $\widetilde{\mathcal{E}}(\mathcal{G})$ .

We will examine the map $\widetilde{\mathcal{D}}(\mathcal{G})\to\widetilde{\mathcal{E}}(\mathcal{G})$ and observe that the nonempty fibers of this map can also be identified with the restricted deformation spaces.

Let $\mathcal{G}$ be a labeled combinatorial 3-polytope. Recall that the elements of $\mathcal{C}(\mathcal{G})$ are of the form $[(Q,r_{1},\cdots,r_{f})]$ , where $Q\subset\mathbb{S}^{3}$ is a convex 3-polytope combinatorially equivalent to $\mathcal{G}$ , and $r_{1},\cdots,r_{f}\in\text{SL}_{\pm}(V)$ are linear reflections in the facets $Q_{1},\cdots,Q_{f}$ of $Q$ corresponding to the facets $\mathcal{G}_{1},\cdots,\mathcal{G}_{f}$ of $\mathcal{G}$ , respectively. The projection $(Q,r_{1},\cdots,r_{f})\mapsto Q$ descends to the map $\mathcal{C}(\mathcal{G})\to\mathcal{RS}(\mathcal{G})$ given by $[(Q,r_{1},\cdots,r_{f})]\mapsto[(q_{v})_{v\in\mathcal{V}}]$ , where each vertex $v\in\mathcal{V}$ is a vertex of the facet $\mathcal{G}_{j}$ if and only if the vertex $q_{v}$ of $Q$ is fixed by the reflection $r_{j}$ .

Recall that by Lemma 2.4, we have an embedding $\Phi:\mathcal{C}(\mathcal{G})\to((V^{*})^{f}\times V^{f})/\mathbb{G}$ . Let $\mathcal{D}(\mathcal{G}):=\Phi(\mathcal{C}(\mathcal{G}))$ denote the image of this embedding (see the diagram in Figure 2). Let

\eta:(V^{*})^{f}\times V^{f}\to((V^{*})^{f}\times V^{f})/\mathbb{G}

be the projection induced by the action of $\mathbb{G}$ on $(V^{*})^{f}\times V^{f}$ , and define $\widetilde{\mathcal{D}}(\mathcal{G}):=\eta^{-1}(\mathcal{D}(\mathcal{G}))$ . An element $(\alpha_{1},\cdots,\alpha_{f},v_{1},\cdots,v_{f})\in(V^{*})^{f}\times V^{f}$ belongs to $\widetilde{\mathcal{D}}(\mathcal{G})$ if and only if it satisfies the conditions of Remark 3.4.

Now, the map $\mathcal{C}(\mathcal{G})\to\mathcal{RS}(\mathcal{G})$ can be described in terms of the images of the embeddings $\Phi$ and $\iota$ . This map is simply the projection $\overline{\rho}$ induced by the natural projection $\rho:(V^{*})^{f}\times V^{f}\to(V^{*})^{f}$ given by

[(\alpha_{1},\cdots,\alpha_{f},v_{1},\cdots,v_{f})]\mapsto[(\alpha_{1},\cdots,\alpha_{f})].

We thus have the following commuting diagram:

Figure 2.

Recall from Section 4 that for each convex 3-polytope $Q\subset\mathbb{S}^{3}$ representing the underlying combinatorial polytope of $\mathcal{G}$ , the restricted deformation space $\mathcal{C}_{Q}(\mathcal{G})$ is defined to be the subspace consisting of $[(Q^{\prime},r_{1}^{\prime},\cdots,r_{f}^{\prime})]\in\mathcal{C}(\mathcal{G})$ such that $Q$ and $Q^{\prime}$ are projectively equivalent. In other words, the restricted deformation spaces in $\mathcal{C}(\mathcal{G})$ are precisely the nonempty fibers of the map $\mathcal{C}(\mathcal{G})\to\mathcal{RS}(\mathcal{G})$ .

We abuse the notation by letting $\rho:\widetilde{\mathcal{D}}(\mathcal{G})\to\widetilde{\mathcal{E}}(\mathcal{G})$ to denote the restriction of the natural projection $\rho:(V^{*})^{f}\times V^{f}\to(V^{*})^{f}$ as well. The map $\rho$ is $\mathbb{G}$ -equivariant, so it induces a map $\overline{\rho}:\mathcal{D}(\mathcal{G})\to\mathcal{E}(\mathcal{G})$ . By the above commuting diagram, the restricted deformation spaces can be identified with the nonempty fibers of the map $\overline{\rho}$ . Moreover, each nonempty fiber $\rho^{-1}(\alpha)$ of the map $\rho$ is mapped homeomorphically onto the fiber $\overline{\rho}^{-1}(\pi(\alpha))$ , since the action of $\mathbb{G}$ on $\widetilde{\mathcal{E}}(\mathcal{G})$ is free and the projection map $\eta$ is open. Therefore, each nonempty fiber $\rho^{-1}(\alpha)$ is homeomorphic to the restricted deformation space $\mathcal{C}_{Q}(\mathcal{G})$ , where

Q:=\{[x]\in\mathbb{S}^{3}\ |\ \alpha_{i}(x)\geq 0\ \text{for all}\ i\}

is the convex 3-polytope determined by $\alpha$ .

We conclude this subsection by discussing the smooth structure on each fiber of $\rho$ . In the forthcoming discussions, the nonempty fibers $\rho^{-1}(\alpha)\subset\{\alpha\}\times V^{f}$ will often be identified with the subspace $V^{f}$ via the homeomorphism $\{\alpha\}\times V^{f}\to V^{f}$ . By Theorem 4.5, each nonempty fiber $\rho^{-1}(\alpha)$ is a smooth manifold. In fact, the theorem’s proof shows that $\rho^{-1}(\alpha)$ is an embedded submanifold of $V^{f}$ . To explain this, we briefly outline the key idea behind the theorem’s proof.

Since we are considering the restricted deformation space, we fix a convex 3-polytope $P$ with facets $P_{1},\ldots,P_{f}$ and determine all possible reflections in the $f$ facets that make $P$ a Coxeter 3-polytope. As the convex 3-polytope $P$ and its facets are fixed, we only need to find all possible choices of the vectors $v_{1},\ldots,v_{f}$ defining the reflections. We may assume that the polytope $P$ and the vectors $v_{1},\ldots,v_{f}$ lie in $\mathbb{R}^{3}$ .

There exist $f$ affine functions $d_{j}:\mathbb{R}^{3}\to\mathbb{R}$ , for $j=1,\ldots,f$ , such that $P=\{x\in\mathbb{R}^{3}\mid d_{j}(x)\geq 0\}$ and $d_{j}(x)=0$ on the facet $P_{j}$ . The map $D:=(d_{1},\ldots,d_{f}):\mathbb{R}^{3}\to\mathbb{R}^{f}$ is an affine function mapping onto a 3-dimensional affine subspace $A:=D(\mathbb{R}^{3})\subset\mathbb{R}^{f}$ . Each vector $v_{j}$ is mapped to $D(v_{j})$ , whose $j$ -th coordinate must be negative by the first condition of Remark 3.4. Under the projection $\mathbb{R}^{f}\to\mathbb{S}^{f-1}$ , it is mapped to $[(v_{j1},\ldots,-1,\ldots,v_{jf})]\in\mathbb{S}^{f-1}$ , where the $j$ -th coordinate is $-1$ . By removing the $j$ -th coordinate, we obtain an element $v_{j}^{\prime}\in\mathbb{R}^{f-1}$ .

In this manner, the tuple $(v_{1},\ldots,v_{f})$ can be viewed as an element of $\mathbb{R}^{(f-1)f}$ by stacking $v_{1}^{\prime},\ldots,v_{f}^{\prime}$ . The entries of $v_{1},\ldots,v_{f}$ must satisfy the equalities in Remark 3.4 as well as the defining equations of the affine subspace $A$ . Thus, the set of such $(v_{1}^{\prime},\ldots,v_{f}^{\prime})\in\mathbb{R}^{(f-1)f}$ is the zero-level set of a polynomial map $\mathbb{R}^{(f-1)f}\to\mathbb{R}^{e+e_{2}+(f-4)f}$ , where $e+e_{2}+(f-4)f$ represents the total number of equalities in Remark 3.4 and the defining equations of $A$ .

In the proof of Theorem 4.5, Choi proves that the polynomial map has full rank. Hence, the set of $(v_{1}^{\prime},\ldots,v_{f}^{\prime})$ , which represents the restricted deformation space, is an embedded submanifold of $\mathbb{R}^{f(f-1)}$ . This observation will be employed to verify the smoothness of maps between nonempty fibers of $\rho$ .

Specifically, if there exists a smooth map between two open subsets of $V^{f}$ containing nonempty fibers of $\rho$ , and if this smooth map sends one nonempty fiber of $\rho$ to another, then the restriction of the smooth map to these fibers is also smooth. This fact will be crucial in constructing a smooth atlas on $\widetilde{\mathcal{D}}(\mathcal{G})$ .

6.3. Construction of a smooth structure on $\widetilde{\mathcal{D}}(\mathcal{G})$

In this section, we construct a smooth atlas on $\widetilde{\mathcal{D}}(\mathcal{G})$ . The existence of this smooth structure, along with Lemma 6.9, will be used to define a smooth structure on $\mathcal{C}(\mathcal{G})$ .

The construction proceeds as follows. To define a coordinate chart around each point $(\alpha,v)$ in $\widetilde{\mathcal{D}}(\mathcal{G})$ (note that $v\in\rho^{-1}(\alpha)$ ), it suffices to find a homeomorphism between an open neighborhood of $(\alpha,v)$ in $\widetilde{\mathcal{D}}(\mathcal{G})$ and an open subset of $\widetilde{\mathcal{E}}(\mathcal{G})\times\rho^{-1}(\alpha)$ , as the latter space is proven to be a smooth manifold by Lemma 6.4 and Theorem 4.5. We do it by choosing some open neighborhoods $U_{\alpha}\subset\widetilde{\mathcal{E}}(\mathcal{G})$ and $W_{v}\subset\rho^{-1}(\alpha)$ of $\alpha$ and $v$ , respectively, and constructing a chart $\Phi_{\alpha,v}:U_{\alpha}\times W_{v}\to\widetilde{\mathcal{D}}(\mathcal{G})$ which maps $(\alpha,v)\in U_{\alpha}\times W_{v}$ to $(\alpha,v)\in\widetilde{\mathcal{D}}(\mathcal{G})$ .

For each element $(\beta,w)\in U_{\alpha}\times W_{v}$ , we will construct $\Phi(\beta,w)$ so that it is of the form $(\beta,w^{\prime})$ . Geometrically, we have perturbed the convex 3-polytope $\alpha$ to obtain another convex 3-polytope $\beta$ within the same realization space, and we need to find the tuple $w^{\prime}=(w_{1}^{\prime},\cdots,w_{f}^{\prime})\in V^{f}$ of vectors so that the resulting endomorphisms $\text{Id}-\beta_{j}\otimes w_{j}^{\prime}$ are linear reflections and satisfy the Vinberg’s relations in Remark 3.4. Note that for each $(\beta,w)\in U_{\alpha}\times W_{v}$ , we always have $(\alpha,w)\in\widetilde{\mathcal{D}}(\mathcal{G})$ because $w\in W_{v}\subset\rho^{-1}(\alpha)$ but $(\beta,w)$ itself may not lie in $\widetilde{\mathcal{D}}(\mathcal{G})$ in general.

We construct $w_{1}^{\prime},\cdots,w_{f}^{\prime}$ in succession to satisfy the conditions in Remark 3.4. In each step, $w_{i}^{\prime}$ is determined as the solution to a system of up to 4 linear equations. If there are fewer than 4 equations, $w_{i}^{\prime}$ may not be uniquely determined, so we add extra equations as needed. For this purpose, we assign 3 indices $k_{i,1},k_{i,2},k_{i,3}\subset\{1,\cdots,f\}$ according to the orderability condition and add necessary linear equations to some of these indices. This process allows us to determine a unique $w^{\prime}$ such that $(\beta,w^{\prime})\in\widetilde{\mathcal{D}}(\mathcal{G})$ .

Finally, we verify in Lemma 6.8 that the transition maps between the charts $\{\Phi_{\alpha,v}\ |\ (\alpha,v)\in\widetilde{\mathcal{D}}(\mathcal{G})\}$ , are smooth, and this completes the construction of the smooth atlas on $\widetilde{\mathcal{D}}(\mathcal{G})$ .

We begin with giving an explicit description of the elements of $\widetilde{\mathcal{D}}(\mathcal{G})$ . Recall that the subspace $\widetilde{\mathcal{D}}(\mathcal{G})\subset(V^{*})^{f}\times V^{f}$ is defined as the preimage $\eta^{-1}(\mathcal{D}(\mathcal{G}))$ under the natural projection

\eta:(V^{*})^{f}\times V^{f}\to((V^{*})^{f}\times V^{f})/\mathbb{G}.

Moreover, the space $\mathcal{D}(\mathcal{G})\subset((V^{*})^{f}\times V^{f})/\mathbb{G}$ is defined as the image of the embedding $\Phi:\mathcal{C}(\mathcal{G})\to((V^{*})^{f}\times V^{f})/\mathbb{G}$ from Lemma 2.4. According to Theorem 3.1 and Lemma 6.2, an element

(\alpha,v)=(\alpha_{1},\cdots,\alpha_{f},v_{1},\cdots,v_{f})\in(V^{*})^{f}\times V^{f}

belongs to $\widetilde{\mathcal{D}}(\mathcal{G})$ if and only if it satisfies the following conditions:

(V1)

$\alpha_{i}(v_{i})=2$ for $i=1,\cdots,f$ ;
(V2)

$\alpha_{i}(v_{j})\leq 0$ if $i\neq j$ ;
(V3)

$\alpha_{i}(v_{j})=0$ if and only if $\alpha_{j}(v_{i})=0$ ;
(V4)

if $\mathcal{G}_{i}$ and $\mathcal{G}_{j}$ are adjacent and $m_{i,j}$ is the integer assigned to the edge $\mathcal{G}_{i}\cap\mathcal{G}_{j}$ , then $\alpha_{i}(v_{j})\alpha_{j}(v_{i})=4\cos^{2}\left(\frac{\pi}{m_{i,j}}\right)$ ;
(V5)

if $\mathcal{G}_{i}$ and $\mathcal{G}_{j}$ are not adjacent, then $\alpha_{i}(v_{j})\alpha_{j}(v_{i})\geq 4$ ;
(V6)

the set

$Q:=\{[x]\in\mathbb{S}^{3}\mid\alpha_{i}(x)\geq 0\ \text{for all}\ i\}$

is a convex 3-polytope combinatorially equivalent to $\mathcal{G}$ , and there exists a face-lattice isomorphism $\textup{FL}(\mathcal{G})\to\textup{FL}(Q)$ mapping each facet $\mathcal{G}_{i}$ to the facet of $Q$ supported by $\alpha_{i}$ .

As the first step in constructing a smooth structure on $\widetilde{\mathcal{D}}(\mathcal{G})$ , we need to select indices according to the orderability condition. Since $\mathcal{G}$ is orderable, we reindex the facets $\mathcal{G}_{1},\cdots,\mathcal{G}_{f}$ if necessary, so that for each facet $\mathcal{G}_{i}$ , there are at most three facets $\mathcal{G}_{j}$ adjacent to $\mathcal{G}_{i}$ such that either:

-

$j<i$ , or
-

$m_{i,j}=2$ .

Then, for each $i\in\{1,\cdots,f\}$ , we choose three distinct indices $k_{i,1},k_{i,2},k_{i,3}\in\{1,\cdots,f\}$ such that:

(i)

For $j=k_{i,1},k_{i,2},k_{i,3}$ , the facets $\mathcal{G}_{i}$ and $\mathcal{G}_{j}$ are adjacent.
(ii)

The first $a_{i}$ indices $j\in\{k_{i,1},\cdots,k_{i,a_{i}}\}$ satisfy $m_{i,j}=2$ .
(iii)

The next $b_{i}$ indices $j\in\{k_{i,a_{i}+1},\cdots,k_{i,a_{i}+b_{i}}\}$ satisfy $m_{i,j}\geq 3$ and $j<i$ .
(iv)

The last $3-a_{i}-b_{i}$ indices $j\in\{k_{i,a_{i}+b_{i}+1},\cdots,k_{i,3}\}$ satisfy $m_{i,j}\geq 3$ and $j>i$ .
(v)

If $\mathcal{G}_{i}$ and $\mathcal{G}_{j}$ are adjacent and satisfy either $j<i$ or $m_{i,j}=2$ , then $j$ is included in $\{k_{i,1},k_{i,2},k_{i,3}\}$ .

Since $\mathcal{G}$ is orderable, for each $i$ , there are at most three facets $\mathcal{G}_{j}$ adjacent to $\mathcal{G}_{i}$ such that either $m_{i,j}=2$ or $j<i$ . Therefore, the indices $k_{i,1},\cdots,k_{i,a_{i}+b_{i}}$ with $a_{i}+b_{i}\leq 3$ in the above requirement do exist. Note that the requirement (v) indicates that $k_{i,1},\cdots,k_{i,a_{i}+b_{i}}$ include all such indices. For the remaining facets $\mathcal{G}_{j}$ adjacent to $\mathcal{G}_{i}$ , we must have $j>i$ . In (iv), we arbitrarily select $3-a_{i}-b_{i}$ indices $k_{i,a_{i}+b_{i}+1},\cdots,k_{i,3}$ with this property.

We make such a choice due to a technical reason, and the choice of indices $k_{i,1},k_{i,2},k_{i,3}$ achieves the following.

Given a tuple $\beta=(\beta_{1},\cdots,\beta_{f})\in\widetilde{\mathcal{E}}(\mathcal{G})$ , we will frequently construct $w_{1},\cdots,w_{f}$ such that $\alpha_{i},w_{j}$ satisfy the conditions (V1), (V3), and (V4). We will construct them inductively, i.e. we assume that $w_{1},\cdots,w_{i-1}$ are determined and then construct $w_{i}$ . The conditions (V1), (V3), and (V4) indicates the values of $\beta_{j}(w_{i})$ for $j\in\{i,k_{i,1},\cdots,k_{i,a_{i}+b_{i}}\}$ . For instance, if $j\in\{k_{i,a_{i}+1},\cdots,k_{i,a_{i}+b_{i}}\}$ , then (V4) implies $\beta_{j}(w_{i})=\frac{4\cos^{2}\left(\frac{\pi}{m_{i,j}}\right)}{\beta_{i}(w_{j})}$ . (Note that $w_{j}$ on the right-hand side is given by induction hypothesis.)

Since $a_{i}+b_{i}\leq 3$ , we thus have at most 4 linear equations for $w_{i}$ . We add $3-a_{i}-b_{i}$ complementary linear equations by assigning the values $\beta_{j}(w_{i})$ for $j\in\{a_{i}+b_{i}+1,\cdots,3\}$ . The resulting system consists of 4 linear equations and thus determine a unique vector $w_{i}$ .

Now let $(\alpha,v)=(\alpha_{1},\cdots,\alpha_{f},v_{1},\cdots,v_{f})\in\widetilde{\mathcal{D}}(\mathcal{G})$ . We construct an open embedding $\Phi_{\alpha,v}:U_{\alpha}\times W_{v}\to\widetilde{\mathcal{D}}(\mathcal{G})$ of the product of some smooth manifolds $U_{\alpha}$ and $W_{v}$ such that:

-

$U_{\alpha}\subset\widetilde{\mathcal{E}}(\mathcal{G})$ is an open neighborhood of $\alpha$ ;
-

$W_{v}\subset\rho^{-1}(\alpha)\subset\{\alpha\}\times V^{f}\cong V^{f}$ is an open neighborhood of $v$ in $\rho^{-1}(\alpha)$ ; and
-

$\Phi_{\alpha,v}(\alpha,v)=(\alpha,v)\in\widetilde{\mathcal{D}}(\mathcal{G})$ .

Since $U_{\alpha}$ and $W_{v}$ are open in $\widetilde{\mathcal{E}}(\mathcal{G})$ and $\rho^{-1}(\alpha)$ , respectively, and since the latter spaces are proven to admit smooth structures, the sets $U_{\alpha}$ and $W_{v}$ inherit these smooth structures. It will be shown that the transition maps $\Phi_{\beta,w}^{-1}\circ\Phi_{\alpha,v}$ are smooth with respect to these smooth structures. The smooth structure on $\widetilde{\mathcal{D}}(\mathcal{G})$ will be the maximal atlas induced from the open embeddings of this form.

We temporarily let $U_{\alpha}\subset\widetilde{\mathcal{E}}(\mathcal{G})$ and $W_{v}\subset\rho^{-1}(\alpha)$ be arbitrary open neighborhoods of $\alpha$ and $v$ , respectively. These neighborhoods will be shrunk as needed. Let $(\beta,w)\in U_{\alpha}\times W_{v}$ (where we identify $W_{v}\subset\{\alpha\}\times V^{f}$ as a subspace of $V^{f}$ ). Note that $(\beta,w)$ is not an element of $\widetilde{\mathcal{D}}(\mathcal{G})$ in general, since $w\notin\rho^{-1}(\alpha)$ unless $\alpha=\beta$ . We define $\Phi_{\alpha,v}(\beta,w)\in\widetilde{\mathcal{D}}(\mathcal{G})$ to be the element obtained by the following lemma.

Lemma 6.5.

There exist sufficiently small neighborhoods $U_{\alpha}\subset\widetilde{\mathcal{E}}(\mathcal{G})$ and $W_{v}\subset\rho^{-1}(\alpha)$ of $\alpha$ and $v$ , respectively, such that there is a unique continuous map $w^{\prime}:U_{\alpha}\times W_{v}\to V$ satisfying the following properties:

(i)

$(\beta,w^{\prime})\in\widetilde{\mathcal{D}}(\mathcal{G})$ ,
(ii)

$w^{\prime}\to v$ as $(\beta,w)\to(\alpha,v)$ , so that $w^{\prime}(\alpha,v)=v$ ,
(iii)

$\beta_{j}(w_{i}^{\prime})=\alpha_{j}(w_{i})$ for $i\in\{1,\cdots,f\}$ , $j\in\{k_{i,a_{i}+b_{i}+1},\cdots,k_{i,3}\}$ .

Proof.

We construct $w_{1}^{\prime}=w_{1}^{\prime}(\beta,w),\cdots,w_{f}^{\prime}=w_{f}^{\prime}(\beta,w)$ in this order, shrinking $U_{\alpha}$ and $W_{v}$ during the process if necessary.

By requirement (i), the element $(\beta,w^{\prime})$ must satisfy conditions (V1) through (V6). We construct $w_{1}^{\prime},\cdots,w_{f}^{\prime}$ so that these conditions are not violated at each step.

We define $w_{1}^{\prime}\in V$ as the unique solution to the following system of four linear equations:

$\displaystyle\beta_{1}(w_{1}^{\prime})$	$\displaystyle=2,$
$\displaystyle\beta_{j}(w_{1}^{\prime})$	$\displaystyle=0$	for	$\displaystyle j\in\{k_{1,1},\cdots,k_{1,a_{1}}\},$
$\displaystyle\beta_{j}(w_{1}^{\prime})$	$\displaystyle=\alpha_{j}(w_{1})\quad$	for	$\displaystyle j\in\{k_{1,a_{1}+1},\cdots,k_{1,3}\}.$

Note that $b_{1}=0$ since 1 is the smallest index. The solution $w_{1}^{\prime}$ exists and is unique: since $\mathcal{G}_{k_{1,1}},\mathcal{G}_{k_{1,2}},\mathcal{G}_{k_{1,3}}$ are adjacent to $\mathcal{G}_{1}$ , the four linear functionals $\beta_{k_{1,1}},\beta_{k_{1,2}},\beta_{k_{1,3}},\beta_{1}$ are linearly independent by Lemma 6.2. By Cramer’s rule, $w_{1}^{\prime}$ is a continuous function of $\beta$ and $w$ . Moreover, if we substitute $(\beta,w)=(\alpha,v)$ , then it follows from the uniqueness of the solution that $w_{1}^{\prime}=v_{1}$ , implying that $w_{1}^{\prime}\to v_{1}$ as $(\beta,w)\to(\alpha,v)$ .

The element $(\alpha,w)$ may not belong to $\widetilde{\mathcal{D}}(\mathcal{G})$ , and in particular it is possible that $\alpha_{j}(w_{1})\geq 0$ for some $j\in\{k_{1,a_{1}+1},\cdots,k_{1,3}\}$ , so $(\beta,w^{\prime})$ might violate condition (V2) even if $w_{2}^{\prime},\cdots,w_{f}^{\prime}$ are not yet defined. However, since $\alpha_{j}(v_{1})<0$ for $j\in\{k_{a_{1}+1},\cdots,k_{1,3}\}$ , we can use the continuity of $w_{1}^{\prime}$ to choose smaller neighborhoods $U_{\alpha}$ and $W_{v}$ so that $\beta_{j}(w_{1}^{\prime})=\alpha_{j}(w_{1})<0$ for $j\in\{k_{a_{1}+1},\cdots,k_{1,3}\}$ and $(\beta,w)\in U_{\alpha}\times W_{v}$ , ensuring that condition (V2) is no longer violated.

We define $w_{2}^{\prime}$ in a similar way, as a continuous function of $\beta$ , $w$ , and $w_{1}^{\prime}$ . Since $w_{1}^{\prime}$ depends continuously on $\beta$ and $w$ , the resulting $w_{2}^{\prime}$ will also depend continuously only on $\beta$ and $w$ .

Let $w_{2}^{\prime}\in V$ be the unique solution of the following system of four linear equations:

$\displaystyle\beta_{2}(w_{2}^{\prime})$	$\displaystyle=2,$
$\displaystyle\beta_{j}(w_{2}^{\prime})$	$\displaystyle=0\quad$	for	$\displaystyle j\in\{k_{2,1},\cdots,k_{2,a_{2}}\},$
$\displaystyle\beta_{j}(w_{2}^{\prime})$	$\displaystyle=\frac{4\cos^{2}\left(\frac{\pi}{m_{i,j}}\right)}{\beta_{2}(w_{j}^{\prime})}\quad$	for	$\displaystyle j\in\{k_{2,a_{2}+1},\cdots,k_{2,a_{2}+b_{2}}\},$
$\displaystyle\beta_{j}(w_{2}^{\prime})$	$\displaystyle=\alpha_{j}(w_{2})\quad$	for	$\displaystyle j\in\{k_{2,a_{2}+b_{2}+1},\cdots,k_{2,3}\}.$

Using similar logic as with $w_{1}^{\prime}$ , we confirm that $\beta_{2},\beta_{k_{2,1}},\beta_{k_{2,2}},\beta_{k_{2,3}}$ are linearly independent. Since $b_{2}$ is either 0 or 1 (because $j=1$ is the only index with $j<2$ ) and $\beta_{2}(w_{1}^{\prime})<0$ , $w_{2}^{\prime}$ is well-defined and continuous.

We proceed similarly to define $w_{3}^{\prime},\cdots,w_{f}^{\prime}$ by iterating this construction process, ensuring each $w_{i}^{\prime}$ is continuous and meets the conditions (V1) through (V6) as required. Fix $i\geq 3$ and assume that the neighborhoods $U_{\alpha}$ and $W_{v}$ of $\alpha$ and $v$ , along with the continuous functions $w_{k}^{\prime}:U_{\alpha}\times W_{v}\to V$ for $k=1,\cdots,i-1$ , are defined so that $w_{k}^{\prime}(\alpha,v)=v$ .

We define $w_{i}^{\prime}\in V$ to be the solution to the following system of four linear equations:

$\displaystyle\beta_{i}(w_{i}^{\prime})$	$\displaystyle=2,$
$\displaystyle\beta_{j}(w_{i}^{\prime})$	$\displaystyle=0\quad$	for	$\displaystyle j\in\{k_{i,1},\cdots,k_{i,a_{i}}\},$
$\displaystyle\beta_{j}(w_{i}^{\prime})$	$\displaystyle=\frac{4\cos^{2}\left(\frac{\pi}{m_{i,j}}\right)}{\beta_{i}(w_{j}^{\prime})}\quad$	for	$\displaystyle j\in\{k_{i,a_{i}+1},\cdots,k_{i,a_{i}+b_{i}}\},$
$\displaystyle\beta_{j}(w_{i}^{\prime})$	$\displaystyle=\alpha_{j}(w_{i})\quad$	for	$\displaystyle j\in\{k_{i,a_{i}+b_{i}+1},\cdots,k_{i,3}\}.$

The continuity of $w_{i}^{\prime}=w_{i}^{\prime}(\beta,w)$ follows from the continuities of $w_{1}^{\prime},\cdots,w_{i-1}^{\prime}$ . Since $w_{1}^{\prime}(\alpha,v)=\cdots=w_{i-1}^{\prime}(\alpha,v)=v$ , we also obtain $w_{i}^{\prime}(\alpha,v)=v$ by the uniqueness of the solution and the fact that $(\alpha,v)$ satisfies (V1) through (V6). Since $\alpha_{j}(v_{i})<0$ for $j\in\{k_{i,a_{i}+b_{i}+1},\cdots,k_{i,3}\}$ , the continuity of $w_{i}^{\prime}$ ensures that we can choose small neighborhoods $U_{\alpha}$ and $W_{v}$ so that $\beta_{j}(w_{i}^{\prime})<0=\alpha_{j}(w_{i})<0$ for $j\in\{k_{i,a_{i}+b_{i}+1},\cdots,k_{i,3}\}$ . In this way, we obtain continuous functions $w_{1}^{\prime},\cdots,w_{f}^{\prime}$ on $U_{\alpha}\times W_{v}$ satisfying the condition (ii).

Finally, since $(\alpha,v)$ satisfies conditions (V2) and (V3), we can select a sufficiently small neighborhood $\mathcal{U}$ of $(\alpha,v)\in(V^{*})^{f}\times V^{f}$ such that

\beta_{i}(w_{j})<0\quad\text{for}\ i,j\ \text{with}\ i\neq j\ \text{and}\ m_{i,j}\neq 2,\text{and for}\ (\beta,w)\in\mathcal{U}.

By choosing smaller neighborhoods $U_{\alpha}$ and $W_{v}$ , we ensure that $(\beta,w^{\prime})\in\mathcal{U}$ for $w^{\prime}=w^{\prime}(\beta,w)$ with $(\beta,w)\in U_{\alpha}\times W_{v}$ .

It remains to show that $(\beta,w^{\prime})\in\widetilde{\mathcal{D}}(\mathcal{G})$ , i.e., $(\beta,w^{\prime})$ satisfies conditions (V1) through (V6). Conditions (V1), (V3), and (V4) are satisfied by construction. Since $\beta\in\widetilde{\mathcal{E}}(\mathcal{G})$ , condition (V6) is automatically satisfied by Lemma 6.2. Condition (V2) holds because if $m_{i,j}=2$ , it is satisfied by the construction of $w_{i}^{\prime}$ , and if $m_{i,j}\geq 3$ , it holds by our choice of neighborhoods $\mathcal{U},U_{\alpha},W_{v}$ .

Condition (V5) follows from Lemma 6.6. According to this lemma, if $\mathcal{G}_{i}$ and $\mathcal{G}_{j}$ are not adjacent, we can find constants $c_{l}\geq 0$ for $l\in S_{i}$ (the set of indices $k$ such that $\mathcal{G}_{k}$ and $\mathcal{G}_{i}$ are adjacent) and $c_{i}>0$ such that

\beta_{j}=-c_{i}\beta_{i}+\sum_{l\in S_{i}}c_{l}\beta_{l}.

Using this, we obtain

	$\displaystyle\beta_{i}(w_{j}^{\prime})\beta_{j}(w_{i}^{\prime})$	$\displaystyle=\beta_{i}(w_{j}^{\prime})\left(-2c_{i}+\sum_{l\in S_{i}}c_{l}\beta_{l}(w_{i}^{\prime})\right)$
		$\displaystyle\geq-2c_{i}\beta_{i}(w_{j}^{\prime})\quad(\text{since}\ \beta_{a}(w_{b})\leq 0\ \text{if}\ a\neq b)$
		$\displaystyle=2\left(\beta_{j}(w_{j}^{\prime})-\sum_{l\in S_{i}}c_{l}\beta_{l}(w_{j}^{\prime})\right)$
		$\displaystyle=2\left(2-\sum_{l\in S_{i}}c_{l}\beta_{l}(w_{j}^{\prime})\right)$
		$\displaystyle\geq 4.$

Therefore, $(\beta,w^{\prime})$ satisfies (V5).

In conclusion, the neighborhoods $U_{\alpha},W_{v}$ and the functions $w_{i}^{\prime}:U_{\alpha}\times W_{v}\to V$ meet all the required conditions. ∎

Lemma 6.6.

([15], Proposition 14) Let $W$ be a finite dimensional real vector space, and let $\mu_{1},\cdots,\mu_{s}\in W^{*}\setminus\{0\}$ be linear functionals such that the cone

K:=\{w\in W\ |\ \mu_{j}(w)\geq 0\ \text{for}\ j=1,\cdots,f\}

has nonempty interior and no inequality $\mu_{j}\geq 0$ is implied by any combinations of the other inequalities $\mu_{k}\geq 0$ with $k\neq j$ . Let $K_{1},\cdots,K_{f}\subset K$ be the facets of $K$ supported by the functionals $\mu_{1},\cdots,\mu_{f}$ respectively. Let $i\neq j\in\{1,\cdots,f\}$ be two indices such that the facets $K_{i},K_{j}$ are not adjacent. Let

(6.4)

\displaystyle S_{i}:=\{l\in\{1,\cdots,f\}\ |\ K_{l}\ \text{is adjacent to}\ K_{i}\}.

Then there are $c_{l}\geq 0$ for $l\in S_{i}$ and $c_{i}>0$ such that

\mu_{j}=-c_{i}\mu_{i}+\sum_{l\in S_{i}}c_{l}\mu_{l}.

For each $(\alpha,v)\in\widetilde{\mathcal{D}}(\mathcal{G})$ , let $w^{\prime}:U_{\alpha}\times W_{v}\to V$ be the map given by Lemma 6.5. Then we define a map $\Phi_{\alpha,v}:U_{\alpha}\times W_{v}\to\widetilde{\mathcal{D}}(\mathcal{G})$ by $\Phi_{\alpha,v}(\beta,w):=(\beta,w^{\prime})$ . We refer to such maps as standard charts around the point $(\alpha,v)$ .

Lemma 6.7.

Let $(\alpha,v)\in\widetilde{\mathcal{D}}(\mathcal{G})$ and let $U_{\alpha}\subset\widetilde{\mathcal{E}}(\mathcal{G}),W_{v}\subset\rho^{-1}(\alpha)$ be open neighborhoods of $\alpha,v$ as in Lemma 6.5. Then the standard chart $\Phi_{\alpha,v}:U_{\alpha}\times W_{v}\to\widetilde{\mathcal{D}}(\mathcal{G})$ is a topological embedding onto an open subset of $\widetilde{\mathcal{D}}(\mathcal{G})$ .

Proof.

The map $\Phi_{\alpha,v}$ is continuous by Lemma 6.5. It remains to show that $\Phi_{\alpha,v}$ is injective and open.

We first show that $\Phi_{\alpha,v}$ is injective. Let $(\beta,w),(\gamma,x)\in U_{\alpha}\times W_{v}$ and suppose that $\Phi_{\alpha,v}(\beta,w)=\Phi_{\alpha,v}(\gamma,x)$ . From the definition, it follows immediately that $\beta=\gamma$ .

Write $(\beta,w^{\prime})=\Phi_{\alpha,v}(\beta,w)$ and $(\gamma,x^{\prime})=\Phi_{\alpha,v}(\gamma,x)$ . The construction in Lemma 6.5 includes the condition

\displaystyle\beta_{j}(w_{1}^{\prime})=\alpha_{j}(w_{1})\quad\text{for}\ j\in\{k_{1,a_{1}+b_{1}+1},\cdots,k_{1,3}\}.

(Note that $b_{1}=0$ .)

On the other hand, since $w\in W_{v}\subset\rho^{-1}(\alpha)$ , we have $(\alpha,w)\in\widetilde{\mathcal{D}}(\mathcal{G})$ . In particular, $(\alpha,w)$ satisfies conditions (V1) and (V3). These conditions imply that

	$\displaystyle\alpha_{1}(w_{1})$	$\displaystyle=2,$
	$\displaystyle\alpha_{j}(w_{1})$	$\displaystyle=0\quad\text{for}\ j\in\{k_{1,1},\cdots,k_{1,a_{1}}\}.$

To summarize, the vector $w_{1}\in V$ is the unique solution of the following system of four linear equations:

$\displaystyle\alpha_{1}(w_{1})$	$\displaystyle=2,$
$\displaystyle\alpha_{j}(w_{1})$	$\displaystyle=0\quad$	for	$\displaystyle j\in\{k_{1,1},\cdots,k_{1,a_{1}}\},$
$\displaystyle\alpha_{j}(w_{1})$	$\displaystyle=\beta_{j}(w_{1}^{\prime})\quad$	for	$\displaystyle j\in\{k_{1,a_{1}+1},\cdots,k_{1,3}\}.$

Since $\beta=\gamma$ and $w^{\prime}=x^{\prime}$ , we see that $x_{1}\in V$ is also a solution of this same system of equations. Therefore, we conclude that $w_{1}=x_{1}$ .

Next, we prove that $w_{2}=x_{2}$ . The construction of $w_{2}^{\prime}$ in Lemma 6.5 includes the condition

\beta_{j}(w_{2}^{\prime})=\alpha_{j}(w_{2})\quad\text{for}\ j\in\{k_{2,a_{2}+b_{2}+1},\cdots,k_{2,3}\}.

On the other hand, since $(\alpha,w)\in\widetilde{\mathcal{D}}(\mathcal{G})$ , it satisfies conditions (V1), (V3), and (V4). These conditions imply that

$\displaystyle\alpha_{2}(w_{2})$	$\displaystyle=2,$
$\displaystyle\alpha_{j}(w_{2})$	$\displaystyle=0\quad$	for	$\displaystyle j\in\{k_{2,1},\cdots,k_{2,a_{2}}\},$
$\displaystyle\alpha_{j}(w_{2})$	$\displaystyle=\frac{4\cos^{2}\left(\frac{\pi}{m_{i,j}}\right)}{\alpha_{2}(w_{j})}\quad$	for	$\displaystyle j\in\{k_{2,a_{2}+1},\cdots,k_{2,a_{2}+b_{2}}\}.$

By combining these systems, we see that $w_{2}$ is the unique solution of the following system:

$\displaystyle\alpha_{2}(w_{2})$	$\displaystyle=2,$
$\displaystyle\alpha_{j}(w_{2})$	$\displaystyle=0\quad$	for	$\displaystyle j\in\{k_{2,1},\cdots,k_{2,a_{2}}\},$
$\displaystyle\alpha_{j}(w_{2})$	$\displaystyle=\frac{4\cos^{2}\left(\frac{\pi}{m_{i,j}}\right)}{\alpha_{2}(w_{j})}\quad$	for	$\displaystyle j\in\{k_{2,a_{2}+1},\cdots,k_{2,a_{2}+b_{2}}\},$
$\displaystyle\alpha_{j}(w_{2})$	$\displaystyle=\beta_{j}(w_{2}^{\prime})\quad$	for	$\displaystyle j\in\{k_{2,a_{2}+b_{2}+1},\cdots,k_{2,3}\}.$

Since $\beta=\gamma$ , $w^{\prime}=x^{\prime}$ , and $w_{1}=x_{1}$ , we conclude that $x_{2}\in V$ is also a solution of this same system of equations. Therefore, we obtain $w_{2}=x_{2}$ .

The equalities $w_{i}=x_{i}$ for $i\geq 3$ can be proved in the same manner. This completes the proof of the injectivity of $\Phi_{\alpha,v}$ .

Now we prove the openness of $\Phi_{\alpha,v}$ . Let $(\beta,w)\in U_{\alpha}\times W_{v}$ and write $(\beta,w^{\prime}):=\Phi_{\alpha,v}(\beta,w)$ . To prove openness, it suffices to construct a continuous local inverse

F:\widetilde{\mathcal{D}}(\mathcal{G})\cap(S_{\beta}\times T_{w^{\prime}})\to U_{\alpha}\times W_{v}

of $\Phi_{\alpha,v}$ for some open neighborhoods $S_{\beta}\subset(V^{*})^{f}$ and $T_{w^{\prime}}\subset V^{f}$ of $\beta$ and $w^{\prime}$ , respectively, such that $\Phi_{\alpha,v}\circ F=\text{id}_{\widetilde{\mathcal{D}}(\mathcal{G})\cap(S_{\beta}\times T_{w^{\prime}})}$ .

Let $S_{\beta}$ and $T_{w^{\prime}}$ be arbitrary open neighborhoods of $\beta$ and $w^{\prime}$ in $(V^{*})^{f}$ and $V^{f}$ , respectively. We choose $S_{\beta}$ small enough so that $S_{\beta}\cap\widetilde{\mathcal{E}}(\mathcal{G})\subset U_{\alpha}$ . Let $(\gamma,x^{\prime})\in\widetilde{\mathcal{D}}(\mathcal{G})\cap(S_{\beta}\times T_{w^{\prime}})$ . We now construct $X_{1}(\gamma,x^{\prime}),\cdots,X_{f}(\gamma,x^{\prime})\in V$ so that $(\gamma,X(\gamma,x^{\prime}))$ with $X(\gamma,x^{\prime})=(X_{1}(\gamma,x^{\prime}),\cdots,X_{f}(\gamma,x^{\prime}))$ belongs to $U_{\alpha}\times W_{v}$ in a manner analogous to the construction of $\Phi_{\alpha,v}$ . Since we require $X(\gamma,x^{\prime})\in W_{v}$ , the element $X(\gamma,x^{\prime})$ must satisfy $X(\gamma,x^{\prime})\in\rho^{-1}(\alpha)$ , i.e., $(\alpha,X(\gamma,x^{\prime}))$ must satisfy conditions (V1) through (V6).

We define $X_{1}(\gamma,x^{\prime})\in V$ to be the unique solution of the system

$\displaystyle\alpha_{1}(X_{1}(\gamma,x^{\prime}))$	$\displaystyle=2,$
$\displaystyle\alpha_{j}(X_{1}(\gamma,x^{\prime}))$	$\displaystyle=0\quad$	for	$\displaystyle j\in\{k_{1,1},\cdots,k_{1,a_{1}}\},$
$\displaystyle\alpha_{j}(X_{1}(\gamma,x^{\prime}))$	$\displaystyle=\gamma_{j}(x_{1}^{\prime})\quad$	for	$\displaystyle j\in\{k_{1,a_{1}+1},\cdots,k_{1,3}\}.$

The inequalities $\alpha_{j}(X_{1}(\gamma,x^{\prime}))=\gamma_{j}(x_{1}^{\prime})<0$ for $j\in\{k_{1,a_{1}+1},\cdots,k_{1,3}\}$ are automatically satisfied since $(\gamma,x^{\prime})\in\widetilde{\mathcal{D}}(\mathcal{G})$ . Thus, the subsequent equations to define $X_{2}(\gamma,x^{\prime}),\cdots,X_{f}(\gamma,x^{\prime})$ are well-defined.

Let $X_{2}(\gamma,x^{\prime}),\cdots,X_{f}(\gamma,x^{\prime})$ be the vectors obtained by applying the same process. Up to this point, there is no need to shrink the neighborhoods $S_{\beta}$ and $T_{w^{\prime}}$ .

To ensure that $\alpha_{i}(X_{j}(\gamma,x^{\prime}))<0$ for all $i,j$ with $i\neq j$ and $m_{i,j}\neq 2$ , we select smaller neighborhoods $S_{\beta}$ and $T_{w^{\prime}}$ of $\beta$ and $w^{\prime}$ , respectively. Since $X(\gamma,x^{\prime})$ depends continuously on $(\gamma,x^{\prime})\in S_{\beta}\times T_{w^{\prime}}$ and $X(\beta,w^{\prime})=(\beta,w)$ by the injectivity of $\Phi_{\alpha,v}$ , the constructions of $X_{1}(\gamma,x^{\prime}),\cdots,X_{f}(\gamma,x^{\prime})$ and $\Phi_{\alpha,v}$ imply that $\Phi_{\alpha,v}\circ F$ is the identity on $\widetilde{\mathcal{D}}(\mathcal{G})\cap(S_{\beta}\times T_{w^{\prime}})$ .

By the same reasoning as in the proof of Lemma 6.5, the element $(\alpha,X(\gamma,x^{\prime}))$ satisfies conditions (V1) through (V6). Using the continuity of $X(\gamma,x^{\prime})$ , we can further refine the neighborhoods $S_{\beta}$ and $T_{w^{\prime}}$ so that the image of $F$ lies within $U_{\alpha}\times W_{v}$ .

Thus, the map $F:\widetilde{\mathcal{D}}(\mathcal{G})\cap(S_{\beta}\times T_{w^{\prime}})\to U_{\alpha}\times W_{v}$ is a continuous local inverse of $\Phi_{\alpha,v}$ . This completes the proof of the openness of $\Phi_{\alpha,v}$ . ∎

The images of the standard charts $\Phi_{\alpha,v}$ for $(\alpha,v)\in\widetilde{\mathcal{D}}(\mathcal{G})$ cover the “lifted” deformation space $\widetilde{\mathcal{D}}(\mathcal{G})$ . We next prove that their transition maps are smooth.

Lemma 6.8.

Let $\Phi^{(l)}:U_{\alpha}^{(l)}\times W_{v}^{(l)}\to\widetilde{\mathcal{D}}(\mathcal{G})$ , $l=1,2$ , be two standard charts around the points $(\alpha^{(l)},v^{(l)})\in\widetilde{\mathcal{D}}(\mathcal{G})$ , for $l=1,2$ . Suppose that the intersection

C:=\Phi^{(1)}\left(U_{\alpha}^{(1)}\times W_{v}^{(1)}\right)\cap\Phi^{(2)}\left(U_{\alpha}^{(2)}\times W_{v}^{(2)}\right)

is nonempty. Then the map

(\Phi^{(2)})^{-1}\circ\Phi^{(1)}:(\Phi^{(1)})^{-1}(C)\to(\Phi^{(2)})^{-1}(C)

is smooth.

In particular, the standard charts of $\widetilde{\mathcal{D}}(\mathcal{G})$ define a smooth structure on $\widetilde{\mathcal{D}}(\mathcal{G})$ . The dimension of $\widetilde{\mathcal{D}}(\mathcal{G})$ is $4f-e_{2}+6$ .

Proof.

Let $(\beta^{(1)},w^{(1)})\in(\Phi^{(1)})^{-1}(C)$ and let $(\beta^{(2)},w^{(2)}):=(\Phi^{(2)})^{-1}\circ\Phi^{(1)}(\beta^{(1)},w^{(1)})$ . Define $(\beta,w^{\prime}):=\Phi^{(1)}(\beta^{(1)},w^{(1)})=\Phi^{(2)}(\beta^{(2)},w^{(2)})$ .

From the definition of standard charts, it follows immediately that $\beta^{(1)}=\beta=\beta^{(2)}$ . Furthermore, the following relations hold due to the fact that $(\alpha^{(2)},w^{(2)})\in\widetilde{\mathcal{D}}(\mathcal{G})$ and from the construction of the standard charts:

$\displaystyle\alpha_{i}^{(2)}(w_{i}^{(2)})$	$\displaystyle=2,$
$\displaystyle\alpha_{j}^{(2)}(w_{i}^{(2)})$	$\displaystyle=0\quad$	for	$\displaystyle j\in\{k_{i,1},\cdots,k_{i,a_{i}}\},$
$\displaystyle\alpha_{j}^{(2)}(w_{i}^{(2)})$	$\displaystyle=\frac{4\cos^{2}\left(\frac{\pi}{m_{i,j}}\right)}{\alpha_{i}^{(2)}(w_{j}^{(2)})}\quad$	for	$\displaystyle j\in\{k_{i,a_{i}+1},\cdots,k_{i,a_{i}+b_{i}}\},$
$\displaystyle\alpha_{j}^{(2)}(w_{i}^{(2)})$	$\displaystyle=\beta_{j}^{(2)}(w_{i}^{\prime})=\alpha_{j}^{(1)}(w_{1}^{(1)})\quad$	for	$\displaystyle j\in\{k_{i,a_{i}+b_{i}+1},\cdots,k_{i,3}\}.$

In particular, $w^{(2)}$ can be expressed as a rational function of the entries of $w^{(1)}$ .

Since $\rho^{-1}(\alpha)$ is an embedded submanifold of $V^{f}$ , we conclude that the transition map is smooth. (See the remark at the end of Section 6.2.)

By Lemma 6.4 and Lemma 4.5, the dimension of $\widetilde{\mathcal{D}}(\mathcal{G})$ can be computed as

	$\displaystyle\dim\widetilde{\mathcal{D}}(\mathcal{G})$	$\displaystyle=\dim\widetilde{\mathcal{E}}(\mathcal{G})+\dim\rho^{-1}(\alpha)$
		$\displaystyle=\dim\widetilde{\mathcal{E}}(\mathcal{G})+\dim\mathcal{C}_{P}(\mathcal{G})$
		$\displaystyle=(e+f+6)+(3f-e-e_{2})$
		$\displaystyle=4f-e_{2}+6,$

where $\alpha\in\widetilde{\mathcal{E}}(\mathcal{G})$ is any element such that $\rho^{-1}(\alpha)$ is nonempty, and $P\subset\mathbb{S}^{3}$ is the convex 3-polytope determined by $\alpha$ . ∎

6.4. Proof of the main theorem

In this section, we finalize the proof of Theorem 1.2.

Recall that the group $\mathbb{G}=\text{SL}_{\pm}(V)\times\mathbb{R}_{+}^{f}$ acts smoothly on $\widetilde{\mathcal{D}}(\mathcal{G})$ by

	$\displaystyle(A,c_{1},\cdots,c_{f})\cdot(\alpha_{1},\cdots,\alpha_{f},v_{1},\cdots,v_{f})$
	$\displaystyle\quad=(c_{1}^{-1}\alpha_{1}\circ A^{-1},\cdots,c_{f}^{-1}\alpha_{f}\circ A^{-1},c_{1}Av_{1},\cdots,c_{f}Av_{f}).$

Lemma 6.9.

Suppose that the underlying combinatorial polytope of $\mathcal{G}$ is not a cone over a polygon. Then the action of $\mathbb{G}$ on $\widetilde{\mathcal{D}}(\mathcal{G})$ is proper and free.

Proof.

Since the group $\mathbb{G}$ acts freely on $\widetilde{\mathcal{E}}(\mathcal{G})$ , it also acts freely on $\widetilde{\mathcal{D}}(\mathcal{G})$ .

We prove the properness of the action of $\mathbb{G}$ on $\widetilde{\mathcal{D}}(\mathcal{G})$ . Since the map $\widetilde{\mathcal{E}}(\mathcal{G})\to\widetilde{\mathcal{E}}(\mathcal{G})/\mathbb{G}$ is a principal bundle and both $\widetilde{\mathcal{E}}(\mathcal{G})$ and $\mathcal{E}(\mathcal{G})$ are smooth manifolds (Lemma 6.4), the action of $\mathbb{G}$ on $\widetilde{\mathcal{E}}(\mathcal{G})$ is proper.

To prove that the action of $\mathbb{G}$ on $\widetilde{\mathcal{D}}(\mathcal{G})$ is proper, let $K$ be a compact subset of $\widetilde{\mathcal{D}}(\mathcal{G})\times\widetilde{\mathcal{D}}(\mathcal{G})$ . Let $\{g_{n}\}_{n\in\mathbb{N}}$ be a sequence in $\mathbb{G}$ , $\{(\alpha^{(n)},v^{(n)})\}_{n\in\mathbb{N}}$ a sequence in $\widetilde{\mathcal{D}}(\mathcal{G})$ , and suppose $(g_{n}\cdot(\alpha^{(n)},v^{(n)}),(\alpha^{(n)},v^{(n)}))\in K$ . We need to find a convergent subsequence of $\{g_{n}\}_{n\in\mathbb{N}}$ . The sequences $g_{n}\cdot\alpha^{(n)}$ and $\alpha^{(n)}\in\widetilde{\mathcal{E}}(\mathcal{G})$ lie in the compact subset $\rho(K)\subset\widetilde{\mathcal{E}}(\mathcal{G})$ . Since the action of $\mathbb{G}$ on $\widetilde{\mathcal{E}}(\mathcal{G})$ is proper, we conclude that $\{g_{n}\}_{n\in\mathbb{N}}$ has a convergent subsequence.

∎

We conclude the proof of our main theorem.

Theorem 1.2. Let $\mathcal{G}$ be a labeled combinatorial 3-polytope. Let $f$ and $e_{2}$ be the numbers of facets and the edges of order 2 of $\mathcal{G}$ . Suppose that $\mathcal{G}$ is orderable and of normal type. Suppose further that the stabilizer subgroup of $\textup{SL}_{\pm}(4,\mathbb{R})$ fixing any polyhedron $P\subset\mathbb{S}^{3}$ representing $\mathcal{G}$ is trivial. Then $\mathcal{C}(\mathcal{G})$ is a smooth manifold of dimension $3f-e_{2}-9$ .

Proof.

By the embedding $\Phi$ in Lemma 2.4 and Section 6.2, the space $\mathcal{C}(\mathcal{G})$ is homeomorphic to $\mathcal{D}(\mathcal{G})=\widetilde{\mathcal{D}}(\mathcal{G})/\mathbb{G}$ . By Lemma 6.9, the space $\widetilde{\mathcal{D}}(\mathcal{G})/\mathbb{G}$ admits a smooth structure such that the quotient map $\widetilde{\mathcal{D}}(\mathcal{G})\to\widetilde{\mathcal{D}}(\mathcal{G})/\mathbb{G}$ is a smooth submersion.

The dimension of the smooth manifold $\widetilde{\mathcal{D}}(\mathcal{G})$ is computed in Lemma 6.8. It follows that $\mathcal{C}(\mathcal{G})$ admits a smooth structure, and its dimension is

\dim\mathcal{C}(\mathcal{G})=\dim\widetilde{\mathcal{D}}(\mathcal{G})-\dim\mathbb{G}=(4f-e_{2}+6)-(15+f)=3f-e_{2}-9.

∎

7. Examples

The deformation space $\mathcal{C}(\mathcal{G})$ is naturally identified with a subspace of $((V^{*})^{f}\times V^{f})/\mathbb{G}$ . In the preceding section, we introduced the map $\overline{\rho}:\mathcal{C}(\mathcal{G})\to\mathcal{RS}(\mathcal{G})$ , which associates each Coxeter 3-polytope $[(\alpha,v)]=[(\alpha_{1},\dots,\alpha_{f},v_{1},\dots,v_{f})]$ with the projective equivalence class $[\alpha]=[(\alpha_{1},\dots,\alpha_{f})]$ of polytopes. Each non-empty fiber of $\overline{\rho}$ corresponds to a restricted deformation space, and it is a smooth manifold in the settings of Theorem 1.2. Furthermore, since the map $\overline{\rho}$ is induced by the projection $(V^{*})^{f}\times V^{f}\to(V^{*})^{f}$ , it is an open map onto an open submanifold of the realization space $\mathcal{RS}(\mathcal{G})$ .

In this section, we examine two examples of labeled combinatorial 3-polytopes $\mathcal{G}$ to illustrate a procedure for computing $\mathcal{C}(\mathcal{G})$ . We explicitly compute the restricted deformation spaces, corresponding to the non-empty fibers of the map $\overline{\rho}:\mathcal{C}(\mathcal{G})\to\mathcal{RS}(\mathcal{G})$ , and describe the global deformation space $\mathcal{C}(\mathcal{G})$

In Example 7.1, we present a case where the deformation space $\mathcal{C}(\mathcal{G})$ is topologically a cell, outlining a method to compute both the realization space $\mathcal{RS}(\mathcal{G})$ and the restricted deformation spaces in detail. In Example 7.2, we apply the same procedure and observe that, while the restricted deformation spaces are smooth manifolds of the same dimension (cf. Theorem 4.5), some of them may not be homeomorphic to each others. In particular, the projection map $\overline{\rho}$ is generally not a fiber bundle. In both examples, we see that the map $\overline{\rho}$ is not surjective. This implies that, given two combinatorially equivalent convex 3-polytopes in $\mathbb{S}^{3}$ , one polytope may admit reflections making it a Coxeter polytope, while the other may not admit such reflections.

Figure 3. An labeled combinatorial 3-polytope

\mathcal{G}

where

\mathcal{C}(\mathcal{G})

is 2-dimensional. (Example 7.1)

Example 7.1.

Let $\mathcal{G}$ be the labeled combinatorial 3-polytope given in Figure 3. In the notations of Theorem 1.2, we have $f=6$ , $e=10$ , and $e_{2}=7$ . Thus, the deformation space $\mathcal{C}(\mathcal{G})$ is a smooth manifold of dimension $3f-e_{2}-9=2$ by Theorem 1.2 and the realization space $\mathcal{RS}(\mathcal{G})\cong\mathcal{E}(\mathcal{G})$ is a smooth manifold of dimension $e-9=1$ by 5.4. Furthermore, the restricted deformation spaces, which can be identified with the nonempty fibers of the map $\mathcal{C}(\mathcal{G})\to\mathcal{RS}(\mathcal{G})$ , are smooth manifolds of dimension $3f-e-e_{2}-k(\mathcal{G})=1$ .

For this example, we first describe the realization space $\mathcal{E}(\mathcal{G})$ , then compute the restricted deformation spaces explicitly, and finally determine the global structure of the deformation space $\mathcal{C}(\mathcal{G})$ .

- The description of the realization space $\mathcal{E}(\mathcal{G})$ .

We first describe the realization space $\mathcal{E}(\mathcal{G})$ explicitly. Here is a brief outline of the computations. For each $\alpha=(\alpha_{1},\cdots,\alpha_{6})\in\widetilde{\mathcal{E}}(\mathcal{G})$ (see Section 6.1 for the notation), we find a “standard” representative $\beta=(\beta_{1},\cdots,\beta_{6})$ of the $\mathbb{G}$ -orbit $[\alpha]\in\mathcal{E}(\mathcal{G})$ of $\alpha$ such that the first five linear functionals $\beta_{1},\cdots,\beta_{5}$ are in a specified form. It can then be easily checked that such a representative is unique in each $\mathbb{G}$ -orbit. Next, we determine the last linear functional $\beta_{6}$ such that the resulting convex 3-polytope, defined by the inequalities $\beta_{i}\geq 0$ , is combinatorially equivalent to $\mathcal{G}$ .

Let $\alpha=(\alpha_{1},\cdots,\alpha_{6})\in\widetilde{\mathcal{E}}(\mathcal{G})$ be an arbitrary element. We note that the first five linear inequalities $\alpha_{1}\geq 0,\cdots,\alpha_{5}\geq 0$ determine a convex 3-polytope, which is a triangular prism. Since any triangular prism has 9 edges, its realization space is 0-dimensional by Theorem 5.4. Indeed, it can be verified that the realization space consists of a single element, defined by the inequalities $\beta_{1}\geq 0,\cdots,\beta_{5}\geq 0$ where

	$\displaystyle\beta_{i}$	$\displaystyle=e_{i}^{*}\quad\text{for}\quad i=1,2,3,4,$
	$\displaystyle\beta_{5}$	$\displaystyle=-e_{1}^{}+e_{2}^{}+e_{3}^{}+e_{4}^{},$

and $\{e_{1}^{*},\cdots,e_{4}^{*}\}$ is the standard basis of the dual vector space $V^{*}=(\mathbb{R}^{4})^{*}$ . Here we have chosen the indices $1,\cdots,5$ so that the facets supported by $\beta_{1},\cdots,\beta_{5}$ match the adjacency relations shown in Figure 3.

We next find $\beta_{6}\in V^{*}$ such that the six inequalities $\beta_{i}\geq 0$ , $i=1,\cdots,6$ , determine a convex polytope combinatorially equivalent to $\mathcal{G}$ . Let $T\subset\mathbb{S}^{3}$ be the triangular prism determined by the first five inequalities $\beta_{1}\geq 0,\cdots,\beta_{5}\geq 0$ , and let $v_{124},v_{235}\in\mathbb{S}^{3}$ be the vertices of $T$ contained in the facets $T_{1},T_{2},T_{4}$ and $T_{2},T_{3},T_{5}$ , respectively.

Then, the sixth supporting hyperplane $\mathbb{S}\{\beta_{6}=0\}\subset\mathbb{S}^{3}$ must pass through the vertices $v_{124}$ and $v_{235}$ . Using the explicit forms of the first five functionals $\beta_{1},\cdots,\beta_{5}$ , we can obtain

v_{124}=[(0,0,1,0)],\quad v_{235}=[(1,0,0,1)].

Substituting these vertices into $\beta_{6}$ , we find that $\beta_{6}$ has the form

(7.1)

\displaystyle\beta_{6}=d_{1}e_{1}^{*}+d_{2}e_{2}^{*}-d_{1}e_{4}^{*}

for some $d_{1},d_{2}\in\mathbb{R}$ .

To obtain possible values of $d_{1}$ , we observe that the vertex $v_{123}$ of $T$ (defined analogously to $v_{235}$ and $v_{124}$ ) must lie inside the half-space $\mathbb{S}\{\beta_{6}>0\}$ . The vertex $v_{123}$ must satisfy $\beta_{1}=\beta_{2}=\beta_{3}=9$ and $\beta_{4}>0$ , so we have $v_{123}=[(0,0,0,1)]$ . Substituting $v_{123}$ in the relation (7.1) we thus obtain $d_{1}<0$ . Since multiplying $\beta_{6}$ by a positive scalar leaves the hyperplane $\mathbb{S}\{\beta_{6}=0\}$ invariant, we may assume

\beta_{6}=-e_{1}^{*}+de_{2}^{*}+e_{4}^{*}

for some $d\in\mathbb{R}$ .

We find the range of $d$ such that the inequalities $\beta_{1}\geq 0,\cdots,\beta_{6}\geq 0$ determine a convex 3-polytope combinatorially equivalent to $\mathcal{G}$ . Let $E_{45}$ denote the edge $T_{4}\cap T_{5}$ of the triangular prism $T$ . The inequalities $\beta_{1}\geq 0,\cdots,\beta_{6}\geq 0$ determine a convex 3-polytope combinatorially equivalent to $\mathcal{G}$ if and only if the hyperplane $\mathbb{S}\{\beta_{6}=0\}$ intersects the interior of the edge $E_{45}$ transversally. This condition holds if and only if the three vertices $v_{134}$ , $v_{123}$ , and $v_{345}$ lie in the open half-space $\mathbb{S}\{\beta_{6}>0\}$ , while the vertex $v_{245}$ lies in the opposite half-space $\mathbb{S}\{\beta_{6}<0\}$ . Using the explicit forms of the first five functionals $\beta_{1},\cdots,\beta_{5}$ , we obtain $v_{134}=[(0,1,0,0)]$ , $v_{123}=[(0,0,0,1)]$ , $v_{345}=[(1,1,0,0)]$ and $v_{245}=[(1,0,1,0)]$ . Substituting these vertices, we find that $d$ determines an element of $\widetilde{\mathcal{E}}(\mathcal{G})$ if and only if $d\in(1,\infty)$ . By Lemma 6.3, we see that different values of $d$ give projectively non-equivalent elements of $\widetilde{\mathcal{E}}(\mathcal{G})$ . Thus, we obtain an infinite path $(1,\infty)\to\widetilde{\mathcal{E}}(\mathcal{G})$ given by $d\mapsto(\beta_{1},\cdots,\beta_{5},\beta_{6}^{d})$ , where

	$\displaystyle\beta_{i}$	$\displaystyle=e_{i}^{*}\quad\text{for}\quad i=1,2,3,4,$
	$\displaystyle\beta_{5}$	$\displaystyle=-e_{1}^{}+e_{2}^{}+e_{3}^{}+e_{4}^{},$
	$\displaystyle\beta_{6}^{d}$	$\displaystyle=-e_{1}^{}+de_{2}^{}+e_{4}^{*},$

and this path descends to a homeomorphism $(1,\infty)\cong\mathcal{E}(\mathcal{G})$ .

- The computation of the restricted deformation spaces.

We compute the restricted deformation spaces for the same labeled combinatorial 3-polytope $\mathcal{G}$ . Recall from Section 6.2 that the restricted deformation spaces in $\mathcal{C}(\mathcal{G})$ can be identified with the nonempty fibers of the map $\mathcal{C}(\mathcal{G})\to\mathcal{RS}(\mathcal{G})$ . In the commuting diagram of Figure 2, we see that they can also be identified with the nonempty fibers of the map $\overline{\rho}:\mathcal{D}(\mathcal{G})\to\mathcal{E}(\mathcal{G})$ . Moreover, since $\eta:\widetilde{\mathcal{D}}(\mathcal{G})\to\mathcal{D}(\mathcal{G})$ maps each nonempty fiber of $\rho:\widetilde{\mathcal{D}}(\mathcal{G})\to\widetilde{\mathcal{E}}(\mathcal{G})$ homeomorphically onto a nonempty fiber of $\overline{\rho}$ , it suffices to compute the nonempty fibers of $\rho$ .

Here is the outline of the computation. Let $d\in(1,\infty)$ , and let $\beta:=(\beta_{1},\cdots,\beta_{5},\beta_{6}^{d})\in\widetilde{\mathcal{E}}(\mathcal{G})$ as described in the previous computation. An element $w=(w_{1},\cdots,w_{6})\in V^{6}$ lies in $\rho^{-1}(\beta)$ if and only if $(\beta,w)$ satisfies the conditions (V1) through (V6). The condition (V6) is already given since $\beta\in\widetilde{\mathcal{E}}(\mathcal{G})$ . We first find such $w_{1},\cdots,w_{6}$ successively by solving the linear equations arising from the conditions (V1), (V3), and (V4). Then we sort out the solutions $w_{1},\cdots,w_{6}$ satisfying the condition (V2). Checking (V5) is unnecessary, as it is redundant with (V1) through (V4) by Lemma 6.6 (this redundancy is explained at the end of the proof of Lemma 6.5). In the process, we will also determine all values of $d$ for which the fiber $\rho^{-1}(\beta)$ is nonempty.

We first find the vector $w_{1}$ . By conditions (V1) and (V3), it is uniquely determined by the equalities

	$\displaystyle\beta_{1}(w_{1})$	$\displaystyle=2,$
	$\displaystyle\beta_{i}(w_{1})$	$\displaystyle=0\quad\text{for}\quad i=2,3,4.$

Hence, we obtain $w_{1}=(2,0,0,0)$ .

Similarly, the equations $\beta_{2}(w_{2})=2$ , $\beta_{i}(w_{2})=0$ for $i=1,3$ , and $\beta_{6}^{d}(w_{2})=0$ yield $w_{2}=(0,2,0,-2d)$ .

The third vector $w_{3}$ is not uniquely determined. It is given by only the three equations $\beta_{3}(w_{3})=2$ and $\beta_{i}(w_{3})=0$ for $i=1,2$ . Thus, $w_{3}=(0,0,2,x)$ for some $x\in\mathbb{R}$ . The parameter $x$ will serve as the parameter for the 1-dimensional restricted deformation space, and we will determine the range of possible values for $x$ later.

For $w_{4}$ , we need an additional condition (V4), and the resulting system of equations is

	$\displaystyle\beta_{4}(w_{4})$	$\displaystyle=2,$
	$\displaystyle\beta_{i}(w_{4})$	$\displaystyle=0\quad\text{for}\quad i=1,6,$
	$\displaystyle\beta_{3}(w_{4})$	$\displaystyle=\frac{4\cos^{2}\left(\frac{\pi}{3}\right)}{\beta_{4}(w_{3})}=\frac{1}{x}.$

Thus, we obtain $w_{4}=\left(0,-\frac{2}{d},\frac{1}{x},2\right)$ , which depends uniquely on $d$ and $x$ .

Following the same approach, we find

w_{5}=\left(\frac{d\left(\frac{1}{x+1}-2\right)}{d-1}+\frac{1}{-\frac{2}{d}+\frac{1}{x}+2},\frac{\frac{1}{x+1}-2}{d-1},\frac{1}{x+2},\frac{1}{-\frac{2}{d}+\frac{1}{x}+2}\right),

and $w_{6}=(-2,0,-2,0)$ .

To summarize, we have obtained

	$\displaystyle w_{1}$	$\displaystyle=(2,0,0,0),\quad w_{2}=(0,2,0,-2d),\quad w_{3}=(0,0,2,x),\quad w_{4}=\left(0,-\frac{2}{d},\frac{1}{x},2\right),$
	$\displaystyle w_{5}$	$\displaystyle=\left(\frac{d\left(\frac{1}{x+1}-2\right)}{d-1}+\frac{1}{-\frac{2}{d}+\frac{1}{x}+2},\frac{\frac{1}{x+1}-2}{d-1},\frac{1}{x+2},\frac{1}{-\frac{2}{d}+\frac{1}{x}+2}\right),$
	$\displaystyle w_{6}$	$\displaystyle=(-2,0,-2,0).$

The realization space is parametrized by $d$ , and the restricted deformation space corresponding to the parameter $d$ is then parametrized by $x$ .

Next, we find all $d$ and $x$ such that $(\beta,w)$ satisfies condition (V2). Since the cases $\beta_{i}(w_{j})=0$ are already covered in the construction of $w_{1},\cdots,w_{6}$ , we only need to consider the pairs $i,j$ such that $m_{i,j}\geq 3$ . Furthermore, Lemma 6.6 implies that it suffices to consider pairs $i,j$ with $m_{i,j}\neq\infty$ , $i\neq j$ , and $m_{i,j}\neq 2$ . These pairs correspond to the edges of $\mathcal{G}$ with order $\geq 3$ .

It follows that the values of $d$ and $x$ satisfying condition (V2) must satisfy the inequalities

\beta_{3}(w_{4})<0,\quad\beta_{3}(w_{5})<0,\quad\beta_{4}(w_{5})<0.

These inequalities, combined with the condition $d\in(1,\infty)$ , yield

1<d<\frac{4}{3},\quad\frac{d}{2-2d}<x<-2.

We conclude that the restricted deformation space $\rho^{-1}(\beta)$ over the point $\beta=(\beta_{1},\cdots,\beta_{5},\beta_{6}^{d})\in\widetilde{\mathcal{E}}(\mathcal{G})$ is nonempty if and only if $1<d<\frac{4}{3}$ . Moreover, each nonempty fiber $\rho^{-1}(\beta)$ , parametrized by $x$ , is homeomorphic to an open interval.

From these observations, we conclude that the deformation space $\mathcal{C}(\mathcal{G})$ is homeomorphic to an open 2-cell $\mathbb{R}^{2}$ , which gives an additional information to the statement of Theorem 1.2.

Figure 4. An example where the map

\mathcal{C}(\mathcal{G})\to\mathcal{RS}(\mathcal{G})

neither is surjective nor is a bundle. (Example 7.2)

Example 7.2.

We consider the labeled combinatorial 3-polytope $\mathcal{G}$ given in Figure 4. The global topological structure of $\mathcal{C}(\mathcal{G})$ can be easily obtained without computing the restricted deformation spaces as in the previous example. This polytope is a truncation polytope, so we can apply Theorem 3.16 of [8] to conclude that the deformation space $\mathcal{C}(\mathcal{G})$ is homeomorphic to the space $\mathbb{R}^{4}\sqcup\mathbb{R}^{4}$ , the disjoint union of two copies of $\mathbb{R}^{4}$ .

In Example 7.1, the map $\mathcal{C}(\mathcal{G})\to\mathcal{RS}(\mathcal{G})$ is not surjective, but it is a fiber bundle onto its open image, parametrized by $d\in(1,\frac{4}{3})$ . In the current example, we will see that the map $\mathcal{C}(\mathcal{G})\to\mathcal{RS}(\mathcal{G})$ is not a fiber bundle over its image by observing that the number of connected components of the fibers over a single continuous path in $\mathcal{RS}(\mathcal{G})$ is not constant.

We will compute the realization space and some restricted deformation spaces, but will omit details of computations that are similar to those in Example 7.1.

In the computation of the space $\widetilde{\mathcal{E}}(\mathcal{G})$ , we apply the same argument as in Example 7.1. Specifically, we consider the unique representative $\beta=(\beta_{1},\cdots,\beta_{6})$ in each $\mathbb{G}$ -orbit such that the first five functionals determine the triangular prism $T$ as in Example 7.1. This prism is defined by the five inequalities $\beta_{1}\geq 0,\cdots,\beta_{5}\geq 0$ , where $\beta_{i}=e_{i}^{*}$ for $i=1,2,3,4$ (the standard basis elements of $V^{*}$ ), and $\beta_{5}=e_{1}^{*}+e_{2}^{*}+e_{3}^{*}-e_{4}^{*}$ .

Note that the positions of $\mathcal{G}_{1}$ and $\mathcal{G}_{4}$ are switched compared to Example 7.1. This adjustment ensures that the orderability conditions are satisfied.

Let $v_{125}$ be the vertex $T_{1}\cap T_{2}\cap T_{5}$ of $T$ . For the sixth element $\beta_{6}\in V^{*}$ , the set $\mathbb{S}(\{x\in V\ |\ \beta_{i}(x)\geq 0\ \text{for}\ i=1,\cdots,6\})$ forms a convex 3-polytope combinatorially equivalent to $\mathcal{G}$ if and only if the vertex $v_{125}$ lies in the open half-space $\mathbb{S}(\{\beta_{6}<0\})$ and the remaining five vertices of $T$ lie in the opposite half-space $\mathbb{S}(\{\beta_{6}>0\})$ .

These conditions, along with normalization by positive scalars, yield the parametrization of $\mathcal{E}(\mathcal{G})$ given by

	$\displaystyle\beta_{i}$	$\displaystyle=e_{i}^{*}\quad\text{for}\quad i=1,2,3,4,$
	$\displaystyle\beta_{5}$	$\displaystyle=e_{1}^{}+e_{2}^{}+e_{3}^{}-e_{4}^{},$
	$\displaystyle\beta_{6}$	$\displaystyle=d_{1}e_{1}^{}+d_{2}e_{2}^{}+d_{3}e_{3}^{}-e_{4}^{},$

with $(d_{1},d_{2},d_{3})\in(1,\infty)\times(1,\infty)\times(0,1)$ . The number of parameters also coincides with Theorem 5.4 since we have $e=12$ .

Next, we consider a specific path inside $\mathcal{E}(\mathcal{G})$ using the parametrization. Let $\gamma(s)\in\widetilde{\mathcal{E}}(\mathcal{G})$ , $s\in(1,\infty)$ be the (infinite) path given by

d_{1}=s,\quad d_{2}=2,\quad d_{3}=\frac{1}{2}

in the above parametrization.

We examine the fiber $\rho^{-1}(\gamma(s))$ over the point $\gamma(s)$ , i.e., the restricted deformation space associated with the polytope determined by $\gamma(s)$ . Following the same argument as in Example 7.1, we compute the vectors $w_{1},\cdots,w_{6}$ as

	$\displaystyle w_{1}$	$\displaystyle=\left(2,t,-4s-4t,0\right),$
	$\displaystyle w_{2}$	$\displaystyle=\left(\frac{1}{t},2,-\frac{2(s+4t)}{t},0\right),$
	$\displaystyle w_{3}$	$\displaystyle=\left(-\frac{1}{2(2s+2t)},-\frac{t}{s+4t},2,-\frac{\frac{s}{2}-4s^{2}+2t-18st-14t^{2}}{(2s+2t)(s+4t)}\right),$
	$\displaystyle w_{4}$	$\displaystyle=\left(0,0,\frac{(2s+2t)(s+4t)}{-\frac{s}{2}+4s^{2}-2t+18st+14t^{2}},2\right),$
	$\displaystyle w_{5}$	$\displaystyle=\left(-\frac{1}{2\left(-1+2s+\frac{3t}{2}\right)},\frac{t}{2\left(\frac{1}{2}-s-3t\right)},0,-\frac{\frac{3}{4}-\frac{7s}{2}+4s^{2}-\frac{13t}{2}+16st+\frac{39t^{2}}{4}}{\left(\frac{1}{2}-s-3t\right)\left(1-2s-\frac{3t}{2}\right)}\right),$
	$\displaystyle w_{6}$	$\displaystyle=\left(0,0,\frac{2\left(-1+\frac{9s}{2}-5s^{2}+\frac{29t}{4}-\frac{35st}{2}-12t^{2}\right)}{\frac{3}{4}-\frac{13s}{4}+\frac{7s^{2}}{2}-\frac{11t}{2}+\frac{25st}{2}+\frac{33t^{2}}{4}},\frac{2\left(-\frac{5}{4}+\frac{11s}{2}-6s^{2}+\frac{73t}{8}-\frac{85st}{4}-\frac{57t^{2}}{4}\right)}{\frac{3}{4}-\frac{13s}{4}+\frac{7s^{2}}{2}-\frac{11t}{2}+\frac{25st}{2}+\frac{33t^{2}}{4}}\right),$

where $t\in\mathbb{R}$ parametrizes the 1-dimensional restricted deformation space for each $s\in(1,\infty)$ .

To satisfy the conditions (V1) through (V6), the parameters $s$ and $t$ must meet the following inequalities:

	$\displaystyle\beta_{3}(w_{4})$	$\displaystyle<0,\quad\beta_{2}(w_{5})<0,\quad\beta_{1}(w_{5})<0,\quad\beta_{1}(w_{2})<0,$
	$\displaystyle\beta_{2}(w_{3})$	$\displaystyle<0,\quad\beta_{1}(w_{3})<0,\quad\beta_{5}(w_{6})<0.$

The complete solution to this system of inequalities is complicated. For our purposes, it suffices to note the following partial solutions:

	$\displaystyle t$	$\displaystyle\in\left(\frac{1}{3}(2-4s),\frac{1}{14}(1-9s)-\frac{1}{14}\sqrt{1-11s+25s^{2}}\right)$
		$\displaystyle\qquad\text{if}\ \frac{1}{176}\left(193+3\sqrt{697}\right)<s\leq\frac{4}{19}\left(5+\sqrt{6}\right),$
	$\displaystyle t$	$\displaystyle\in\left(\frac{1}{3}(2-4s),\frac{1}{14}\left(1-9s\right)-\frac{1}{14}\sqrt{1-11s+25s^{2}}\right)$
		$\displaystyle\qquad\cup\left(\frac{1}{33}\left(11-25s\right)+\frac{1}{33}\sqrt{22-121s+163s^{2}},-\frac{s}{4}\right)$
		$\displaystyle\qquad\text{if}\ \frac{4}{19}\left(5+\sqrt{6}\right)<s\leq 2,\ \cdots.$

In particular, the number of connected components of the restricted deformation spaces $\rho^{-1}(\gamma(s))$ changes from 1 to 2 as the parameter $s$ passes the value $s=\frac{4}{19}\left(5+\sqrt{6}\right)$ . Moreover, the system of inequalities has no solution if

1<s\leq\frac{1}{176}\left(193+3\sqrt{697}\right),

so the fibers $\rho^{-1}(\gamma(s))$ for such $s$ are empty. Thus, the map $\mathcal{C}(\mathcal{G})\to\mathcal{RS}(\mathcal{G})$ is not surjective, as in Example 7.1, but in this example, the map is not a fiber bundle either.

References

[1] Suhyoung Choi. The deformation spaces of projective structures on 3-dimensional Coxeter orbifolds. Geom. Dedicata, 119, 2006.
[2] Suhyoung Choi, Craig D. Hodgson, and Gye-Seon Lee. Projective deformations of hyperbolic Coxeter 3-orbifolds. Geom. Dedicata, 159:125–167, 2012.
[3] Suhyoung Choi and Gye-Seon Lee. Projective deformations of weakly orderable hyperbolic Coxeter orbifolds. Geom. Topol., 19(4):1777–1828, 2015.
[4] Suhyoung Choi, Gye-Seon Lee, and Ludovic Marquis. Deformation spaces of Coxeter truncation polytopes. J. Lond. Math. Soc. (2), 106(4):3822–3864, 2022.
[5] Jeffrey Danciger, François Guéritaud, Fanny Kassel, Gye-Seon Lee, and Ludovic Marquis. Convex cocompactness for coxeter groups. Journal of the European Mathematical Society, 2024. Published online first.
[6] Jean Paul Filpo Molina. On the realization space of convex polyhedra. Master’s thesis, Korea Advanced Institute of Science and Technology, Daejeon, South Korea, 2022.
[7] G. A. Margulis and È.B̃. Vinberg. Some linear groups virtually having a free quotient. J. Lie Theory, 10(1):171–180, 2000.
[8] Ludovic Marquis. Espace des modules de certains polyèdres projectifs miroirs. Geom. Dedicata, 147:47–86, 2010.
[9] Ludovic Marquis. Coxeter group in Hilbert geometry. Groups Geom. Dyn., 11(3):819–877, 2017.
[10] Seung-Yeol Park. The mathematica code for counting orderable labeled polytopes. https://sites.google.com/view/seungyeolpark/resources, 2024. Accessed on 2024-07-21.
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The smoothness of the real projective deformation spaces of orderable Coxeter 3-polytopes

Abstract.

Key words and phrases:

2020 Mathematics Subject Classification:

1. Introduction

Question 1.1.

Theorem 1.2.

1.1. Outline of the Paper

2. Deformation spaces of Coxeter polytopes

2.1. Labeled combinatorial polytopes

Definition 2.1.

2.2. The deformation space 𝒞​(𝒢)\mathcal{C}(\mathcal{G}) of Coxeter polytopes

Definition 2.2.

Definition 2.3.

Lemma 2.4.

Proof.

3. The theory of Vinberg

Theorem 3.1.

Theorem 3.2.

Remark 3.3.

Remark 3.4.

4. Restricted deformation spaces

Definition 4.1.

Definition 4.2.

Remark 4.3.

Proposition 4.4.

Proof.

Case 1: rank​(A)=4\text{rank}(A)=4.

Case 2: rank​(A)=3\text{rank}(A)=3.

Theorem 4.5.

Remark 4.6.

Remark 4.7.

5. Realization spaces

Definition 5.1.

Definition 5.2.

Definition 5.3.

Theorem 5.4.

Remark 5.5.

6. The proof of the main theorem

6.1. Reparametrizing the realization space

Lemma 6.1.

Proof.

Lemma 6.2.

Proof.

Lemma 6.3.

Proof.

Lemma 6.4.

Proof.

6.2. The projection 𝒞​(𝒢)→ℛ​𝒮​(𝒢)\mathcal{C}(\mathcal{G})\to\mathcal{RS}(\mathcal{G}) and the restricted deformation spaces

6.3. Construction of a smooth structure on 𝒟~​(𝒢)\widetilde{\mathcal{D}}(\mathcal{G})

Lemma 6.5.

Proof.

Lemma 6.6.

Lemma 6.7.

Proof.

Lemma 6.8.

Proof.

6.4. Proof of the main theorem

Lemma 6.9.

Proof.

Proof.

7. Examples

Example 7.1.

Example 7.2.

References

2.2. The deformation space $\mathcal{C}(\mathcal{G})$ of Coxeter polytopes

Case 1: $\text{rank}(A)=4$ .

Case 2: $\text{rank}(A)=3$ .

6.2. The projection $\mathcal{C}(\mathcal{G})\to\mathcal{RS}(\mathcal{G})$ and the restricted deformation spaces

6.3. Construction of a smooth structure on $\widetilde{\mathcal{D}}(\mathcal{G})$