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

Suhyoung Choi, Seungyeol Park

TL;DR

The paper investigates deformation spaces of real projective Coxeter 3-polytopes realizing a labeled combinatorial polytope $\mathcal{G}$. It introduces the notions of orderable and normal-type polytopes, and shows that under these conditions with trivial stabilizers, the global deformation space $\mathcal{C}(\mathcal{G})$ is a smooth manifold of dimension $3f - e_2 - 9$ by relating it to a real-analytic realization space $\mathcal{RS}(\mathcal{G})$ via a natural map whose fibers are restricted deformation spaces. The authors develop a detailed framework: embedding $\mathcal{C}(\mathcal{G})$ into a quotient of facet-data, reparametrizing realization spaces, and constructing a smooth atlas on an augmented space $\widetilde{\mathcal{D}}(\mathcal{G})$ to implement a clean quotient. They further specialize to Vinberg theory to control the reflection groups and provide explicit examples that illustrate both the manifold structure and nontrivial fiber behavior, including cases where the projection is not a fiber bundle or surjective. Overall, the work strengthens the understanding of moduli of real projective Coxeter orbifolds, connecting deformation theory with realization spaces and Vinberg’s linear reflection groups, and yielding precise dimension counts under combinatorial hypotheses. The results have implications for constructing new real projective orbifolds via truncation and gluing and for assessing the local-global structure of deformation spaces in 3-dimensional Coxeter settings.

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.

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

TL;DR

The paper investigates deformation spaces of real projective Coxeter 3-polytopes realizing a labeled combinatorial polytope . It introduces the notions of orderable and normal-type polytopes, and shows that under these conditions with trivial stabilizers, the global deformation space is a smooth manifold of dimension by relating it to a real-analytic realization space via a natural map whose fibers are restricted deformation spaces. The authors develop a detailed framework: embedding into a quotient of facet-data, reparametrizing realization spaces, and constructing a smooth atlas on an augmented space to implement a clean quotient. They further specialize to Vinberg theory to control the reflection groups and provide explicit examples that illustrate both the manifold structure and nontrivial fiber behavior, including cases where the projection is not a fiber bundle or surjective. Overall, the work strengthens the understanding of moduli of real projective Coxeter orbifolds, connecting deformation theory with realization spaces and Vinberg’s linear reflection groups, and yielding precise dimension counts under combinatorial hypotheses. The results have implications for constructing new real projective orbifolds via truncation and gluing and for assessing the local-global structure of deformation spaces in 3-dimensional Coxeter settings.

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 associated to such orbifolds, and study the deformation space of Coxeter polytopes realizing . We prove that if is orderable and of normal type then the deformation space of real projective Coxeter 3-polytopes realizing is a smooth manifold. This result is achieved by analyzing a natural map of into a smooth manifold called the realization space.

Paper Structure

This paper contains 16 sections, 16 theorems, 90 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Outline of the Paper
  3. Deformation spaces of Coxeter polytopes
  4. Labeled combinatorial polytopes
  5. The deformation space $\mathcal{C} (\mathcal{G})$ of Coxeter polytopes
  6. The theory of Vinberg
  7. Restricted deformation spaces
  8. Case 1: $\text{rank}(A) = 4$.
  9. Case 2: $\text{rank}(A) = 3$.
  10. Realization spaces
  11. The proof of the main theorem
  12. Reparametrizing the realization space
  13. The projection $\mathcal{C} (\mathcal{G}) \to \mathcal{RS} (\mathcal{G})$ and the restricted deformation spaces
  14. Construction of a smooth structure on $\widetilde{\mathcal{D}} (\mathcal{G})$
  15. Proof of the main theorem
  16. ...and 1 more sections

Key Result

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

Figures (4)

  • Figure 1: A truncation of a convex polytope $P$ by a hyperplane $\Pi \subset \mathbb{S}^3$.
  • Figure 2:
  • Figure 3: An labeled combinatorial 3-polytope $\mathcal{G}$ where $\mathcal{C} (\mathcal{G})$ is 2-dimensional. (Example \ref{['Example 7.1']})
  • Figure 4: An example where the map $\mathcal{C} (\mathcal{G}) \to \mathcal{RS} (\mathcal{G})$ neither is surjective nor is a bundle. (Example \ref{['counterexample']})

Theorems & Definitions (43)

  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.3
  • Remark 3.4
  • ...and 33 more