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Collapsing Flat ${\rm{SU}}(2)$-Bundles to Spherical 3-Manifolds

Eder M. Correa

Abstract

We present a geometric mechanism for the emergence of spherical $3$-manifolds from the superspace of Riemannian metrics associated with flat ${\rm{SU}}(2)$-bundles over closed orientable hyperbolic surfaces. Our main result shows that any spherical 3-manifold $(S,g_{S})$ can be realized as a boundary point in the Gromov-Hausdorff closure of a superspace $\mathcal{S}(P)$, where $P$ is a flat ${\rm{SU}}(2)$-bundle over a closed orientable hyperbolic surface $(Σ,h_Σ)$. We show that the convergence of the sequence of metric spaces towards the spherical limit is controlled by the order of the fundamental group of $S$ and the metric invariant of the hyperbolic base provided by the ratio between its area and its systole. In this framework, the problem of obtaining the sharpest upper bound error reduces to the classical problem of maximizing the systole function over the moduli space of hyperbolic Riemann surfaces. As a byproduct, we observe that certain arithmetic surfaces provide the best possible error estimates within this family. To illustrate these results, we show that, according to our mechanism, the Bolza surface yields the optimal error bound for the convergence toward the Poincaré homology sphere.

Collapsing Flat ${\rm{SU}}(2)$-Bundles to Spherical 3-Manifolds

Abstract

We present a geometric mechanism for the emergence of spherical -manifolds from the superspace of Riemannian metrics associated with flat -bundles over closed orientable hyperbolic surfaces. Our main result shows that any spherical 3-manifold can be realized as a boundary point in the Gromov-Hausdorff closure of a superspace , where is a flat -bundle over a closed orientable hyperbolic surface . We show that the convergence of the sequence of metric spaces towards the spherical limit is controlled by the order of the fundamental group of and the metric invariant of the hyperbolic base provided by the ratio between its area and its systole. In this framework, the problem of obtaining the sharpest upper bound error reduces to the classical problem of maximizing the systole function over the moduli space of hyperbolic Riemann surfaces. As a byproduct, we observe that certain arithmetic surfaces provide the best possible error estimates within this family. To illustrate these results, we show that, according to our mechanism, the Bolza surface yields the optimal error bound for the convergence toward the Poincaré homology sphere.
Paper Structure (8 sections, 19 theorems, 111 equations, 5 figures, 1 table)

This paper contains 8 sections, 19 theorems, 111 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Generalities on Hyperbolic Surfaces
  3. Flat Principal Bundles
  4. Spherical 3-manifolds and Flat ${\rm{SU}}(2)$-bundles
  5. Gromov-Hausdorff Space
  6. Superspace of Riemannian Metrics
  7. Proof of Main Result
  8. From Bolza Surface to Poincaré Homology Sphere

Key Result

Theorem A

Let $(S,g_{S})$ be an orientable spherical $3$-manifold. Then, there exist a closed orientable hyperbolic surface $\Sigma \in \mathcal{M}_{g}$ and $[P,\theta] \in \mathcal{M}_{{\text{flat}}}(\Sigma,{\rm{SU}}(2))$, such that where $\overline{\mathcal{S}(P)}^{{\rm{GH}}}$ denotes the Gromov-Hausdorff closure of the superspace $\mathcal{S}(P)$. Moreover, we have where $h_{\Sigma}$ is a hyperbolic me

Figures (5)

  • Figure 1: ADE type Dynkin diagrams.
  • Figure 2: The symmetry groups of the Platonic solids correspond, via the McKay correspondence, to the exceptional Dynkin diagrams $E_{6}$, $E_{7}$, and $E_{8}$ of the ADE classification dechant2018trinity.
  • Figure 3: The arrows above indicate a map which generates the group of deck transformations $\mathbbm{Z}_{5}$. In this case, we have a covering $\Sigma_{6} \to \Sigma_{2}$, such that $\Sigma_{2} = \Sigma_{6}/\mathbbm{Z}_{5}$.
  • Figure 4: The concatenation of $\beta$ with $\alpha$ provides a path in $P$ joining $a$ and $b$.
  • Figure 5: It is immediate from the presentation in Eq. (\ref{['affineBolza']}) that the branch points are $0, \pm 1, \pm i,\infty$.

Theorems & Definitions (57)

  • Theorem A
  • Corollary A
  • Corollary B
  • Theorem 2.1: Killing-Hopf
  • Theorem 2.2
  • Definition 2.3
  • Corollary 2.4
  • Remark 2.5
  • Theorem 2.6
  • Lemma 2.7
  • ...and 47 more