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

TL;DR

This paper introduces a geometric process where spherical 3-manifolds emerge as limits of superspaces associated with flat SU(2)-bundles over hyperbolic surfaces, linking topology, geometry, and moduli space optimization.

Contribution

It demonstrates that any homogeneous spherical 3-manifold can be realized as a boundary point in the Gromov-Hausdorff closure of a superspace linked to flat SU(2)-bundles over hyperbolic surfaces, connecting geometric limits with systolic invariants.

Findings

Spherical 3-manifolds appear as boundary points in superspace closures.

Convergence depends on the fundamental group order and hyperbolic surface invariants.

Arithmetic surfaces provide optimal error estimates in convergence bounds.

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 homogeneous 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 $(\Sigma,h_{\Sigma})$. 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 Poincar\'e homology sphere.