On the existence of universal links in three-manifolds

TL;DR

This paper investigates the existence of universal links in closed 3-manifolds, establishing that only spherical manifolds admit such links and exploring distinctions between universal and complement universal links.

Contribution

It proves that only spherical 3-manifolds admit universal links, distinguishes between universal and complement universal links, and characterizes manifolds with certain branched coverings.

Findings

Only spherical 3-manifolds admit universal links.

Infinite examples of complement universal links not being universal.

No closed aspherical 3-manifold covers all aspherical 3-manifolds.

Abstract

We study the existence of branched coverings between closed $3$-manifolds, with emphasis on universal knots and links. We prove that the only closed $3$-manifolds that admit a universal link are spherical. Furthermore, we distinguish between universal links and complement universal links and show that these notions do not coincide in general, by exhibiting infinitely many examples of complement universal links that are not universal. Also, we prove that there is no closed aspherical $3$-manifold, such that every closed, aspherical $3$-manifold is a branched covering over it. Finally, we characterize the closed $3$-manifolds admitting branching coverings from $P^3 \# P^3$, and deduce that there is no closed reducible $3$-manifold, such that every closed reducible $3$-manifold is a branched covering over it.