On the existence of universal links in three-manifolds
Francisco González-Acuña, Araceli Guzmán-Tristán, Jesús Rodríguez-Viorato, José Andrés Rodríguez Migueles
TL;DR
The paper classifies when closed 3-manifolds admit universal knots/links, proving this occurs if and only if the manifold is spherical. It shows how to construct universal links in spherical manifolds by transferring a known $S^3$-universal link via a local homeomorphism, and it distinguishes universal links from complement universal links by constructing infinitely many examples that are complement universal but not universal. It also establishes a nonexistence result: there is no closed aspherical 3-manifold that dominates all aspherical manifolds via branched covers, using transfer maps and Hopf-type index obstructions. The work clarifies the relationship between branched coverings, universality notions, and geometric structures, and outlines several open questions for future study.
Abstract
We prove that the only closed $3$-manifolds that admit a universal link are spherical. Also, we prove the non-existence of branched covers between closed aspherical $3$-manifolds. Moreover, we show infinitely many examples of complement universal links that are not universal.