3-manifolds of rank 3 have filling links

TL;DR

This paper proves that every closed, orientable 3-manifold with fundamental group rank 3 contains a filling link, confirming a conjecture and expanding understanding of the topology of such manifolds.

Contribution

It establishes that all 3-manifolds with fundamental group rank 3 have filling links, answering an open question left by Freedman and Krushkal.

Findings

All closed, orientable 3-manifolds with rank 3 contain filling links.

Filling links exist in the 3-torus as a special case.

The result generalizes previous weaker forms of filling links.

Abstract

M. Freedman and V. Krushkal introduced the notion of a "filling" link in a 3-manifold: a link $L$ is filling in $M$ if for any spine $G$ of $M$ disjoint from $L$, $\pi_1(G)$ injects into $\pi_1(M \setminus L )$. Freedman and Krushkal show that there exist links in the $3$-torus $T^3$ that satisfy a weaker form of filling, but they leave open the question of whether $T^3$ contains an actual filling link. We answer this question affirmatively by proving in fact that every closed, orientable 3-manifold $M$ with $\operatorname{rank}(\pi_1(M)) = 3$ contains a filling link.