3-manifolds of rank 3 have 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 $\mathrm{rank}({\pi }_1(M))=3$ contains a filling link.

10 pages

References in corpus (2)

• Tameness of hyperbolic 3-manifolds
• Technical Report: All Principal Congruence Link Groups