Filling links and spines in 3-manifolds

We introduce and study the notion of filling links in $3$-manifolds: a link $L$ is filling in $M$ if for any $1$-spine $G$ of $M$ which is disjoint from $L$, $\pi _{1}(G)$ injects into $\pi _{1}(M\smallsetminus L)$. A weaker "$k$-filling" version concerns injectivity modulo $k$-th term of the lower central series. For each $k\geq 2$ we construct a $k$-filling link in the $3$-torus. The proof relies on an extension of the Stallings theorem which may be of independent interest. We discuss notions related to "filling" links in $3$-manifolds, and formulate several open problems. The appendix by C. Leininger and A. Reid establishes the existence of a filling hyperbolic link in any closed orientable $3$-manifold with $\pi _{1}(M)$ of rank $2$.