Filling links and spines in 3-manifolds

TL;DR

This paper introduces the concept of filling links in 3-manifolds, explores their properties, constructs examples in the 3-torus, and discusses related open problems, extending classical theorems like Stallings.

Contribution

It defines and studies filling links and k-filling links in 3-manifolds, extending Stallings theorem, and constructs explicit examples in the 3-torus.

Findings

Existence of k-filling links in the 3-torus for each k>1.

Extension of Stallings theorem related to filling links.

Existence of filling hyperbolic links in any closed orientable 3-manifold.

Abstract

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>1 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.