Communications in Analysis and Geometry

Volume 31 (2023)

Number 10

Filling links and spines in $3$-manifolds

Pages: 2307 – 2333

DOI: https://dx.doi.org/10.4310/CAG.2023.v31.n10.a1

Authors

Michael Freedman (Microsoft Research, Station Q, and Department of Mathematics, University of California, Santa Barbara, Calif., U.S.A.)

Vyacheslav Krushkal (Department of Mathematics, University of Virginia, Charlottesville, Va., U.S.A.)

Christopher J. Leininger (Department of Mathematics, Rice University, Houston, Texas, U.S.A.)

Alan W. Reed (Department of Mathematics, Rice University, Houston, Texas, U.S.A.)

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 $\pi_1(M \setminus L)$. 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 $2$.

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Appendix by Christopher J. Leininger and Alan W. Reid.

V.K. was supported in part by the Miller Institute for Basic Research in Science at UC Berkeley, Simons Foundation fellowship 608604, and NSF Grant DMS-1612159.

Received 29 October 2020

Accepted 9 November 2021

Published 13 August 2024