Homotopy ribbon discs with a fixed group

TL;DR

This paper advances the classification of homotopy ribbon discs in topology by establishing conditions under which exteriors are s-cobordant, leading to new classifications for specific knot groups and groups satisfying Farrell-Jones conjecture.

Contribution

It proves that for geometrically 2-dimensional groups satisfying Farrell-Jones, exteriors of aspherical homotopy ribbon discs are s-cobordant, enabling classification of such discs.

Findings

Classified homotopy ribbon discs for knots with good groups.

Extended classification to groups like $BS(m,n)$ with $|m-n|=1$.

Established s-cobordism conditions for exteriors based on fundamental group properties.

Abstract

In the topological category, the classification of homotopy ribbon discs is known when the fundamental group $G$ of the exterior is $\mathbb{Z}$ and the Baumslag-Solitar group $BS(1,2)$. We prove that if a group $G$ is geometrically $2$-dimensional and satisfies the Farrell-Jones conjecture, then a condition involving the fundamental group ensures that exteriors of aspherical homotopy ribbon discs with fundamental group $G$ are s-cobordant rel.\ boundary. When $G$ is good, this leads to the classification of such discs. As an application, for any knot $J \subset S^3$ whose knot group $G(J)$ is good, we classify the homotopy ribbon discs for $J \# -J$ whose complement has group $G(J)$. A similar application is obtained for $BS(m,n)$ when $|m-n|=1$.