Homotopy ribbon discs with a fixed group

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 $|m-n|=1$.

9 pages, 1 figure. v2: minor modifications made to the introduction to emphasise which results hold regardless of whether the group $G$ is good. v3: Further improvements, corrected some inaccuracies that had appeared in v2; to appear in AGT

References in corpus (2)

• Relative Khovanov-Jacobsson classes
• A mixed invariant of non-orientable surfaces in equivariant Khovanov homology

Cited by in corpus (1)

• Action of the Mazur pattern up to topological concordance