Homotopy ribbon discs with a fixed group
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866915077309857792 |
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| author | Conway, Anthony |
| author_facet | Conway, Anthony |
| contents | 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 $|m-n|=1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_04465 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Homotopy ribbon discs with a fixed group Conway, Anthony Geometric Topology 57N35, 57R67, 57K10, 57N70 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 $|m-n|=1$. |
| title | Homotopy ribbon discs with a fixed group |
| topic | Geometric Topology 57N35, 57R67, 57K10, 57N70 |
| url | https://arxiv.org/abs/2201.04465 |