Ribbonness of Kervaire's sphere-link in homotopy 4-sphere and its consequences to 2-complexes
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arXiv
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866914731085791232 |
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| author | Kawauchi, Akio |
| author_facet | Kawauchi, Akio |
| contents | M. A. Kervaire showed that every group of deficiency $d$ and weight $d$ is the fundamental group of a smooth sphere-link of $d$ components in a smooth homotopy 4-sphere. In the use of the smooth unknotting conjecture and the smooth 4D Poincar{é} conjecture, any such sphere-link is shown to be a sublink of a free ribbon sphere-link in the 4-sphere. Since every ribbon sphere-link in the 4-sphere is also shown to be a sublink of a free ribbon sphere-link in the 4-sphere, Kervaire's sphere-link and the ribbon sphere-link are equivalent concepts. By applying this result to a ribbon disk-link in the 4-disk, it is shown that the compact complement of every ribbon disk-link in the 4-disk is aspherical. By this property, a ribbon disk-link presentation for every contractible finite 2-complex is introduced. By using this presentation, it is shown that every connected subcomplex of a contractible finite 2-complex is aspherical (meaning partially yes for Whitehead aspherical conjecture). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_02617 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Ribbonness of Kervaire's sphere-link in homotopy 4-sphere and its consequences to 2-complexes Kawauchi, Akio Geometric Topology 57Q45, 57M20 M. A. Kervaire showed that every group of deficiency $d$ and weight $d$ is the fundamental group of a smooth sphere-link of $d$ components in a smooth homotopy 4-sphere. In the use of the smooth unknotting conjecture and the smooth 4D Poincar{é} conjecture, any such sphere-link is shown to be a sublink of a free ribbon sphere-link in the 4-sphere. Since every ribbon sphere-link in the 4-sphere is also shown to be a sublink of a free ribbon sphere-link in the 4-sphere, Kervaire's sphere-link and the ribbon sphere-link are equivalent concepts. By applying this result to a ribbon disk-link in the 4-disk, it is shown that the compact complement of every ribbon disk-link in the 4-disk is aspherical. By this property, a ribbon disk-link presentation for every contractible finite 2-complex is introduced. By using this presentation, it is shown that every connected subcomplex of a contractible finite 2-complex is aspherical (meaning partially yes for Whitehead aspherical conjecture). |
| title | Ribbonness of Kervaire's sphere-link in homotopy 4-sphere and its consequences to 2-complexes |
| topic | Geometric Topology 57Q45, 57M20 |
| url | https://arxiv.org/abs/2212.02617 |