Classical Poincar{é} conjecture via 4D topology
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866929296539385856 |
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| author | Kawauchi, Akio |
| author_facet | Kawauchi, Akio |
| contents | The classical Poincar{é} conjecture that every homotopy 3-sphere is diffeomorphic to the 3-sphere is confirmed by Perelman in arXiv papers solving Thurston's program on geometrizations of 3-manifolds. A new confirmation of this conjecture is given by a method of 4D topology. For this proof, the spun torus-knot of every knot in every homotopy 3-sphere is observed to be a ribbon torus-knot in the 4-sphere, where Smooth 4D Poincar{é} Conjecture and Ribbonness of a sphere-link with (not necessarily meridian-based) free fundamental group are used. By examining a disk-chord system of a ribbon solid torus bounded by the spun torus-knot,it is proved that the knot belongs to a 3-ball in the homotopy 3-sphere. Then by Bing's result, it is confirmed that the homotopy 3-sphere is diffeomorphic to the 3-sphere. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_16001 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Classical Poincar{é} conjecture via 4D topology Kawauchi, Akio Geometric Topology 57M40, 57N13, 57Q45 The classical Poincar{é} conjecture that every homotopy 3-sphere is diffeomorphic to the 3-sphere is confirmed by Perelman in arXiv papers solving Thurston's program on geometrizations of 3-manifolds. A new confirmation of this conjecture is given by a method of 4D topology. For this proof, the spun torus-knot of every knot in every homotopy 3-sphere is observed to be a ribbon torus-knot in the 4-sphere, where Smooth 4D Poincar{é} Conjecture and Ribbonness of a sphere-link with (not necessarily meridian-based) free fundamental group are used. By examining a disk-chord system of a ribbon solid torus bounded by the spun torus-knot,it is proved that the knot belongs to a 3-ball in the homotopy 3-sphere. Then by Bing's result, it is confirmed that the homotopy 3-sphere is diffeomorphic to the 3-sphere. |
| title | Classical Poincar{é} conjecture via 4D topology |
| topic | Geometric Topology 57M40, 57N13, 57Q45 |
| url | https://arxiv.org/abs/2103.16001 |