Orthogonal 2-sphere basis of stable 4-sphere
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918475128111104 |
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| author | Kawauchi, Akio |
| author_facet | Kawauchi, Akio |
| contents | Every stable 4-sphere is identified with the double branched covering space of a trivial surface-knot space. As a result of Wall, it is known that any two orthogonal bases of every stable 4-sphere are transformed into each other by an orientation-preserving diffeomorphism of the stable 4-sphere. In this paper another proof of Wall's result is presented, strengthened in the sense that the lift of an equivalence of the trivial surface-knot space can be taken as the diffeomorphism. Two applications are made. The first shows that every orientation-preserving diffeomorphism of every stable 4-sphere is nothing but the double branched covering lift of an equivalence of a trivial surface-knot space up to a smooth isotopy and a composition with an identity-shift. The second gives a similar result for TOP stable 4-spheres. Here, even if it is a smooth 4-manifold, unless it is diffeomorphic to the stable 4-sphere, the TOP trivial surface-knot space cannot be smooth. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_01507 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Orthogonal 2-sphere basis of stable 4-sphere Kawauchi, Akio Geometric Topology 57K45, 57K40 Every stable 4-sphere is identified with the double branched covering space of a trivial surface-knot space. As a result of Wall, it is known that any two orthogonal bases of every stable 4-sphere are transformed into each other by an orientation-preserving diffeomorphism of the stable 4-sphere. In this paper another proof of Wall's result is presented, strengthened in the sense that the lift of an equivalence of the trivial surface-knot space can be taken as the diffeomorphism. Two applications are made. The first shows that every orientation-preserving diffeomorphism of every stable 4-sphere is nothing but the double branched covering lift of an equivalence of a trivial surface-knot space up to a smooth isotopy and a composition with an identity-shift. The second gives a similar result for TOP stable 4-spheres. Here, even if it is a smooth 4-manifold, unless it is diffeomorphic to the stable 4-sphere, the TOP trivial surface-knot space cannot be smooth. |
| title | Orthogonal 2-sphere basis of stable 4-sphere |
| topic | Geometric Topology 57K45, 57K40 |
| url | https://arxiv.org/abs/2602.01507 |