The intersection of spheres in a sphere and a new geometric meaning of the Arf invariant
| dc.creator | Ogasa, Eiji | |
| dc.date | 2000-03-15 | |
| dc.date.accessioned | 2026-07-07T04:34:19Z | |
| dc.date.available | 2026-07-07T04:34:19Z | |
| dc.description | Let S^3_i be a 3-sphere embedded in the 5-sphere S^5 (i=1,2). Let S^3_1 and S^3_2 intersect transversely. Then the intersection C of S^3_1 and S^3_2 is a disjoint collection of circles. Thus we obtain a pair of 1-links, C in S^3_i (i=1,2), and a pair of 3-knots, S^3_i in S^5 (i=1,2). Conversely let (L_1,L_2) be a pair of 1-links and (X_1,X_2) be a pair of 3-knots. It is natural to ask whether the pair of 1-links (L_1,L_2) is obtained as the intersection of the 3-knots X_1 and X_2 as above. We give a complete answer to this question. Our answer gives a new geometric meaning of the Arf invariant of 1-links. Let f be a smooth transverse immersion S^3 into S^5. Then the self-intersection C consists of double points. Suppose that C is a single circle in S^5. Then f^{-1}(C) in S^3 is a 1-knot or a 2-component 1-link. There is a similar realization problem. We give a complete answer to this question. | |
| dc.description | 15 pages 13 figures | |
| dc.identifier | https://arxiv.org/abs/math/0003089 | |
| dc.identifier | http://arxiv.org/abs/math/0003089 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/58855 | |
| dc.subject | Geometric Topology | |
| dc.subject | Mathematical Physics | |
| dc.subject | 57M25, 57Q45 | |
| dc.title | The intersection of spheres in a sphere and a new geometric meaning of the Arf invariant | |
| dc.type | text |