The intersection of spheres in a sphere and a new geometric meaning of the Arf invariant
Ogasa, Eiji
arXiv, 0003089 / Harvested from arXiv
Text on arXiv
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.
Publié le : 2000-03-15
Classification:  Mathematics - Geometric Topology,  Mathematical Physics,  57M25, 57Q45 
@article{0003089,
     author = {Ogasa, Eiji},
     title = {The intersection of spheres in a sphere and a new geometric meaning of
  the Arf invariant},
     journal = {arXiv},
     volume = {2000},
     number = {0},
     year = {2000},
     language = {en},
     url = {http://dml.mathdoc.fr/item/0003089}
}

Ogasa, Eiji. The intersection of spheres in a sphere and a new geometric meaning of
  the Arf invariant. arXiv, Tome 2000 (2000) no. 0, . http://gdmltest.u-ga.fr/item/0003089/