THE INTERSECTION OF SPHERES IN A SPHERE AND A NEW GEOMETRIC MEANING OF THE ARF INVARIANT

Let be a 3-sphere embedded in the 5-sphere S⁵ (i = 1,2). Let and intersect transversely. Then the intersection is a disjoint collection of circles. Thus we obtain a pair of 1-links, C in , and a pair of 3-knots, in S⁵ (i = 1, 2). Conversely let (L₁, L₂) be a pair of 1-links and (X₁, X₂) be a pair of 3-knots. It is natural to ask whether the pair of 1-links (L₁, L₂) is obtained as the intersection of the 3-knots X₁ and X₂ 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 : S³ → S⁵ be a smooth transverse immersion such that the self-intersection C consists of double points. Suppose that C is a single circle in S⁵. Then f^-1(C) in S³ is a 1-knot or a 2-component 1-link. There is a similar realization problem. We give a complete answer to this question.