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The intersection of three spheres in a sphere
and a new application
of the Sato-Levine invariant

Author: Eiji Ogasa
Journal: Proc. Amer. Math. Soc. 126 (1998), 3109-3116
MSC (1991): Primary 57M25, 57Q45
MathSciNet review: 1452817
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Abstract: Take transverse immersions $f:S^{4}_{1}\amalg $ $S^{4}_{2}\amalg $ $S^{4}_{3}\looparrowright S^{6}$ such that (1) $f\vert S^{4}_{i}$ is an embedding, (2) $f(S^{4}_{i})\cap f(S^{4}_{j})\neq \varnothing$ and $f(S^{4}_{i})\cap f(S^{4}_{j})$ is connected, and (3) $f(S^{4}_{1})\cap f(S^{4}_{2})\cap f(S^{4}_{3})$ $=\varnothing$. Then we obtain three surface-links $L_{i}$= ($f^{-1}(f(S^{4}_{i})\cap f(S^{4}_{j}))$, $f^{-1}(f(S^{4}_{i})\cap f(S^{4}_{k}))$ ) in $S^{4}_{i}$, where $(i,j,k)$=(1,2,3), (2,3,1), (3,1,2). We prove that, we have the equality $\beta (L_{1})+$ $\beta (L_{2})+$ $\beta (L_{3})=0,$ where $\beta (L_{i})$ is the Sato-Levine invariant of $L_{i}$, if all $L_{i}$ are semi-boundary links.

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Additional Information

Eiji Ogasa
Affiliation: Department of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 153, Japan

Keywords: Surface-knot, surface-link, spin cobordism group, the Sato-Levine invariant, realizable triple of surface-links
Received by editor(s): July 23, 1996
Received by editor(s) in revised form: February 26, 1997
Additional Notes: This research was partially suppported by Research Fellowships of the Promotion of Science for Young Scientists.
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 1998 American Mathematical Society

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