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

TL;DR

This paper explores the intersection properties of three spheres immersed in a higher-dimensional sphere and introduces a novel application of the Sato-Levine invariant to analyze their linked surface structures.

Contribution

It establishes a new relationship involving the Sato-Levine invariant for certain intersecting sphere configurations in higher dimensions.

Findings

Proves the equality of Sato-Levine invariants sum to zero for semi-boundary links.

Introduces a new application of the Sato-Levine invariant in higher-dimensional sphere intersections.

Provides conditions under which the invariants relate in a specific linear sum.

Abstract

Take transverse immersions f from a disjoint unin of the three 4-spheres $S^4_1$, $S^4_2$, and $S^4_3$ into $S^6$ with the following properties: (1) The restriction of $f$ to $S^4_i$ is an embedding, (2) The intersection of $f(S^4_i)$ and $f(S^4_j)$ is not empty and connected, (3)The intersection among $f(S^4_1)$, $f(S^4_2)$, and $f(S^4_3)$ is not empty. Then we obtain three surface-links $L_i=(S^4_i\cap S^4_j, S^4_i\cap S^4_k)$ 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.