The triviality of a certain invariant of link maps in the four-sphere

TL;DR

This paper investigates the relationship between two invariants, $\sigma$ and $\omega$, in the context of link maps in four-dimensional spheres, showing that $\omega$ offers no new obstruction beyond $\sigma$.

Contribution

The paper proves that the invariant $\omega$ is completely determined by $\sigma$, establishing that $\omega$ is a weaker invariant and not useful for constructing counterexamples.

Findings

$\omega$ is determined by $\sigma$

$\omega$ is a strictly weaker invariant than $\sigma$

No new obstructions are provided by $\omega$

Abstract

It is an open problem whether Kirk's $\sigma$ invariant is the complete obstruction to a link map $S^2\cup S^2\to S^4$ being link homotopically trivial. With the objective of constructing counterexamples, Li proposed a link homotopy invariant $\omega$ that is defined on the kernel of $\sigma$ and also obstructs link nullhomotopy. We show that $\omega$ is determined by $\sigma$, and is a strictly weaker invariant.