arXiv subjects
Publications and source records attributed to Ash Lightfoot.
It is an open problem whether Kirk's $σ$ 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 $ω$ that is defined on the kernel of $σ$ and also obstructs link nullhomotopy. We show that $ω$ is determined by $σ$, and is a strictly weaker invariant.
We affirmatively address the question of whether the proposed link homotopy invariant $ω$ of Li is well-defined. It is also shown that if one wishes to adapt the homotopy invariant $τ$ of Schneiderman-Teichner to a link homotopy invariant of link maps, the result coincides with $ω$.
It is shown that Schneiderman and Teichner's invariant $τ$ detects the example due to Kirk of a link map $f:S^2_+\cup S^2_-\to S^4$ for which the restricted map $f|S^2_+:S^2_+\to S^4-f(S^2_-)$ has vanishing Wall self-intersection but is not homotopic to an embedding.