Narrow Results

We fill a gap in the proof that the proposed link homotopy invariant $\omega$ of Li is well defined. It is also shown that if the homotopy invariant $\tau$ of Schneiderman–Teichner is to be adapted to a link homotopy invariant of link maps, the result coincides with $\omega$.

It is an open problem whether Kirk’s $\sigma$-invariant is the complete obstruction to a link map $f:S_{+}^2\cup S_{-}^2\to S^4$ being link homotopic to the trivial link. The link homotopy invariant associates to such a link map $f$ a pair $\sigma (f)=({\sigma }_{+}(f),{\sigma }_{-}(f))$, and we write $\sigma =({\sigma }_{+},{\sigma }_{-})$. With the objective of constructing counterexamples, Li proposed a link homotopy invariant $\omega =({\omega }_{+},{\omega }_{-})$ such that ${\omega }_{\pm }$ is defined on the kernel of ${\sigma }_{\pm }$ and which also obstructs link null-homotopy. We show that, when defined, the invariant ${\omega }_{\pm }$ is determined by ${\sigma }_{\mp }$, and is strictly weaker. In particular, this implies that if a link map $f$ has $\sigma (f)=(0,0)$, then after a link homotopy the self-intersections of $f(S_{+}^2)$ may be equipped with framed, immersed Whitney disks in $S^4\setminus f(S_{-}^2)$ whose interiors are disjoint from $f(S_{+}^2)$.

We use Kirk’s invariant of link maps $S^2\bigsqcup S^2\to S^4$ and its variations due to Koschorke and Kirk–Livingston to deduce results about classical links. Namely, we give a new proof of the Nakanishi–Ohyama classification of two-component links in $S^3$ up to $\mathrm{\Delta }$-link homotopy. We also prove its version for string links, which is due (in a slightly different form) to Fleming–Yasuhara. The proofs do not use Clasper Theory.