$\Delta$-link homotopy of links in $S^3$ and invariants of link maps in $S^4$

In 2003, Nakanishi and Ohyama obtained a classification of $2$-component links up to $\Delta$-link homotopy. Namely, they are classified by the linking number and the generalized Sato-Levine invariant. Using Kirk's invariant of link maps $S^2\sqcup S^2\to S^4$ and its variation due to Koschorke, we obtain a simple proof of the Nakanishi–Ohyama theorem, and also its version for string links. We also prove that $3$-component links that are trivial up to link homotopy are classified up to weak $\Delta$-link homotopy by $\bar\mu$-invariants of length $\le 4$. The proof uses a computation of the image of Koschorke's $\tilde\beta$-invariant of link maps $S^2\sqcup S^2\sqcup S^2\to S^4$ (which is strictly stronger than Gui-Song Li's version of Kirk's invariant). This computation in is turn based on Yasuhara's results about $\Delta$-link homotopy. This talk is based on joint work with Yuka Kotorii.