Link homotopy with many components
The author proves that in a metastable range \(q:=\max \{p_ 2,...,p_ r\}\leq (2/3)m-1\) and \(p_ 1\leq 3(m-q-2)\) (excluded is \(m=3\leq p_ 1+...+p_ r)\) based link homotopy of based link maps naturally splits up into two- and three-component linking phenomena. For the two-component case the author has proved existence of a long exact sequence, which relates link homotopy to standard homotopy [Math. Ann. 286, 753-782 (1990; Zbl 0662.57013)]. Three-component linking (in this range) is measured by the author's generalization of Milnor's \(\mu\) (123). It should be noted that the concept of based link homotopy for \(m=3\), \(p_ 1=...=p_ r=1\) generalizes the string links of \textit{N. Habegger} and \textit{X. Lin} used in their classification of link homotopy in the 3- sphere [J. Am. Math. Soc. 3, No.2, 389-419 (1990; Zbl 0704.57016)]. Finally there is an interesting relation to Haefliger's exact sequence for links which, in a metastable range, also ``splits up'' to only two- and three-component linking obstructions.
scientific article; zbMATH DE number 4214961