Link homotopy with many components

Abstract

(i.e. the component mapsf,,, . . . ./, are continuous and have pairwise disjoint imapcs) up to link homotopy (ic. up to deformations through such link maps). This rather crude equivalence relation was introduced in I954 by J. Milnor [I?] in the dimension setting m = 3. pI = pz = . . . = p, = I in an attempt to pet a first rough undcrstanding of the ovcrwhclming multitude of classical links, Milnor gave already a prccisc triviality criterion for such links in terms of his jc-invariants; only rcccntly. though, the full classification in this setting has been given by N. Habcggcr and X. Lin. In the last few years higher dimensional link homotopy has also attracted much new intcrcst. One lint of the dcvclopment. pursued by W. Massey. D. Rolfscn, R. Fcnn. P. Kirk, the author and U. Kaiser a.~., concentrated on link maps with two components; e.g. in a large mctastable dimension range there is now an exact scqucncc at our disposal which rcduccs our classification problem to standard homotopy questions (see [I I]). A strp in iI diflcrcnt direction was taken in 1984 which led to a whole hierarchy of generalized It-invariants for higher dimensional link maps with an arbitrary number of components (see [7J). Now it turns out that p = p( I. 2.3). i.e. the first of these invariants which is truly of higher order, often fills the remaining gap.