Homotopy invariants of links

Abstract

This paper answers a question originally asked by John Milnor in 1957. It offers significant progress toward another question from that paper. In [-M 1] an [M 2], John Milnor defined a collection of isotopy invariants of links. Stallings showed these were actually concordance i~variants of the ~ink. Stallings' work suggested that Milnor's invariants were related to other algebraically defined invariants, the Massey products in the exterior of the link; a connection made precise in [Tu] and [P]. Thus, the results in this paper can be reinterpreted as results about Massey products in the link exterior. Recently, Mike Freedman has suggested that some of these invariants may be the obstructions to the topological surgery theorem in dimension four, [F I ] , IF2], [FL]. In this paper, the approach to studying higher dimensional link concordance presented in [O 1] is extended to the study of links in the 3-sphere. A collection of new link invariants is defined and related to Milnor's invariants. One fundamental advantage of this approach is that these new invariants make explicit use of the three sphere containing the link. The link exterior contains all the relations among Milnor's invariants. By accounting for the ambient sphere, our invariants eliminate redundancy (see Theorem 15). We answer this question originally asked by Milnor: 1) To what extent are Milnor's invariants independent of each other? (Theorem 15). Milnor also asked this question: 2) Can one extract invariants from the group rc/n~,, where 7r is the link group and n~, is the intersection of the lower central series of the link group? Theorem 15 answers question 1. Partial results can be found in [C 1], [Ma I], [On] and [T]. The only previously known results for two component links were k = 2 . . . . . 8 where the number of linearly independent invariants are 0, 1, 0, 5, 0, 6, and 4 respectively. For links with three or more components tess was known.