Characterization of the reduced peripheral system of links

The reduced peripheral system was introduced by Milnor [18] in the 1950s for the study of links up to link-homotopy, that is, up to homotopies leaving distinct components disjoint; this invariant, however, fails to classify links up to link-homotopy for links of four or more components. The purpose of this paper is to show that the topological information which is detected by Milnor's reduced peripheral system is actually 4-dimensional. The main result gives indeed a complete characterization of links having the same reduced peripheral system, in terms of ribbon solid tori in 4-space up to ribbon link-homotopy. The proof relies on an intermediate characterization given in terms of welded diagrams up to self-virtualization, hence providing a purely topological application of the combinatorial theory of welded links.