On the geometry of homotopy invariants of links

Abstract

One of the central problems in higher-dimensional knot theory is the classification of links up to concordance. In 14, Le Dimet constructed a universal model for (disk) link complements, which allowed him to formulate this problem in the framework of surgery theory by applying the Cappell-Shaneson program for studying codimension two embeddings of manifolds 1. The concordance classification was reduced to questions in L-theory (-groups 1) and homotopy theory (of Vogel local spaces 14). While recent results of Cochran and Orr2 (see also 18) provide rich information on the -theoretic part of the problem (in particular, they settle the question of the existence of links not concordant to boundary links), little is known about Le Dimet's homotopy invariant of links; for example, it is not known whether it may ever be non-trivial, or phrasing it more geometrically (according to 19), whether there are links that are not concordant to sublinks of homology boundary links. This motivated us to look at simpler classes of links, for which a more direct geometric approach to the problem is also possible, in an attempt to get some insight on the geometry carried by the homotopy invariants.