Linking and coincidence invariants

Abstract

Given a link map f into a manifold of the form Q = N ×R, when can it be deformed to an “unlinked” position (i.e. where its components map to disjoint R-levels)? Using the language of normal bordism theory as well as the path space approach of Hatcher and Quinn we define obstructions ωe(f), e = + or e = −, which often answer this question completely and which, in addition, turn out to distinguish a great number of different link homotopy classes. In certain cases they even allow a complete link homotopy classification. Our development parallels recent advances in Nielsen coincidence theory and leads also to the notion of Nielsen numbers of link maps. In the special case when N is a product of spheres sample calculations are carried out. They involve the homotopy theory of spheres and, in particular, James–Hopf– invariants.