Linking and coincidence invariants
Volume 184 / 2004
Abstract
Given a link map $f$ into a manifold of the form $Q = N \times {{\mathbb R}}$, when can it be deformed to an “unlinked” position (in some sense, e.g. where its components map to disjoint ${{\mathbb R}}$-levels)? Using the language of normal bordism theory as well as the path space approach of Hatcher and Quinn we define obstructions $\widetilde \omega _\varepsilon (f), \varepsilon = +$ or $\varepsilon = -$, 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 also leads 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.
