Linking and coincidence invariants

Volume 184 / 2004

Ulrich Koschorke Fundamenta Mathematicae 184 (2004), 187-203 MSC: Primary 55P35, 55S35, 57Q45, 57R90; Secondary 55M20, 55Q25, 55Q45. DOI: 10.4064/fm184-0-12


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.


  • Ulrich KoschorkeUniversität Siegen
    Emmy Noether Campus
    Walter-Flex-Str. 3
    D-57068 Siegen, Germany

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