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Link Concordance Implies Link Homotopy by Borodzik, Maciej; Powell, Mark; Teichner, Peter is a scholarly article available to read on EtoBox.

What is Link Concordance Implies Link Homotopy about?

We show that link concordance implies link homotopy for immersions of codimension at least two. As a consequence, we prove that every link $\sqcup^r S^n \hookrightarrow S^{n+2}$ is link homotopically trivial for $n\geq 2$, that is, there is a {\em link} homotopy that (for each time parameter) maps distinct component $n$-spheres disjointly into $S^{n+2}$. In other words, beyond the classical dimension (of embedded circles in $S^3$) there are no `linking modulo knotting' phenomena in codimension two. To date, this was only known for $n=2$. In our proofs we follow, expand, and complete unpublished notes of the third author developing stratified Morse theory for generic immersions, where the $d$-th stratum is given by points that have $d$ preimages under the generic immersion. We discuss gradient like vector fields, their strata preserving flows, and Cerf theory. This generalizes the case of embeddings, as studied by Perron, Sharpe, and Rourke, and expanded by the first two authors. Two vital operations in Cerf theory are the rearrangement and cancellation of critical points. In the setting of stratified theory, there are additional rearrangement and cancellation obstructions arising f

Author

Borodzik, Maciej; Powell, Mark; Teichner, Peter

Published

2025

Language

EN