Link concordance implies link homotopy

TL;DR

The paper proves that in codimension at least two, link concordance implies link homotopy for immersions, extending classical low-dimensional results to higher dimensions. The authors develop a comprehensive framework built on immersed Morse theory, stratified Morse data, and Cerf-theoretic path analysis, culminating in a Path Lifting Theorem that converts concordances into level-preserving link homotopies. Central innovations include grim vector fields, membranes, and finger-move techniques that resolve cancellation obstructions while preserving disjointness of components, ensuring the link-homotopy class is preserved. The results yield broad corollaries: in codimension two, every smooth embedding S^n ⊔ … ⊔ S^n into S^{n+2} is link-homotopically trivial for n ≥ 2, and the framework provides a geometric, constructive route to regular homotopies from concordances. Overall, the work deepens our understanding of linking phenomena beyond knots and links in S^3, showing concordance collapses to homotopy in high codimension via refined Morse-theoretic machinery.

Abstract

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 In our proofs we follow, expand, and complete unpublished notes of the third author developing stratified Morse theory for generic immersions, where the $d$-th stratum is given by points that have $d$ preimages under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 from intersections of ascending and descending membranes for critical points of the Morse function restricted to various strata. We show that both additional obstructions vanish in codimension at least three, implying a smooth proof of Hudson's result that embedded concordance implies isotopy. In codimension at least two, we show that only the rearrangement obstruction vanishes and we introduce finger moves that eliminate the cancellation obstruction. This is done carefully and only at the expense of introducing new self-intersection points into the components of the immersion. Therefore, our moves keep distinct components disjoint and hence preserve the link homotopy class.

Figures (63)

• Figure 1: Any rooted tree arises as dual tree in dimensions $n\geq 1$.
• Figure 2: Immersed concordance does not imply regular homotopy in codimension one. Going from left to right: the singular link $S^1 \looparrowright \mathbb R^2$ to the left is transformed by a saddle move to obtain a singular $2$-component link. Then, a Whitney move transforms the singular link to a $2$-component unlink. Finally, a minimum cancels one of the components. The trace of these moves provides a singular concordance between the singular link on the left and the unknot on the right. The winding numbers of the immersions are different so they are not regularly homotopic.
• Figure 3: A nontrivial cancellation obstruction in codimension two. The intersection of membranes with the planar level presented as a horizontal dashed line is shown in Figure \ref{['fig:trefoil in the plane']}.
• Figure 4: Intersecting membranes in a planar level for the minimum and maximum in Figure \ref{['fig:trefoil']}.
• Figure 5:

Theorems & Definitions (430)

• Theorem I.1.1
• Corollary I.1.2
• Definition I.1.3
• Proposition I.1.4
• Proposition I.1.5
• proof
• Theorem I.2.1
• Remark I.2.2
• Proposition I.2.3
• Theorem I.3.1: Path Lifting