Link concordances as surfaces in 4-space and the 4-dimensional Milnor invariants

TL;DR

This paper explores the space of concordances between links in 3-space as surfaces in 4-space, introducing Milnor-type invariants to classify these concordance groups, especially for slice links.

Contribution

It defines new Milnor-type invariants for concordance groups of links in 4-space, providing a classification method up to link-homotopy for slice links.

Findings

Milnor-type invariants classify concordance groups for slice links.

The set of concordances forms a group under surface-concordance.

Invariants are defined modulo a specific indeterminacy related to classical Milnor invariants.

Abstract

Fixing two concordant links in $3$--space, we study the set of all embedded concordances between them, as knotted annuli in $4$--space. When regarded up to surface-concordance or link-homotopy, the set $\mathcal{C}(L)$ of concordances from a link $L$ to itself forms a group. In order to investigate these groups, we define Milnor-type invariants of $\mathcal{C}(L)$, which are integers defined modulo a certain indeterminacy given by Milnor invariants of $L$. We show in particular that, for a slice link $L$, these invariants classify $\mathcal{C}(L)$ up to link-homotopy.