Homotopy Brunnian links and the $\kappa$-invariant

TL;DR

This paper offers a new proof of the injectivity of Koschorke's $$-invariant for link homotopy classes of Brunnian links, connecting it to homotopy theory and configuration spaces.

Contribution

It presents an alternative proof using homotopy theoretic methods, linking Milnor invariants to homotopy periods of configuration spaces.

Findings

Injectivity of $$-invariant for Brunnian links established

Milnor's $$-invariants expressed as homotopy periods

Connection made between link invariants and rational homotopy theory

Abstract

We provide an alternative proof that Koschorke's $\kappa$-invariant is injective on the set of link homotopy classes of $n$-component homotopy Brunnian links $BLM(n)$. The existing proof (by Koschorke \cite{Koschorke97}) is based on the Pontryagin--Thom theory of framed cobordisms, whereas ours is closer in spirit to techniques based on Habegger and Lin's string links. We frame the result in the language of Fox's torus homotopy groups and the rational homotopy Lie algebra of the configuration space $\text{Conf}(n)$ of $n$ points in $\mathbb{R}^3$. It allows us to express the relevant Milnor's $\mu$--invariants as homotopy periods of $\text{Conf}(n)$.