Homotopy Brunnian links and the 𝜅-invariant

Cohen, F.; Komendarczyk, R.; Shonkwiler, C.

doi:10.1090/S0002-9939-2014-12331-8

Homotopy Brunnian links and the $\kappa$-invariant
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by F. R. Cohen, R. Komendarczyk and C. Shonkwiler PDF

Proc. Amer. Math. Soc. 143 (2015), 1347-1362 Request permission

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 in 1997) 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 $\pi _\ast (\Omega \mathrm {Conf}(n))\otimes \mathbb {Q}$ of the configuration space. It allows us to express the relevant Milnor’s $\mu$–invariants as homotopy periods of $\mathrm {Conf}(n)$.

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