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Homotopy Brunnian links and the $ \kappa$-invariant

Authors: F. R. Cohen, R. Komendarczyk and C. Shonkwiler
Journal: Proc. Amer. Math. Soc. 143 (2015), 1347-1362
MSC (2010): Primary 57M25, 55Q25; Secondary 57M27
Published electronically: November 24, 2014
MathSciNet review: 3293747
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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 \textup {Conf}(n))\otimes \mathbb{Q}$ of the configuration space. It allows us to express the relevant Milnor's $ \mu $-invariants as homotopy periods of $ \textup {Conf}(n)$.

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Additional Information

F. R. Cohen
Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627

R. Komendarczyk
Affiliation: Department of Mathematics, Tulane University, New Orleans, Louisiana 70118

C. Shonkwiler
Affiliation: Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523

Received by editor(s): February 12, 2013
Received by editor(s) in revised form: July 13, 2013
Published electronically: November 24, 2014
Additional Notes: The second author acknowledges support of DARPA YFA N66001-11-1-4132 and NSF DMS 1043009.
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.