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Volume 30, Issue 5


Rational homology and homotopy of high-dimensional string links

Paul Arnaud Songhafouo Tsopméné
  • Corresponding author
  • Department of Mathematics and Statistics, University of Regina, 3737 Wascana Pkwy, Regina, SK, S4S 0A2, Canada
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/ Victor Turchin
Published Online: 2018-03-08 | DOI: https://doi.org/10.1515/forum-2016-0192


Arone and the second author showed that when the dimensions are in the stable range, the rational homology and homotopy of the high-dimensional analogues of spaces of long knots can be calculated as the homology of a direct sum of finite graph-complexes that they described explicitly. They also showed that these homology and homotopy groups can be interpreted as the higher-order Hochschild homology, also called Hochschild–Pirashvili homology. In this paper, we generalize all these results to high-dimensional analogues of spaces of string links. The methods of our paper are applicable in the range when the ambient dimension is at least twice the maximal dimension of a link component plus two, which in particular guarantees that the spaces under study are connected. However, we conjecture that our homotopy graph-complex computes the rational homotopy groups of link spaces always when the codimension is greater than two, i.e. always when the Goodwillie–Weiss calculus is applicable. Using Haefliger’s approach to calculate the groups of isotopy classes of higher-dimensional links, we confirm our conjecture at the level of π0.

Keywords: String links; Goodwillie calculus; formality

MSC 2010: 57Q45


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About the article

Received: 2016-09-07

Revised: 2017-09-27

Published Online: 2018-03-08

Published in Print: 2018-09-01

This work has been supported by Fonds de la Recherche Scientifique-FNRS (F.R.S.-FNRS). It has also been supported by the Kansas State University (KSU), where this paper was partially written during the stay of the first author, and which he thanks for hospitality. The second author is partially supported by the Simons Foundation “Collaboration grant for mathematicians” (award ID: 519474).

Citation Information: Forum Mathematicum, Volume 30, Issue 5, Pages 1209–1235, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2016-0192.

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