Diagrams for primitive cycles in spaces of pure braids and string links

Komendarczyk, Rafal; Koytcheff, Robin; Volić, Ismar

doi:10.5802/aif.3616

Diagrams for primitive cycles in spaces of pure braids and string links
[Diagrammes correspondant aux cycles primitifs dans les espaces de tresses pures et longs entrelacs]

Komendarczyk, Rafal ¹ ; Koytcheff, Robin ² ; Volić, Ismar ³

¹ Department of Mathematics, Tulane University, 6823 St. Charles Ave, New Orleans, LA 70118
² Department of Mathematics, University of Louisiana at Lafayette, PO Box 43568, Lafayette, LA 70504
³ Department of Mathematics, Wellesley College, 106 Central Street, Wellesley, MA 02481

Annales de l'Institut Fourier, Tome 74 (2024) no. 4, pp. 1745-1807.

Résumé
Abstract

L’espace des lacets basés d’un espace de configurations des points dans un espace euclidien peut être vu comme un espace de tresses pures dans un espace euclidien d’une dimension superieure. Nous continuons notre travail sur ces espaces du point de vue de l’ADGC de Kontsevich des diagrammes et des intégrales de Chen. Nous construisons une connexion série puissance qui donne une isomorphisme d’algèbre de Hopf entre l’homologie de l’espace des tresses pures et la construction cobar sur les diagrammes. Il mappe les classes primitives aux arbres trivalents modulo la relation IHX. En conséquence, nous établissons une bijection entre les invariants de Milnor des entrelacs sphériques bruniens et certaines intégrales de Chen. Enfin nous montrons que le graphe induit des injections d’un certain sous-module de l’homotopie des espaces de configurations dans l’homotopie des espaces de longs entrelacs. Nous conjecturons qu’il induit des injections de toute l’homotopie rationelle des espaces de configurations.

The based loop space of a configuration space of points in a Euclidean space can be viewed as a space of pure braids in a Euclidean space of one dimension higher. We continue our study of such spaces in terms of Kontsevich’s CDGA of diagrams and Chen’s iterated integrals. We construct a power series connection which yields a Hopf algebra isomorphism between the homology of the space of pure braids and the cobar construction on diagrams. It maps iterated Whitehead products to trivalent trees modulo the IHX relation. As an application, we establish a correspondence between Milnor invariants of Brunnian spherical links and certain Chen integrals. Finally we show that graphing induces injections of a certain submodule of the homotopy of configuration spaces into the homotopy of many spaces of string links. We conjecture that graphing is injective on all rational homotopy classes.

Reçu le : 2022-03-29
Accepté le : 2022-12-16
Accepté après révision le : 2023-01-13
Première publication : 2024-05-17
Publié le : 2024-07-03

DOI : 10.5802/aif.3616

Classification : 55P35, 55R80, 57K45, 20F36, 58D10, 81Q30
Keywords: Spaces of braids, loop spaces, bar and cobar constructions, configuration space integrals, Chen’s iterated integrals, formality, graph complexes, spaces of high-dimensional string links, generalized Milnor invariants.
Mot clés : Espaces de tresses, espaces de lacets, constructions bar et cobar, intégrales d’espaces de configurations, intégrales itérées de Chen, formalité, complexes de graphes, espaces de longs entrelacs de grande dimension, invariants de Milnor généralisés.

Affiliations des auteurs :

Komendarczyk, Rafal ¹ ; Koytcheff, Robin ² ; Volić, Ismar ³

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Komendarczyk, Rafal; Koytcheff, Robin; Volić, Ismar. Diagrams for primitive cycles in spaces of pure braids and string links. Annales de l'Institut Fourier, Tome 74 (2024) no. 4, pp. 1745-1807. doi : 10.5802/aif.3616. https://aif.centre-mersenne.org/articles/10.5802/aif.3616/

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