Exponential growth in the rational homology of free loop spaces and in torsion homotopy groups

Huang, Ruizhi; Theriault, Stephen

doi:10.5802/aif.3627

Exponential growth in the rational homology of free loop spaces and in torsion homotopy groups
[Croissance exponentielle de l’homologie rationnelle de l’espace des lacets libres et des groupes d’homotopie de torsion]

Huang, Ruizhi ¹ ; Theriault, Stephen ²

¹ Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190 (China)
² School of Mathematics, University of Southampton, Southampton SO17 1BJ (United Kingdom)

Annales de l'Institut Fourier, Tome 74 (2024) no. 4, pp. 1365-1382.

Résumé
Abstract

En utilisant des méthodes intégrales, nous retrouvons et généralisons certains résultats de Félix, Halperin et Thomas sur la croissance des groupes d’homologie rationnelle des espaces des lacets libres, et obtenons une nouvelle famille d’espaces dont la $p$ -torsion dans les groupes d’homotopie croît exponentiellement et satisfait la conjecture de Moore pour tous les nombres premiers, sauf un nombre fini. Au vu des résultats, nous conjecturons qu’il devrait y avoir un lien étroit entre la croissance exponentielle dans les groupes d’homotopie rationnelle et les groupes d’homotopie de $p$ -torsion pour tout nombre premier $p$ .

Using integral methods we recover and generalize some results by Félix, Halperin and Thomas on the growth of the rational homology groups of free loop spaces, and obtain a new family of spaces whose $p$ -torsion in homotopy groups grows exponentially and satisfies Moore’s Conjecture for all but finitely many primes. In view of the results, we conjecture that there should be a strong connection between exponential growth in the rational homotopy groups and the $p$ -torsion homotopy groups for any prime $p$ .

Reçu le : 2022-05-11
Accepté le : 2022-11-17
Première publication : 2024-05-17
Publié le : 2024-07-03

DOI : 10.5802/aif.3627

Classification : 55P35, 55P62
Keywords: Exponential growth, free loop space, homotopy exponent, Moore’s conjecture.
Mot clés : Croissance exponentielle, espace des lacets libre, exposant d’homotopie, conjecture de Moore.

Affiliations des auteurs :

Huang, Ruizhi ¹ ; Theriault, Stephen ²

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Huang, Ruizhi; Theriault, Stephen. Exponential growth in the rational homology of free loop spaces and in torsion homotopy groups. Annales de l'Institut Fourier, Tome 74 (2024) no. 4, pp. 1365-1382. doi : 10.5802/aif.3627. https://aif.centre-mersenne.org/articles/10.5802/aif.3627/

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