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, Online first, 18 p.

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

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

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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, Online first, 18 p.

Bibliographie
Cité par

[1] Anick, David J. The smallest singularity of a Hilbert series, Math. Scand., Volume 51 (1982) no. 1, pp. 35-44 | DOI | MR | Zbl

[2] Ballmann, Werner; Ziller, Wolfgang On the number of closed geodesics on a compact Riemannian manifold, Duke Math. J., Volume 49 (1982) no. 3, pp. 629-632 | DOI | MR | Zbl

[3] Beben, Piotr; Theriault, Stephen The loop space homotopy type of simply-connected four-manifolds and their generalizations, Adv. Math., Volume 262 (2014), pp. 213-238 | DOI | MR | Zbl

[4] Beben, Piotr; Theriault, Stephen Homotopy groups of highly connected Poincaré duality complexes, Doc. Math., Volume 27 (2022), pp. 183-211 | DOI | MR | Zbl

[5] Boyde, Guy $p$ -hyperbolicity of homotopy groups via $K$ -theory, Math. Z., Volume 301 (2022) no. 1, pp. 977-1009 | DOI | MR | Zbl

[6] Chachólski, Wojciech; Pitsch, Wolfgang; Scherer, Jérôme; Stanley, Don Homotopy exponents for large $H$ -spaces, Int. Math. Res. Not. (2008) no. 16, rnn061, 5 pages | DOI | MR | Zbl

[7] Dwyer, William G. Tame homotopy theory, Topology, Volume 18 (1979) no. 4, pp. 321-338 | DOI | MR | Zbl

[8] Félix, Yves; Halperin, Stephen; Thomas, Jean-Claude The homotopy Lie algebra for finite complexes, Publ. Math., Inst. Hautes Étud. Sci. (1982) no. 56, pp. 179-202 | DOI | Numdam | MR | Zbl

[9] Félix, Yves; Halperin, Stephen; Thomas, Jean-Claude Rational homotopy theory, Graduate Texts in Mathematics, 205, Springer, 2001, xxxiv+535 pages | DOI | MR | Zbl

[10] Félix, Yves; Halperin, Stephen; Thomas, Jean-Claude Exponential growth and an asymptotic formula for the ranks of homotopy groups of a finite 1-connected complex, Ann. Math., Volume 170 (2009) no. 1, pp. 443-464 | DOI | MR | Zbl

[11] Félix, Yves; Halperin, Stephen; Thomas, Jean-Claude On the growth of the homology of a free loop space, Pure Appl. Math. Q., Volume 9 (2013) no. 1, pp. 167-187 | DOI | MR | Zbl

[12] Félix, Yves; Halperin, Stephen; Thomas, Jean-Claude On the growth of the homology of a free looop space II, Ann. Inst. Fourier, Volume 67 (2017) no. 6, pp. 2519-2531 | DOI | MR | Zbl

[13] Gromov, Mikhael Homotopical effects of dilatation, J. Differ. Geom., Volume 13 (1978) no. 3, pp. 303-310 | DOI | MR | Zbl

[14] Halperin, Stephen; Lemaire, Jean-Michel Suites inertes dans les algèbres de Lie graduées (“Autopsie d’un meurtre. II”), Math. Scand., Volume 61 (1987) no. 1, pp. 39-67 | DOI | MR | Zbl

[15] Hao, Yanlong; Sun, Qianwen; Theriault, Stephen Moore’s conjecture for polyhedral products, Math. Proc. Camb. Philos. Soc., Volume 167 (2019) no. 1, pp. 23-33 | DOI | MR | Zbl

[16] Hatcher, Allen Algebraic topology, Cambridge University Press, 2002, xii+544 pages | MR | Zbl

[17] Huang, Ruizhi; Wu, Jie Exponential growth of homotopy groups of suspended finite complexes, Math. Z., Volume 295 (2020) no. 3-4, pp. 1301-1321 | DOI | MR | Zbl

[18] Lambrechts, Pascal The Betti numbers of the free loop space of a connected sum, J. Lond. Math. Soc., Volume 64 (2001) no. 1, pp. 205-228 | DOI | MR | Zbl

[19] Lambrechts, Pascal On the Betti numbers of the free loop space of a coformal space, J. Pure Appl. Algebra, Volume 161 (2001) no. 1-2, pp. 177-192 | DOI | MR | Zbl

[20] Long, J. Two contributions to the homotopy theory of $H$ -spaces, Ph. D. Thesis, Princeton University (1978)

[21] Neisendorfer, Joseph A. The exponent of a Moore space, Algebraic topology and algebraic $K$ -theory (Princeton, N.J., 1983) (Annals of Mathematics Studies), Volume 113, Princeton University Press, 1987, pp. 35-71 | MR | Zbl

[22] Selick, Paul On conjectures of Moore and Serre in the case of torsion-free suspensions, Math. Proc. Camb. Philos. Soc., Volume 94 (1983) no. 1, pp. 53-60 | DOI | MR | Zbl

[23] Stanley, Don Exponents and suspension, Math. Proc. Camb. Philos. Soc., Volume 133 (2002) no. 1, pp. 109-116 | DOI | MR | Zbl

[24] Theriault, S. Homotopy fibrations with a section after looping (2020) (https://arxiv.org/abs/2005.11570, to appear in Mem. Amer. Math. Soc.)

[25] Vigué-Poirrier, Micheline Homotopie rationnelle et croissance du nombre de géodésiques fermées, Ann. Sci. Éc. Norm. Supér., Volume 17 (1984) no. 3, pp. 413-431 | DOI | Numdam | MR | Zbl

[26] Wall, Charles T. C. Classification of $(n - 1)$ -connected $2 n$ -manifolds, Ann. Math., Volume 75 (1962), pp. 163-189 | DOI | MR | Zbl

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