On The Growth of the Homology of a Free Loop Space II

Félix, Yves; Halperin, Steve; Thomas, Jean-Claude

doi:10.5802/aif.3141

On The Growth of the Homology of a Free Loop Space II
[Sur la croissance de l’homologie des espaces de lacets II]

Félix, Yves ¹ ; Halperin, Steve ² ; Thomas, Jean-Claude ³

¹ Université Catholique de Louvain, Institut de Mathématique, 2, Chemin du cyclotron, 1348 Louvain-La-Neuve (Belgium)
² University of Maryland, Department of Mathematics, Mathematics Building, College Park, MD 20742 (USA)
³ Université d’Angers, LAREMA, 2 Bd Lavoisier, 49045 Angers Cedex (France)

Annales de l'Institut Fourier, Tome 67 (2017) no. 6, pp. 2519-2531.

Résumé
Abstract

La croissance exponentielle controlée est une version forte de la croissance exponentielle. Nous prouvons que les nombres de Betti de l’espace des lacets libres sur un espace $X$ ont une croissance exponentielle controlée dans deux cas : lorsque $X$ est la somme connexe de variétés dont la cohomologie n’est pas monogène, et lorsque l’algèbre de Lie $L_{X}$ a une croissance exponentielle strictement plus grande que ses indécomposables.

Controlled exponential growth is a stronger version of exponential growth. We prove that the homology of the free loop space $ℒ X$ has controlled exponential growth in two important situations : (1) when $X$ is a connected sum of manifolds whose rational cohomologies are not monogenic, (2) when the rational homotopy Lie algebra $L_{X}$ contains an inert element and $ρ (L_{X}) < ρ (L_{X} / [L_{X}, L_{X}])$ , where $ρ (V)$ denotes the radius of convergence of $V$ .

Reçu le : 2015-04-16
Révisé le : 2017-05-27
Accepté le : 2017-08-07
Publié le : 2017-12-14

DOI : 10.5802/aif.3141

Classification : 55P62
Keywords: free loop space, exponential growth, inert attachment
Mot clés : espace des lacets libres, croissance exponentielle, attachement inerte

Affiliations des auteurs :

Félix, Yves ¹ ; Halperin, Steve ² ; Thomas, Jean-Claude ³

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Félix, Yves; Halperin, Steve; Thomas, Jean-Claude. On The Growth of the Homology of a Free Loop Space II. Annales de l'Institut Fourier, Tome 67 (2017) no. 6, pp. 2519-2531. doi : 10.5802/aif.3141. http://archive.numdam.org/articles/10.5802/aif.3141/

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