On The Growth of the Homology of a Free Loop Space II
[Sur la croissance de l’homologie des espaces de lacets II]
Annales de l'Institut Fourier, Tome 67 (2017) no. 6, pp. 2519-2531.

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 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 :
Révisé le :
Accepté le :
Publié le :
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
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)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
@article{AIF_2017__67_6_2519_0,
     author = {F\'elix, Yves and Halperin, Steve and Thomas, Jean-Claude},
     title = {On {The} {Growth} of the {Homology} of a {Free} {Loop} {Space} {II}},
     journal = {Annales de l'Institut Fourier},
     pages = {2519--2531},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {67},
     number = {6},
     year = {2017},
     doi = {10.5802/aif.3141},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3141/}
}
TY  - JOUR
AU  - Félix, Yves
AU  - Halperin, Steve
AU  - Thomas, Jean-Claude
TI  - On The Growth of the Homology of a Free Loop Space II
JO  - Annales de l'Institut Fourier
PY  - 2017
SP  - 2519
EP  - 2531
VL  - 67
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3141/
DO  - 10.5802/aif.3141
LA  - en
ID  - AIF_2017__67_6_2519_0
ER  - 
%0 Journal Article
%A Félix, Yves
%A Halperin, Steve
%A Thomas, Jean-Claude
%T On The Growth of the Homology of a Free Loop Space II
%J Annales de l'Institut Fourier
%D 2017
%P 2519-2531
%V 67
%N 6
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3141/
%R 10.5802/aif.3141
%G en
%F AIF_2017__67_6_2519_0
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. https://aif.centre-mersenne.org/articles/10.5802/aif.3141/

[1] Anick, David J. The smallest singularity of a Hilbert series, Math. Scand, Volume 51 (1982), pp. 35-44 | DOI | Zbl

[2] Ballmann, Werner; Ziller, Wolfgang On the number of closed geodesics on a compact riemannian manifold, Duke Math. J., Volume 49 (1982), pp. 629-632 | DOI | Zbl

[3] Félix, Yves; Halperin, Stephen; Jean-Claude, Thomas The homotopy Lie algebra for finite complexes, Publ. Math., Inst. Hautes Étud. Sci., Volume 56 (1983), pp. 387-410 | Zbl

[4] Félix, Yves; Halperin, Stephen; Jean-Claude, Thomas Rational Homotopy Theory, Graduate Texts in Mathematics, 205, Springer, 2001, xxxii+535 pages | Zbl

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

[6] Félix, Yves; Halperin, Stephen; Jean-Claude, Thomas Rational Homotopy II, World Scientific, 2015, xxxvi+412 pages | Zbl

[7] Gromov, Mikhael Homotopical effects of dilatations, J. Differ. Geom., Volume 13 (1978), pp. 303-310 | DOI | Zbl

[8] Halperin, Stephen; Jean-Michel, Lemaire Suites inertes dans les algèbres de Lie graduées, Math. Scand., Volume 61 (1987), pp. 39-67 | DOI | Zbl

[9] Halperin, Stephen; Levin, Gerson High skeleta of CW complexes, Algebra, Algebraic Topology and their Interactions (Stockholm 1983) (Lecture Notes in Math.), Volume 1183 (1986), pp. 211-217 | Zbl

[10] Lambrechts, Pascal Analytic properties of Poincaré series of spaces, Topology, Volume 37 (1998), pp. 1363-1370 | DOI | Zbl

[11] Lambrechts, Pascal The Betti numbers of the free loop space of a connected sum, J. Lond. Math. Soc., Volume 64 (2001), pp. 205-228 | DOI | Zbl

[12] Lambrechts, Pascal On the Betti numbers of the free loop space of a coformal space, J. Pure Appl. Algebra, Volume 161 (2001), pp. 177-192 | DOI | Zbl

[13] Mather, Michael Pull-backs in homotopy theory, Can. J. Math., Volume 28 (1976), pp. 225-263 | DOI | Zbl

[14] Milnor, John W.; Moore, J. On the structure of Hopf algebras, Ann. Math., Volume 81 (1965), pp. 211-264 | DOI | Zbl

[15] Sullivan, Dennis Infinitesimal Computations in Topology, Publ. Math., Inst. Hautes Étud. Sci., Volume 47 (1977), pp. 269-331 | DOI | Zbl

[16] Vigué-Poirrier, Micheline Homotopie rationnelle et nombre de géodésiques fermées, Ann. Sci. Éc. Norm. Supér., Volume 17 (1984), pp. 413-431 | DOI | Zbl

Cité par Sources :