Geodesic
Exponential growth and an asymptotic formula for the ranks of homotopy groups of a finite 1-connected complex
Yves Felix1 ; Steve Halperin2 ; Jean-Claude Thomas3
1 Département de Mathématique<br/>Université catholique de Louvain<br/>Bât. M. de Hemptinne<br/>Chemin du Cyclotron, 2<br/>1348 Louvain-la-Neuve<br/>Belgium
2 College of Computer, Mathematical and Physical Sciences<br/>AV Williams Building<br/>University of Maryland<br/>College Park, MD 20742<br/>United States
3 Université d’Angers and CNRS<br/>2 Bd Lavoisier<br/>49045 Angers Cedex<br/>France
Annals of mathematics, Tome 170 (2009) no. 1, pp. 443-464.

Voir la notice de l'article provenant de la source Annals of Mathematics website

Let $X$ be an $n$-dimensional, finite, simply connected CW complex and set $\alpha_X =\limsup_i (\log\text{ rank}\, \pi_i(X))/i$. We prove that either $\text{rank}\, \pi_i(X) = 0\,, i\geq 2n\,,$ or else that $0\lt \alpha_X\lt \infty$ and that for any $\varepsilon>0$ there is a $K=K(\varepsilon )$ such that \[e^{(\alpha_X -\varepsilon)k}\leq \sum_{i=k+2}^{k+n} \text{rank}\, \pi_i(X) \, \leq e^{(\alpha_X + \varepsilon)k}\,, \quad \mbox{for all } k\geq K\,. \] In particular, this sum grows exponentially in $k$.
DOI : 10.4007/annals.2009.170.443

Yves Felix 1 ; Steve Halperin 2 ; Jean-Claude Thomas 3

1 Département de Mathématique<br/>Université catholique de Louvain<br/>Bât. M. de Hemptinne<br/>Chemin du Cyclotron, 2<br/>1348 Louvain-la-Neuve<br/>Belgium
2 College of Computer, Mathematical and Physical Sciences<br/>AV Williams Building<br/>University of Maryland<br/>College Park, MD 20742<br/>United States
3 Université d’Angers and CNRS<br/>2 Bd Lavoisier<br/>49045 Angers Cedex<br/>France 
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Yves Felix; Steve Halperin; Jean-Claude Thomas. Exponential growth and an asymptotic formula for the ranks of homotopy groups of a finite 1-connected complex. Annals of mathematics, Tome 170 (2009) no. 1, pp. 443-464. doi : 10.4007/annals.2009.170.443. http://geodesic.mathdoc.fr/articles/10.4007/annals.2009.170.443/

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