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# The Ranks of the Homotopy Groups of a Finite Dimensional Complex

Published:2012-11-13
Printed: Feb 2013
• Yves Félix,
Université Catholique de Louvain, 1348, Louvain-La-Neuve, Belgium
• Steve Halperin,
University of Maryland, College Park, MD 20742-3281, USA
• Jean-Claude Thomas,
CNRS.UMR 6093-Université d'Angers, 49045 Bd Lavoisier, Angers, France
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## Abstract

Let $X$ be an $n$-dimensional, finite, simply connected CW complex and set $\alpha_X =\limsup_i \frac{\log\mbox{ rank}\, \pi_i(X)}{i}$. When $0\lt \alpha_X\lt \infty$, we give upper and lower bound for $\sum_{i=k+2}^{k+n} \textrm{rank}\, \pi_i(X)$ for $k$ sufficiently large. We show also for any $r$ that $\alpha_X$ can be estimated from the integers rk$\,\pi_i(X)$, $i\leq nr$ with an error bound depending explicitly on $r$.
 Keywords: homotopy groups, graded Lie algebra, exponential growth, LS category
 MSC Classifications: 55P35 - Loop spaces 55P62 - Rational homotopy theory 17B70 - Graded Lie (super)algebras