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The Ranks of the Homotopy Groups of a Finite Dimensional Complex


Published:20121113
Printed: Feb 2013
Yves Félix,
Université Catholique de Louvain, 1348, LouvainLaNeuve, Belgium
Steve Halperin,
University of Maryland, College Park, MD 207423281, USA
JeanClaude Thomas,
CNRS.UMR 6093Université d'Angers, 49045 Bd Lavoisier, Angers, France
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 $r$.