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The Ranks of the Homotopy Groups of a Finite Dimensional Complex

Published online by Cambridge University Press:  20 November 2018

Yves Félix
Affiliation:
Université Catholique de Louvain 1348, Louvain-La-Neuve, Belgium, e-mail: felix@math.ucl.ac.be
Steve Halperin
Affiliation:
University of Maryland, College Park, MD 20742-3281, USA, e-mail: shalper@umd.edu
Jean-Claude Thomas
Affiliation:
CNRS.UMR 6093-Université d'Angers, 49045 Bd Lavoisier, Angers, France, e-mail: jean-claude.thomas@univ-angers.fr
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Abstract

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Let $X$ be an $n$ -dimensional, finite, simply connected $\text{CW}$ complex and set

$${{\alpha }_{X}}\,=\,\underset{i}{\mathop{\lim \,\sup }}\,\frac{\log \,\text{rank}\,{{\pi }_{i}}\left( X \right)}{i}$$

When $0<{{\alpha }_{X}}<\infty $ , we give upper and lower bounds for $\sum\nolimits_{i=k+2}^{k+n}{\,\text{rank}}\text{ }{{\pi }_{i}}\left( X \right)$ for $k$ sufficiently large. We also show for any $r$ that $\alpha x$ can be estimated from the integers $\text{rk }{{\pi }_{i}}\left( X \right),\,i\,\le \,nr$ with an error bound depending explicitly on $r$ .

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2013

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