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The Ranks of the Homotopy Groups of a Finite Dimensional Complex
Published online by Cambridge University Press: 20 November 2018
Abstract
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
$r$.
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- Research Article
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- Copyright
- Copyright © Canadian Mathematical Society 2013