The Ranks of the Homotopy Groups of a Finite Dimensional Complex

http://dx.doi.org/10.4153/CJM-2012-050-x
Canad. J. Math. 65(2013), 82-119

  Published:2012-11-13
 Printed: Feb 2013

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 $r$.