Exponential growth and an asymptotic formula for the ranks of homotopy groups of a finite 1-connected complex

Abstract

Let $X$ be an $n$-dimensional, finite, simply connected CW complex and set $\alpha_X =\limsup_i (\log\text{ rank}\, \pi_i(X))/i$. We prove that either $\text{rank}\, \pi_i(X) = 0\,, i\geq 2n\,,$ or else that $0\lt \alpha_X\lt \infty$ and that for any $\varepsilon>0$ there is a $K=K(\varepsilon )$ such that \[e^{(\alpha_X -\varepsilon)k}\leq \sum_{i=k+2}^{k+n} \text{rank}\, \pi_i(X) \, \leq e^{(\alpha_X + \varepsilon)k}\,, \quad \mbox{for all } k\geq K\,. \] In particular, this sum grows exponentially in $k$.