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 $k$.
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TITLE = {On the chain algebra of a loop space},
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AUTHOR = {Avramov, Luchezar L.},
TITLE = {Local algebra and rational homotopy},
BOOKTITLE = {Algebraic Homotopy and Local Algebra},
VENUE = {{L}uminy, 1982},
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ISBN = {0-521-41057-6; 0-521-42260-4},
MRCLASS = {05-01 (05Axx 05Exx 90B10 94B25)},
MRNUMBER = {94g:05003},
MRREVIEWER = {George E. Andrews},
}