Geodesic
The Ranks of the Homotopy Groups of a Finite Dimensional Complex
Canadian journal of mathematics, Tome 65 (2013) no. 1, pp. 82-119

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DOI

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 $r$ .
DOI : 10.4153/CJM-2012-050-x
Mots-clés : 55P35, 55P62, 17B70, homotopy groups, graded Lie algebra, exponential growth, LS category 
Félix, Yves; Halperin, Steve; Thomas, Jean-Claude. The Ranks of the Homotopy Groups of a Finite Dimensional Complex. Canadian journal of mathematics, Tome 65 (2013) no. 1, pp. 82-119. doi: 10.4153/CJM-2012-050-x
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     title = {The {Ranks} of the {Homotopy} {Groups} of a {Finite} {Dimensional} {Complex}},
     journal = {Canadian journal of mathematics},
     pages = {82--119},
     year = {2013},
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