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Exponential growth and an asymptotic formula for the ranks of homotopy groups of a finite 1-connected complex. (English) Zbl 1197.55006

Rational homotopy theorists have drawn a distinction between two types of finite complexes; “small” complexes typified by \(S^n\) and “big” complexes typified by \(S^n\vee S^n\). The present authors have called a finite complex elliptic if there is \(N\) such that \(\pi_i(X)\otimes\mathbb{Q}=0\) for \(i>N\) and hyperbolic if \(\pi_i(X)\otimes\mathbb{Q}\neq 0\) for infinitely many \(n\). In 1982 the present authors [Publ. Math., Inst. Hautes Étud. Sci. 56, 179–202 (1982; Zbl 0504.55005)] in fact showed that elliptic spaces are very small and hyperbolic ones are very large in ways not apparent from the definition. Roughly speaking they showed that if the rational homotopy of \(X\) is not finite then it grows exponentially.
In the paper under review the authors prove an extremely strong generalisation of the above result. For an \(n\)-dimensional, finite, simply-connected CW complex set
\[ \alpha_X=\limsup_{i\to\infty} \frac{\ln \text{rank}\, \pi_i(X)}{i}. \]
The main outcome of the paper is that, either \(\text{rank}\, \pi_i(X)=0, \;i\geq 2n\) or else that \(0<\alpha_X <\infty\) and that for any \(\varepsilon>0\), there exists \(K(\varepsilon)\) such that
\[ e^{(\alpha_X - \varepsilon)k} \leqslant \sum_{i=k+2}^{k+n} \text{rank}\, \pi_i(X) \leqslant e^{(\alpha_X + \varepsilon)k}\quad \text{for all\;} k > K(\varepsilon). \]
The proof is based on a series of algebraic results about a remarkable growth property of graded Lie algebras.

MSC:

55P62 Rational homotopy theory
55Q05 Homotopy groups, general; sets of homotopy classes

Citations:

Zbl 0504.55005
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References:

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