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

Zbl 0504.55005
Full Text:

References:

 [1] J. F. Adams and P. J. Hilton, ”On the chain algebra of a loop space,” Comment. Math. Helv., vol. 30, pp. 305-330, 1956. · Zbl 0071.16403 [2] L. L. Avramov, ”Local algebra and rational homotopy,” in Algebraic Homotopy and Local Algebra, Paris: Soc. Math. France, 1984, vol. 113, pp. 15-43. · Zbl 0552.13003 [3] Y. Félix and S. Halperin, ”Rational LS category and its applications,” Trans. Amer. Math. Soc., vol. 273, iss. 1, pp. 1-38, 1982. · Zbl 0508.55004 [4] Y. Félix, S. Halperin, C. Jacobsson, C. Löfwall, and J. Thomas, ”The radical of the homotopy Lie algebra,” Amer. J. Math., vol. 110, iss. 2, pp. 301-322, 1988. · Zbl 0654.55011 [5] Y. Félix, S. Halperin, and J. Thomas, ”The homotopy Lie algebra for finite complexes,” Inst. Hautes Études Sci. Publ. Math., iss. 56, pp. 179-202 (1983), 1982. · Zbl 0504.55005 [6] Y. Félix, S. Halperin, and J. Thomas, ”Lie algebras of polynomial growth,” J. London Math. Soc., vol. 43, iss. 3, pp. 556-566, 1991. · Zbl 0755.57019 [7] Y. Félix, S. Halperin, and J. Thomas, Rational Homotopy Theory, New York: Springer-Verlag, 2001. · Zbl 0961.55002 [8] J. B. Friedlander and S. Halperin, ”An arithmetic characterization of the rational homotopy groups of certain spaces,” Invent. Math., vol. 53, iss. 2, pp. 117-133, 1979. · Zbl 0396.55010 [9] Y. Félix, S. Halperin, and J. Thomas, ”Growth and Lie brackets in the homotopy Lie algebra,” Homology Homotopy Appl., vol. 4, iss. 2, part 1, pp. 219-225 (electronic), 2002. · Zbl 1006.55008 [10] Y. Félix, S. Halperin, and J. Thomas, ”An asymptotic formula for the ranks of homotopy groups,” Topology Appl., vol. 153, iss. 18, pp. 3430-3436, 2006. · Zbl 1105.55009 [11] Y. Félix, S. Halperin, and J. Thomas, ”Exponential growth of Lie algebras of finite global dimension,” Proc. Amer. Math. Soc., vol. 135, iss. 5, pp. 1575-1578 (electronic), 2007. · Zbl 1111.55008 [12] Y. Félix, S. Halperin, and J. Thomas, ”The ranks of the homotopy groups of a space of finite LS category,” Expo. Math., vol. 25, iss. 1, pp. 67-76, 2007. · Zbl 1125.55005 [13] J. Koszul, ”Homologie et cohomologie des algèbres de Lie,” Bull. Soc. Math. France, vol. 78, pp. 65-127, 1950. · Zbl 0039.02901 [14] P. Lambrechts, ”Analytic properties of Poincaré series of spaces,” Topology, vol. 37, iss. 6, pp. 1363-1370, 1998. · Zbl 0935.55004 [15] J. P. May, ”The cohomology of restricted Lie algebras and of Hopf algebras,” J. Algebra, vol. 3, pp. 123-146, 1966. · Zbl 0163.03102 [16] J. W. Milnor and J. C. Moore, ”On the structure of Hopf algebras,” Ann. of Math., vol. 81, pp. 211-264, 1965. · Zbl 0163.28202 [17] D. Sullivan, ”Infinitesimal computations in topology,” Inst. Hautes Études Sci. Publ. Math., iss. 47, pp. 269-331 (1978), 1977. · Zbl 0374.57002 [18] J. H. van Lint and R. M. Wilson, A Course in Combinatorics, Cambridge: Cambridge University Press, 1992. · Zbl 0769.05001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.