Hostname: page-component-7d684dbfc8-hffkp Total loading time: 0 Render date: 2023-09-24T12:33:29.617Z Has data issue: false Feature Flags: { "corePageComponentGetUserInfoFromSharedSession": true, "coreDisableEcommerce": false, "coreDisableSocialShare": false, "coreDisableEcommerceForArticlePurchase": false, "coreDisableEcommerceForBookPurchase": false, "coreDisableEcommerceForElementPurchase": false, "coreUseNewShare": true, "useRatesEcommerce": true } hasContentIssue false

The Ranks of the Homotopy Groups of a Finite Dimensional Complex

Published online by Cambridge University Press:  20 November 2018

Yves Félix
Université Catholique de Louvain 1348, Louvain-La-Neuve, Belgium, e-mail:
Steve Halperin
University of Maryland, College Park, MD 20742-3281, USA, e-mail:
Jean-Claude Thomas
CNRS.UMR 6093-Université d'Angers, 49045 Bd Lavoisier, Angers, France, e-mail:
Rights & Permissions [Opens in a new window]


Core share and HTML view are not possible as this article does not have html content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 $r$.

Research Article
Copyright © Canadian Mathematical Society 2013


[1] Adams, J. F. and Hilton, P. J., On the chain algebra of a loop space. Comment. Math. Helvetici. 30(1956), 305330. Google Scholar
[2]Anick, D. J., The smallest singularity of a Hilbert series. Math. Scand. 51(1982), no. 1, 3544.Google Scholar
[3]Babenko, I. K., Analytical properties of Poincaré series of a loop space. Mat. Zametki. 27 (1980), no. 5, 751765, 830.Google Scholar
[4] Félix, Y. and Halperin, S., Rational LS category and its applications. Trans. Amer. Math. Soc. 273(1982), no. 1, 138.Google Scholar
[5] Félix, Y., Halperin, S., and Thomas, J.-C., The homotopy Lie algebra for finite complexes. Inst. Hautes Études Sci. Publ. Math. 56(1983), 179202.Google Scholar
[6] Thomas, J.-C., Rational homotopy theory. Graduate Texts in Mathematics, 205, Springer-Verlag, New York, 2001.Google Scholar
[7] Thomas, J.-C., Exponential growth and an asymptoptic formula for the ranks of homotopy groups of a finite1-connected complex. Ann. of Math. 170(2009), no. 1, 443464. Google Scholar
[8] Thomas, J.-C., Lie algebra of polynomial growth. J. London Math. Soc. 43(1991), no. 3, 556566. Google Scholar
[9] Friedlander, J. and Halperin, S., An arithmetic characterization of the rational homotopy groups of certain spaces. Invent. Math. 53(1979), no. 2, 117133. Google Scholar
[10] Gelfand, I. M. and Kirillov, A. A., Sur les corps liés aux algèbres enveloppantes des algèbres de Lie. Inst. Hautes Études Sci. Publ. Math. 31(1966),519 Google Scholar
[11]Gottlieb, D., Evaluation subgroups of homotopy groups. Amer. J. Math. 91(1969), 729756. Google Scholar
[12]Halperin, S., Torsion gaps in the homotopy of finite complexes. II. Topology 30(1991), no. 3, 471478. Google Scholar
[13]Krause, G. R. and Lenagan, T. H., Growth of algebras and Gel’fand-Kirillov dimension. Research Notes in Mathematics, 116, Pitman, Boston, MA, 1985.Google Scholar
[14] Lambrechts, P., Analytic properties of Poincaré series of spaces. Topology 37(1998), no. 6, 13631370. Google Scholar
[15] Lambrechts, P., On the ranks of homotopy groups of two-cones. Topology Appl. 108(2000), no. 3, 303314. Google Scholar
[16]Milnor, J. and Moore, J. C., On the structure of Hopf algebras. Annals of Math. 81(1965), 211264. Google Scholar
[17] van Lint, J. H. and Wilson, R.M., A course in combinatorics. Second ed., Cambridge University Press, Cambridge, 2001.CrossRefGoogle Scholar