An asymptotic formula for the ranks of homotopy groups
type: Item
instance of: scholarly article
title: An asymptotic formula for the ranks of homotopy groups
zbMATH Open document ID: 1105.55009
author: Stephen Halperin
author: Yves Félix
author: Jean-Claude Thomas
published in: Topology and its Applications
publication date: 2007-01-09T00:00:00Z
review text: If \(X\) is a finite simply connected CW-complex of dimension \(n\), then either \(\pi_i(X)\) is finite for \(i\geq 2n\) (the rationally elliptic case) or else for all \(k\geq 1\), \(\sum^{n-1}_{i=1} rk\,\pi_{k+i}(X)>0\) (the rationally hyperbolic case). The authors show that with an additional hypothesis this sum grows exponentially in \(k\). In a subsequent paper they intend to identify a large class of spaces for which the additional hypothesis holds, and they conjecture that the result holds for all finite simply connected CW-complexes. The proof is a reduction by Lie algebra arguments to facts about the growth and depth of the Lie algebra \(\pi_\ast(\Omega X)\otimes{\mathbb Q}\) established by the authors in previous papers.
Mathematics Subject Classification ID: 55Q99
Mathematics Subject Classification ID: 55P62
Mathematics Subject Classification ID: 55P35
Mathematics Subject Classification ID: 17B70
zbMATH DE Number: 5083213
zbMATH Keywords: homotopy Lie algebra
zbMATH Keywords: graded Lie algebra
zbMATH Keywords: exponential growth
reviewed by: Rainer Vogt
MaRDI profile type: Publication
full work available at URL: j.topol.2006.03.004
OpenAlex ID: W2079536353
cites work: On the chain algebra of a loop space
cites work: Q3892982
cites work: The Radical of the Homotopy Lie Algebra
cites work: Elliptic Hopf Algebras
cites work: The homotopy Lie algebra for finite complexes
cites work: Growth and Lie brackets in the homotopy Lie algebra
cites work: Graded Lie algebras with finite polydepth
cites work: Analytic properties of Poincaré series of spaces
DOI: 10.1016/J.TOPOL.2006.03.004
description: scientific article; zbMATH DE number 5083213