An asymptotic formula for the ranks of homotopy groups

Let X be a finite simply connected CW complex of dimension n. The loop space homology H-*(Omega X;Q) is the universal enveloping algebra of a graded Lie algebra L-X isomorphic with pi(*-1) (X) circle times Q. Let Q(X) subset of L-X be a minimal generating subspace, and set alpha = lim sup(i) logrk pi(i)(X)/i. Theorem: If dim L-X = infinity and lim sup(dim(Q(X))(k))(1/k) < lim sup dim(L-X)(k)(1/k), then [GRAPHICS] In particular Sigma(n-1)(i=1) rk pi(k+i) (X) grows exponentially in k. (C) 2006 Elsevier B.V. All rights reserved.