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∗(ΩX;Q)H∗(ΩX;Q) is the universal enveloping algebra of a graded Lie algebra LXLX isomorphic with π∗−1(X)⊗Qπ∗−1(X)⊗Q. Let QX⊂LXQX⊂LX be a minimal generating subspace, and set α=lim supilogrkπi(X)i.Theorem: If dimLX=∞dimLX=∞ and lim sup(dim(QX)k)1/k