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 ∗ ( Ω X ; Q ) is the universal enveloping algebra of a graded Lie algebra L X isomorphic with π ∗ − 1 ( X ) ⊗ Q . Let Q X ⊂ L X be a minimal generating subspace, and set α = lim sup i log rk π i ( X ) i . m: If dim L X = ∞ and lim sup ( dim ( Q X ) k ) 1 / k < lim sup dim ( L X ) k 1 / k , then ∑ i = 1 n − 1 rk π k + i ( X ) = e ( α + ε k ) k , where  ε k → 0  as  k → ∞ . In particular ∑ i = 1 n − 1 rk π k + i ( X ) grows exponentially in k.