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Let X be an n-dimensional, finite, simply connected CW complex and set αX = lim supi (log rank πi (X))/i. We prove that either rank πi (X) = 0 , i > 2n , or else that 0 < αX < ∞ and that for any ε > 0 there is a K = K(ε) such that k+n e(αX-ε)k ≤ k+nΣ rank πi (X) e(αx+ε)k, for all k ≥ K. i=k+2 In particular, this sum grows exponentially in k

Exponential growth and an asymptotic formula for the ranks of homotopy groups of a finite 1-connected complex