An asymptotic formula for the ranks of homotopy groups (Q860476)

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.