The ranks of the homotopy groups of odd degree of a finite complex

Abstract

Let L be a graded connected Lie algebra of finite type and finite depth (for instance the rational homotopy Lie algebra of a finite simply connected CW complex). Let L(p)={Lpk}k≥1. Then for any prime p, limn⁡log⁡dim⁡L(p)≤nlog⁡dim⁡L≤n=1. In particular for a space X, the Lie algebra LX=π⁎(ΩX)⊗Q and its even dimensional part LX(2) have the same log index.