The ranks of the homotopy groups of odd degree of a finite complex
(2015) Journal of Pure and Applied Algebra — Vol. 219, n° 3, p. 494-501 (2015)
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- 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.
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Félix, Y., Halperin, S., & Thomas, J.-C. (2015). The ranks of the homotopy groups of odd degree of a finite complex. Journal of Pure and Applied Algebra, 219(3), 494-501. https://doi.org/10.1016/j.jpaa.2014.05.008 (Original work published 2015)