Exponential growth of Lie algebras of finite global dimension

Abstract

Let $X$ be a finite simply connected CW complex of dimension $n$. The loop space homology $H\_*(\Omega X;\mathbb Q)$ is the universal enveloping algebra of a graded Lie algebra $L\_X$ isomorphic with $ pi\_{*-1} (X)\otimes \mathbb Q$. Let $Q\_X \subset L\_X$ be a minimal generating subspace, and set $\alpha = \limsup\_i \frac{\log{\scriptsize rk} \pi\_i(X)}{i}$. Theorem: If ${dim} L\_X = \infty$ and $\limsup ({dim} (Q\_X)\_k)^{1/k} < \limsup ({dim} (L\_X)\_k)^{1/k}$ then $$\sum\_{i=1}^{n-1} {rk} \pi\_{k+i}(X) = e^{(\alpha + \epsilon\_k)k} \hspace{1cm} {where} \epsilon\_k \to 0 {as} k\to \infty.$$ In particular $\displaystyle\sum\_{i=1}^{n-1} {rk} \pi\_{k+i}(X)$ grows exponentially in $k$.