Exponential growth of Lie algebras of finite global dimension

Yves Félix (1) , Steve Halperin (2) , Jean-Claude Thomas (3)

1 UCL - Université Catholique de Louvain = Catholic University of Louvain
2 University of Maryland [College Park]
3 LAREMA - Laboratoire Angevin de Recherche en Mathématiques

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 $\alpha = \limsup_i \frac{\log\mbox{\scriptsize rk} \pi_i(X)}{i}$. Theorem: If $\mbox{dim}\, L_X = \infty$ and $\limsup (\mbox{dim} ( Q_X)_k)^{1/k} < \limsup (\mbox{dim} (L_X)_k)^{1/k}$ then $$\sum_{i=1}^{n-1} \mbox{rk} \pi_{k+i}(X) = e^{(\alpha + \varepsilon_k)k} \hspace{1cm} \mbox{where} \varepsilon_k \to 0 \mbox{as} k\to \infty.$$ In particular $\displaystyle\sum_{i=1}^{n-1} \mbox{rk} \pi_{k+i}(X)$ grows exponentially in $k$.

Mots clés: Homotopy Lie algebra; graded Lie algebra; exponential growth

Domaines: Topologie algébrique [math.AT]

Yves Félix, Steve Halperin, Jean-Claude Thomas. Exponential growth of Lie algebras of finite global dimension. 2005. ⟨hal-00008857⟩