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arXiv: math.AT/0509546

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Exponential growth of Lie algebras of finite global dimension

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

(2005)

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 $k$.

1:	Université Catholique de Louvain (UCL)
	Université Catholique de Louvain (UCL) - Belgique
2:	University of Maryland
	University of Maryland
3:	Laboratoire Angevin de REcherche en MAthématiques (LAREMA)
	CNRS : UMR6093 – Université d'Angers

Subject

:

Mathematics/Algebraic Topology

Keyword(s): Homotopy Lie algebra – graded Lie algebra – exponential growth

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From: Secrétariat Math. Angers <>

Submitted on: Monday, 19 September 2005 12:08:15

Updated on: Friday, 23 September 2005 12:54:15