#### Volume 7, issue 4 (2007)

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
The homotopy Lie algebra of the complements of subspace arrangements with geometric lattices

### Gery Debongnie

Algebraic & Geometric Topology 7 (2007) 2007–2020
##### Abstract

A subspace arrangement in ${ℂ}^{l}$ is a finite set $\mathsc{A}$ of subspaces of ${ℂ}^{l}$. The complement space $M\left(\mathsc{A}\right)$ is ${ℂ}^{l}\setminus {\cup }_{x\in \mathsc{A}}x$. If $M\left(\mathsc{A}\right)$ is elliptic, then the homotopy Lie algebra ${\pi }_{\star }\left(\Omega M\left(\mathsc{A}\right)\right)\otimes ℚ$ is finitely generated. In this paper, we prove that if $\mathsc{A}$ is a geometric arrangement such that $M\left(\mathsc{A}\right)$ is a hyperbolic 1–connected space, then there exists an injective map $\mathbb{L}\left(u,v\right)\to {\pi }_{\star }\left(\Omega M\left(\mathsc{A}\right)\right)\otimes ℚ$ where $\mathbb{L}\left(u,v\right)$ denotes a free Lie algebra on two generators.

##### Keywords
homotopy Lie algebra, Subspace arrangements
Primary: 55P62