The homotopy Lie algebra of the complements of subspace arrangements with geometric lattices

Abstract

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