## Algebraic & Geometric Topology

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

Gery Debongnie

#### Abstract

A subspace arrangement in $ℂl$ is a finite set $A$ of subspaces of $ℂl$. The complement space $M(A)$ is $ℂl∖∪x∈Ax$. If $M(A)$ is elliptic, then the homotopy Lie algebra $π⋆(ΩM(A))⊗ℚ$ is finitely generated. In this paper, we prove that if $A$ is a geometric arrangement such that $M(A)$ is a hyperbolic 1–connected space, then there exists an injective map $L(u,v)→π⋆(ΩM(A))⊗ℚ$ where $L(u,v)$ denotes a free Lie algebra on two generators.

#### Article information

Source
Algebr. Geom. Topol., Volume 7, Number 4 (2007), 2007-2020.

Dates
Revised: 9 October 2007
Accepted: 25 October 2007
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796781

Digital Object Identifier
doi:10.2140/agt.2007.7.2007

Mathematical Reviews number (MathSciNet)
MR2366186

Zentralblatt MATH identifier
1144.55013

Subjects
Primary: 55P62: Rational homotopy theory

#### Citation

Debongnie, Gery. The homotopy Lie algebra of the complements of subspace arrangements with geometric lattices. Algebr. Geom. Topol. 7 (2007), no. 4, 2007--2020. doi:10.2140/agt.2007.7.2007. https://projecteuclid.org/euclid.agt/1513796781

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