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

Debongnie, Gery

doi:10.2140/agt.2007.7.2007

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

Gery Debongnie

Algebraic & Geometric Topology 7 (2007) 2007–2020

DOI: 10.2140/agt.2007.7.2007

Abstract

A subspace arrangement in $ℂ^{l}$ is a finite set $A$ of subspaces of $ℂ^{l}$ . The complement space $M (A)$ is $ℂ^{l} ∖ \cup_{x \in A} x$ . If $M (A)$ is elliptic, then the homotopy Lie algebra $π_{⋆} (Ω M (A)) \otimes ℚ$ 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 $L (u, v) \to π_{⋆} (Ω M (A)) \otimes ℚ$ where $L (u, v)$ denotes a free Lie algebra on two generators.

Keywords

homotopy Lie algebra, Subspace arrangements

Mathematical Subject Classification 2000

Primary: 55P62

References

Publication

Received: 10 May 2007

Revised: 9 October 2007

Accepted: 25 October 2007

Published: 26 December 2007

Authors

Gery Debongnie
UCL, Departement de mathematique Chemin du Cyclotron, 2 B-1348 Louvain-la-neuve Belgium
http://www.math.ucl.ac.be/membres/debongnie/

Volume 7, issue 4 (2007)