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 $C^{l}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi>ℂ</mi></mrow><mrow><mi>l</mi></mrow></msup></math>$ is a finite set $A$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="bold-script">A</mi></math>$ of subspaces of $C^{l}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi>ℂ</mi></mrow><mrow><mi>l</mi></mrow></msup></math>$ . The complement space $M (A)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>M</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi mathvariant="bold-script">A</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ is $C^{l} ∖ \cup_{x \in A} x$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi>ℂ</mi></mrow><mrow><mi>l</mi></mrow></msup> <mo class="MathClass-bin">∖</mo><msub><mrow><mo class="MathClass-bin">∪</mo></mrow><mrow><mi>x</mi><mo class="MathClass-rel">∈</mo><mi mathvariant="bold-script">A</mi></mrow></msub><mi>x</mi></math>$ . If $M (A)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>M</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi mathvariant="bold-script">A</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ is elliptic, then the homotopy Lie algebra $π_{⋆} (Ω M (A)) \otimes Q$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>π</mi></mrow><mrow><mo class="MathClass-bin">⋆</mo></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><mi>Ω</mi><mi>M</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi mathvariant="bold-script">A</mi></mrow><mo class="MathClass-close">)</mo></mrow></mrow><mo class="MathClass-close">)</mo></mrow> <mo class="MathClass-bin">⊗</mo> <mi>ℚ</mi></math>$ is finitely generated. In this paper, we prove that if $A$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="bold-script">A</mi></math>$ is a geometric arrangement such that $M (A)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>M</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi mathvariant="bold-script">A</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ is a hyperbolic 1–connected space, then there exists an injective map $L (u, v) \to π_{⋆} (Ω M (A)) \otimes Q$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="double-struck">L</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>u</mi><mo class="MathClass-punc">,</mo><mi>v</mi></mrow><mo class="MathClass-close">)</mo></mrow> <mo class="MathClass-rel">→</mo> <msub><mrow><mi>π</mi></mrow><mrow><mo class="MathClass-bin">⋆</mo></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><mi>Ω</mi><mi>M</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi mathvariant="bold-script">A</mi></mrow><mo class="MathClass-close">)</mo></mrow></mrow><mo class="MathClass-close">)</mo></mrow> <mo class="MathClass-bin">⊗</mo> <mi>ℚ</mi></math>$ where $L (u, v)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="double-struck">L</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>u</mi><mo class="MathClass-punc">,</mo><mi>v</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ 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)