Homotopy Lie algebra of the complements of subspace arrangements with geometric lattices

TL;DR

This paper investigates the homotopy Lie algebra of the complement of certain geometric arrangements, establishing an injective map from a free Lie algebra under specific conditions, thus advancing understanding of their algebraic topology.

Contribution

It proves that for geometric arrangements with rationally hyperbolic complements, an injective map exists from a free Lie algebra to their homotopy Lie algebra, revealing new algebraic structures.

Findings

Existence of an injective map from free Lie algebra to homotopy Lie algebra

Conditions under which the complement space is rationally hyperbolic

Advancement in understanding the algebraic topology of subspace arrangement complements

Abstract

Let A be a geometric arrangement such that codim(x) > 1 for every x in A. We prove that, if the complement space M(A) is rationally hyperbolic, then there exists an injective from a free Lie algebra L(u,v) to the homotopy Lie algebra of M(A).