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

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).

💡 Research Summary

The paper investigates the rational homotopy type of complements of complex subspace arrangements whose intersection lattice is geometric and whose constituent subspaces all have codimension greater than one. The central object of study is the homotopy Lie algebra 𝔏(M(A)) of the complement M(A)=ℂⁿ∖⋃_{x∈A}x, a graded Lie algebra that encodes the rational homotopy groups of M(A) together with their Whitehead products. The authors focus on the dichotomy between rationally elliptic and rationally hyperbolic spaces: a space is rationally hyperbolic if the dimensions of its rational homotopy groups grow exponentially, which is reflected algebraically by the presence of a “large” free Lie subalgebra inside 𝔏(M(A)).

The main theorem states: If A is a geometric arrangement with codim(x)>1 for every subspace x∈A and the complement M(A) is rationally hyperbolic, then there exists an injective Lie algebra homomorphism from the free Lie algebra on two generators L(u,v) into the homotopy Lie algebra 𝔏(M(A)). In other words, the rational hyperbolicity of the complement forces the homotopy Lie algebra to contain a copy of the simplest non‑abelian free Lie algebra.

The proof proceeds through several classical tools of rational homotopy theory. First, the authors construct a Sullivan minimal model (ΛV,d) for M(A). The geometric lattice condition guarantees that the differential d respects the combinatorial structure of the arrangement, while the codimension hypothesis ensures that the generators in V of degree one are insufficient to model the space; higher‑degree generators appear in abundance. Next, they analyze triple Massey products in the cohomology of M(A). Rational hyperbolicity implies the existence of non‑trivial higher order Massey products, which in turn correspond to non‑trivial Lie brackets in 𝔏(M(A)). By selecting two suitable cohomology classes whose Massey product is non‑zero, the authors produce two elements u and v in 𝔏(M(A)) whose iterated brackets generate a free Lie subalgebra.

A key technical step is showing that no hidden relations among u, v, and their brackets arise from the differential d. This is achieved by a careful examination of the lower central series of 𝔏(M(A)) and by comparing the growth of the Betti numbers of the minimal model with the known growth rates of free Lie algebras. The authors verify that the map L(u,v)→𝔏(M(A)) is injective by demonstrating that any linear combination of Lie monomials in u and v that would vanish in 𝔏(M(A)) would force a contradiction with the non‑triviality of the Massey product.

Several examples illustrate the theorem. For instance, the complement of three generic planes in ℂ³ satisfies the geometric lattice condition, has codimension two for each plane, and is known to be rationally hyperbolic; consequently, its homotopy Lie algebra contains a free Lie algebra on two generators. Conversely, arrangements where some subspace has codimension one (e.g., a line arrangement in ℂ²) fail the hypothesis, and their complements are rationally elliptic, so the conclusion does not hold.

The paper concludes by discussing the broader implications of the result. The existence of a free Lie subalgebra signals a rich and complicated rational homotopy structure, suggesting that many invariants (such as the ranks of rational homotopy groups or the growth of the lower central series) are unbounded. The authors propose future directions, including extending the method to detect larger free Lie algebras (e.g., on three or more generators), investigating arrangements with non‑geometric lattices, and exploring connections with the theory of Koszul algebras and the formality of arrangement complements. Overall, the work bridges combinatorial properties of subspace arrangements with deep algebraic features of their topological complements, providing a clear criterion that guarantees the presence of non‑trivial free Lie structure in rational homotopy.