Lower central series and free resolutions of hyperplane arrangements

Abstract

If $M$ is the complement of a hyperplane arrangement, and $A=H^*(M,\k)$ is the cohomology ring of $M$ over a field of characteristic 0, then the ranks, ${\varphi }_k$, of the lower central series quotients of ${\pi }_1(M)$ can be computed from the Betti numbers, $b_{ii}=\dim_{\k} \Tor^A_i(\k,\k)_i$, of the linear strand in a (minimal) free resolution of $\k$ over $A$. We use the Cartan-Eilenberg change of rings spectral sequence to relate these numbers to the graded Betti numbers, $b'_{ij}=\dim_{\k} \Tor^E_i(A,\k)_j$, of a (minimal) resolution of $A$ over the exterior algebra $E$. From this analysis, we recover a formula of Falk for ${\varphi }_3$, and obtain a new formula for ${\varphi }_4$. The exact sequence of low degree terms in the spectral sequence allows us to answer a question of Falk on graphic arrangements, and also shows that for these arrangements, the algebra $A$ is Koszul iff the arrangement is supersolvable. We also give combinatorial lower bounds on the Betti numbers, $b_{i,i+1}^{\prime }$, of the linear strand of the free resolution of $A$ over $E$; if the lower bound is attained for $i=2$, then it is attained for all $i\ge 2$. For such arrangements, we compute the entire linear strand of the resolution, and we prove that all components of the first resonance variety of $A$ are local. For graphic arrangements (which do not attain the lower bound, unless they have no braid sub-arrangements), we show that $b_{i,i+1}^{\prime }$ is determined by the number of triangles and $K_4$ subgraphs in the graph.

25 pages, to appear in Trans. Amer. Math. Soc

References in corpus (2)

• Fundamental groups of line arrangements: Enumerative aspects
• Quadratic Orlik-Solomon algebras of graphic matroids

Cited by in corpus (24)

• Chen Lie algebras
• Resonance, linear syzygies, Chen groups, and the Bernstein-Gelfand-Gelfand correspondence
• When does the associated graded Lie algebra of an arrangement group decompose?
• Broken circuit complexes and hyperplane arrangements
• Homotopy Lie algebras, lower central series and the Koszul property
• Chen ranks and resonance
• The pure braid groups and their relatives
• Homological properties of Orlik-Solomon algebras
• The line geometry of resonance varieties
• Topological Criteria for $k-$Formal Arrangements
• Homology, lower central series, and hyperplane arrangements
• Holonomy Lie algebras and the LCS formula for subarrangements of A_n
• Koszulness and supersolvability for Dirichlet arrangements
• Electrical networks and hyperplane arrangements
• On the second nilpotent quotient of higher homotopy groups, for hypersolvable arrangements
• Conjugation-free groups, lower central series and line arrangements
• Resonance varieties over fields of positive characteristic
• On the Falk invariant of signed graphic arrangements
• On the Falk invariant of Shi and Linial arrangements
• On the Falk invariant of hyperplane arrangements attached to gain graphs
• Complex hyperplane arrangements
• Note on resonance varieties
• The Orlik-Terao algebra and 2-formality
• Efficient computation of resonance varieties via Grassmannians