Lower central series and free resolutions of hyperplane arrangements

Abstract:

If $M$ is the complement of a hyperplane arrangement, and $A=H^*(M,\Bbbk )$ is the cohomology ring of $M$ over a field $\Bbbk$ of characteristic $0$, then the ranks, $\phi _k$, of the lower central series quotients of $\pi _1(M)$ can be computed from the Betti numbers, $b_{ii}=\dim \operatorname {Tor}^A_i(\Bbbk ,\Bbbk )_i$, of the linear strand in a minimal free resolution of $\Bbbk$ over $A$. We use the Cartan-Eilenberg change of rings spectral sequence to relate these numbers to the graded Betti numbers, $b’_{ij}=\dim \operatorname {Tor}^E_i(A,\Bbbk )_{j}$, of a minimal resolution of $A$ over the exterior algebra $E$. From this analysis, we recover a formula of Falk for $\phi _3$, and obtain a new formula for $\phi _4$. 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 if and only if the arrangement is supersolvable. We also give combinatorial lower bounds on the Betti numbers, $b’_{i,i+1}$, 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 subarrangements), we show that $b’_{i,i+1}$ is determined by the number of triangles and $K_4$ subgraphs in the graph.