Cohomology rings and nilpotent quotients of real and complex arrangements

Abstract

For an arrangement with complement $X$ and fundamental group $G$, we relate the truncated cohomology ring, $H^{\le2} (X)$, to the second nilpotent quotient, $G/G_3$. We define invariants of $G/G_3$ by counting normal subgroups of a fixed prime index $p$, according to their abelianization. We show how to compute this distribution from the resonance varieties of the Orlik–Solomon algebra mod $p$. As an application, we establish the cohomology classification of 2-arrangements of $n \le 6$ planes in ${\mathbb{R}}^4$.