Cohomology rings and nilpotent quotients of real and complex arrangements

For an arrangement with complement X and fundamental group G, we relate the truncated cohomology ring, H �2 (X), to the second nilpotent quotient, G/G3. We define invariants of G/G3 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� 6 planes in R 4 .