On the second nilpotent quotient of higher homotopy groups, for hypersolvable arrangements

We examine the first non-vanishing higher homotopy group, ${\pi }_p$, of the complement of a hypersolvable, non–supersolvable, complex hyperplane arrangement, as a module over the group ring of the fundamental group, $\mathbb{Z}{\pi }_1$. We give a presentation for the $I$–adic completion of ${\pi }_p$. We deduce that the second nilpotent $I$–adic quotient of ${\pi }_p$ is determined by the combinatorics of the arrangement, and we give a combinatorial formula for the second associated graded piece, ${\text{gr}}_I^1{\pi }_p$. We relate the torsion of this graded piece to the dimensions of the minimal generating systems of the Orlik–Solomon ideal of the arrangement $\text{A}$ in degree $p+2$, for various field coefficients. When $\text{A}$ is associated to a finite simple graph, we show that ${\text{gr}}_I^1{\pi }_p$ is torsion–free, with rank explicitly computable from the graph.