On the Second Nilpotent Quotient of Higher Homotopy Groups, for Hypersolvable Arrangements

Abstract

We examine the first nonvanishing higher homotopy group, |$\pi _p$|⁠, of the complement of a hypersolvable, nonsupersolvable, 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, |${\rm gr}^1_I \pi _p$|⁠. We relate the torsion of this graded piece to the dimensions of the minimal generating systems of the Orlik–Solomon ideal of the arrangement |$\mathcal {A}$| in degree |$p+2$|⁠, for various field coefficients. When |$\mathcal {A}$| is associated to a finite simple graph, we show that |${\rm gr}^1_I \pi _p$| is torsion-free, with rank explicitly computable from the graph.