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

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, 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, 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 A in degree p+2, for various field coefficients. When A is associated to a finite simple graph, we show that gr(I)(1)pi(p) is torsion-free, with rank explicitly computable from the graph.