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

TL;DR

This paper investigates the structure of higher homotopy groups of hypersolvable hyperplane arrangement complements, providing a combinatorial formula for their second nilpotent quotients and relating torsion to the arrangement's algebraic properties.

Contribution

It offers a new combinatorial description of the second nilpotent quotient of higher homotopy groups for hypersolvable arrangements, including explicit formulas and torsion analysis.

Findings

The second nilpotent I-adic quotient of _p is determined by arrangement combinatorics.

A combinatorial formula for ^1_I _p is provided.

For arrangements from finite simple graphs, ^1_I _p is torsion-free with computable rank.

Abstract

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, $\Z\pi_1$. 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, $\gr^1_I \pi_p$. We relate the torsion of this graded piece to the dimensions of the minimal generating systems of the Orlik--Solomon ideal of the arrangement $\A$ in degree $p+2$, for various field coefficients. When $\A$ is associated to a finite simple graph, we show that $\gr^1_I \pi_p$ is torsion--free, with rank explicitly computable from the graph.