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

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, $\mathbb{Z}{\pi }_1$. 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, $\gr^1_I π_p$. 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 $\gr^1_I π_p$ is torsion--free, with rank explicitly computable from the graph.

11 pages, updated references

References in corpus (1)

• The spectral sequence of an equivariant chain complex and homology with local coefficients