Higher Homotopy Groups of Complements of Complex Hyperplane Arrangements

Abstract

We generalize results of Hattori on the topology of complements of hyperplane arrangements, from the class of generic arrangements, to the much broader class of hypersolvable arrangements. We show that the higher homotopy groups of the complement vanish in a certain combinatorially determined range, and we give an explicit Zπ1-module presentation of πp, the first non-vanishing higher homotopy group. We also give a combinatorial formula for the π1-coinvariants of πp. For affine line arrangements whose cones are hypersolvable, we provide a minimal resolution of π2 and study some of the properties of this module. For graphic arrangements associated to graphs with no 3-cycles, the algorithm for computing π2 is purely combinatorial. The Fitting varieties associated to π2 may distinguish the homotopy 2-types of arrangement complements with the same π1, and the same Betti numbers in low degrees.