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 Zπ_1-module presentation of π_p, the first non-vanishing higher homotopy group. We also give a combinatorial formula for the π_1-coinvariants of π_p. For affine line arrangements whose cones are hypersolvable, we provide a minimal resolution of π_2, and study some of the properties of this module. For graphic arrangements associated to graphs with no 3-cycles, we obtain information on π_2, directly from the graph. The π_1-coinvariants of π_2 may distinguish the homotopy 2-types of arrangement complements with the same π_1, and the same Betti numbers in low degrees.