The fundamental group of the complement of an arrangement of complex hyperplanes (Q1209383)

Let \(A\) be an arrangement (i.e. a finite collection of affine hyperplanes) in a finite dimensional vector space \(V\) over the complex numbers. The purpose of this paper is to give a presentation (i.e. a set of generators and relations) for the fundamental group \(\Pi_ 1\) of the complement in \(V\) of the union of the members of \(A\). From general methods it suffices to consider the case \(V=C^ 2\). The answer was already known when \(A\) is restricted to be the complexification of a real arrangement and the presentation of \(\Pi_ 1\) was seen to be determined by a certain planar graph, namely the underlying real arrangement. In this paper this restriction is removed and the presentation is shown to be determined by a certain planar graph with additional structure. The number of generators in the presentation of \(\Pi_ 1\) is shown to be the same as the number of elements in \(A\) and an algorithm is given that produces the relations necessary to define the desired presentation.