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The fundamental group of the complement of an arrangement of complex hyperplanes. (English) Zbl 0772.57001
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.

MSC:
57M05 Fundamental group, presentations, free differential calculus
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
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