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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)
##### Keywords:
arrangement; presentation; fundamental group; planar graph
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