Topology of factored arrangements of lines

Abstract

A real arrangement of affine lines is a finite family A \mathcal {A} of lines in R 2 {{\mathbf {R}}^2} . A real arrangement A \mathcal {A} of lines is said to be factored if there exists a partition Π = ( Π 1 , Π 2 ) \Pi = ({\Pi _1},{\Pi _2}) of A \mathcal {A} into two disjoint subsets such that the Orlik-Solomon algebra of A \mathcal {A} factors according to this partition. We prove that the complement of the complexification of a factored real arrangement of lines is a K ( π , 1 ) K(\pi ,1) space.