The Extended Picture Group , with Applications to Line Arrangement Complements

We obtain the picture group as the quotient with a torsion subgroup, of an extended picture group, which is isomorphic to the kernel of a precrossed module homomorphism. In addition to expanding the notion of a picture group, the new formulation gives a natural way to construct homomorphisms between picture groups by describing deformations of one-vertex subpictures. The extended picture group thus provides a convenient way to describe generators for the second homotopy group of line arrangement complements as well as homomorphisms between these groups. In particular, we show that the homomorphisms relate to a lattice structure corresponding roughly to the condition of being more nearly in general position. Examples include generators for Falk’s X2 arrangement and for a generic section of braid arrangement A3. Finally, we demonstrate that the C arrangement C(5) is a K(π, 1) space.