#### Volume 11, issue 1 (2011)

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Line arrangements and direct products of free groups

### Kristopher Williams

Algebraic & Geometric Topology 11 (2011) 587–604
##### Abstract

We show that if the fundamental groups of the complements of two line arrangements in the complex projective plane are isomorphic to the same direct product of free groups, then the complements of the arrangements are homotopy equivalent. For any such arrangement $\mathsc{A}$, we also construct an arrangement ${\mathsc{A}}^{\prime }$ such that ${\mathsc{A}}^{\prime }$ is a complexified-real arrangement, the intersection lattices of the arrangements are isomorphic, and the complements of the arrangements are diffeomorphic.

##### Keywords
line arrangement, fundamental group, hyperplane arrangement, direct product of free groups, homotopy type
##### Mathematical Subject Classification 2000
Primary: 52C30
Secondary: 32S22, 14F35