Line arrangements and direct sums of free groups

TL;DR

This paper proves that line arrangements with isomorphic fundamental groups of their complements, which are direct sums of free groups, have homotopy equivalent complements, and constructs real arrangements with isomorphic intersection lattices and diffeomorphic complements.

Contribution

It establishes a link between the algebraic structure of fundamental groups and the topological and geometric properties of line arrangement complements, including homotopy equivalence and diffeomorphism.

Findings

Fundamental groups determine homotopy type of complements.

Constructs real arrangements with same intersection lattice.

Shows diffeomorphic complements for arrangements with isomorphic fundamental groups.

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 sum of free groups, then the complements of the arrangements are homotopy equivalent. For any such arrangement, we construct another arrangement that is complexified-real, the intersection lattices of the arrangements are isomorphic, and the complements of the arrangements are diffeomorphic.