Characterization of Line Arrangement for which the Fundamental Group of the Complement is a Direct Product of Free Groups

TL;DR

This paper proves a conjecture linking the structure of line arrangements' intersection lattices without cycles to their complement's fundamental group being a product of free groups.

Contribution

It establishes the equivalence between the absence of cycles in the intersection lattice and the fundamental group being a direct product of free groups.

Findings

Proves the conjecture that cycle-free intersection lattices correspond to free group products.

Clarifies the topological structure of line arrangements with acyclic intersection lattices.

Enhances understanding of the relationship between combinatorial arrangements and algebraic topology.

Abstract

Kwai Man Fan proved that if the intersection lattice of a line arrangement does not contain a cycle, then the fundamental group of its complement is a direct sum of infinite and cyclic free groups. He also conjectured that the converse is true as well. The main purpose of this paper is to prove this conjecture