Even torsions in the homology group of the Milnor fiber boundary of hyperplane arrangements in $\mathbb{C}^3$

TL;DR

This paper investigates the torsion in the homology group of the Milnor fiber boundary of hyperplane arrangements in three complex dimensions, revealing new lower bounds on even torsion components based on combinatorial data.

Contribution

It establishes a lower bound on the number of even-order torsion summands in the first homology group, linking it to the Euler characteristic of the projectivized complement.

Findings

Number of even torsion summands ≥ Euler characteristic of the projectivized complement

Provides conditions under which the torsion structure can be bounded

Enhances understanding of the homology of Milnor fiber boundaries in complex arrangements

Abstract

We study the homology group of the Milnor fiber boundary of a hyperplane arrangement in $\mathbb{C}^{3}$. By the work of N\'emethi--Szil\'ard, the homeomorphism type of the Milnor fiber boundary is combinatorially determined, and an explicit formula for the first Betti number is known. However, the torsion part of the first homology group is poorly understood. In this paper, under some conditions, we prove that the number of even-order torsion summands of the first homology group is greater than or equal to the Euler characteristic of the projectivized complement.