Even torsions in the homology group of the Milnor fiber boundary of hyperplane arrangements in $\mathbb{C}^3$
Sakumi Sugawara
TL;DR
The paper addresses torsion in the first homology of the Milnor fiber boundary for hyperplane arrangements in $\mathbb{C}^3$, a topic with a known combinatorial description of the boundary but poorly understood torsion. It develops a framework based on a tower of double coverings and mod-2 resonance (Aomoto-Betti numbers) for the boundary manifolds, proving a lower bound on the number of even-order torsion summands: when the number of hyperplanes is a power of two and a codimension-two multiplicity condition holds, $\dim_{\mathbb{Z}_2} (\operatorname{Tor}(H_1(\partial F);\mathbb{Z})\otimes \mathbb{Z}_2) \ge \chi(U)$. The result connects the torsion structure to the Euler characteristic of the projectivized complement and generalizes known results for generic arrangements, with equality in the generic case. The paper combines doubling techniques for cohomology rings with resonance theory and leverages recent bounds on covering-space homology, supported by concrete eight-hyperplane examples. Overall, it advances explicit torsion calculations in Milnor fiber boundaries and clarifies how combinatorial data governs 2-torsion phenomena in these boundary manifolds.
Abstract
We study the homology group of the Milnor fiber boundary of a hyperplane arrangement in $\mathbb{C}^{3}$. By the work of Némethi--Szilárd, 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.