The homology groups of the Milnor fiber associated to a central arrangement of hyperplanes in $\CC^3$

TL;DR

This paper investigates the homology groups of Milnor fibers linked to hyperplane arrangements in three-dimensional complex space, providing bounds and exact calculations under specific combinatorial conditions.

Contribution

It introduces a combinatorial upper bound for the first homology group rank of Milnor fibers and determines exact ranks in certain cases, proving torsion freeness.

Findings

Established a combinatorially determined upper bound for the homology rank.

Identified conditions under which the homology rank can be exactly computed.

Proved the homology group is torsion free under certain combinatorial conditions.

Abstract

We use covering space theory and the fundamental group of complements of complexified-real line arrangements to explore the associated Milnor fiber. This work yields a combinatorially determined upper bound on the rank of the first homology group of the Milnor fiber. Under certain combinatorial conditions, we then show that one may determine the exact rank of the group and show that it is torsion free.