Betti numbers and torsions in homology groups of double coverings

TL;DR

This paper explores the relationship between Betti numbers and torsion in homology groups of double coverings, establishing an equivalence between a specific inequality and the presence of 2-torsion in the Milnor fiber's homology.

Contribution

It proves that the strict inequality in Papadima-Suciu's cohomology bound is equivalent to the existence of 2-torsion in the Milnor fiber's first integral homology.

Findings

The strict inequality holds for the icosidodecahedral arrangement.

The first integral homology of the Milnor fiber has non-trivial 2-torsion.

The inequality and torsion presence are equivalent properties.

Abstract

Papadima and Suciu proved an inequality between the ranks of the cohomology groups of the Aomoto complex with finite field coefficients and the twisted cohomology groups, and conjectured that they are actually equal for certain cases associated with the Milnor fiber of the arrangement. Recently, an arrangement (the icosidodecahedral arrangement) with the following two peculiar properties was found: (i) the strict version of Papadima-Suciu's inequality holds, and (ii) the first integral homology of the Milnor fiber has a non-trivial $2$-torsion. In this paper, we investigate the relationship between these two properties for double covering spaces. We prove that (i) and (ii) are actually equivalent.