# Double coverings of arrangement complements and $2$-torsion in Milnor fiber homology

## TL;DR

This paper shows that the mod 2 Betti numbers of double coverings of complex hyperplane arrangement complements are determined by combinatorial data, and demonstrates the existence of 2-torsion in the Milnor fiber homology for a specific arrangement.

## Contribution

It establishes a combinatorial method to compute mod 2 Betti numbers of certain coverings and reveals 2-torsion in Milnor fiber homology for the icosidodecahedral arrangement.

## Key findings

- Mod 2 Betti numbers are combinatorially determined.
- Non-trivial 2-torsion exists in the Milnor fiber homology.
- Application to the icosidodecahedral arrangement confirms the presence of 2-torsion.

## Abstract

We prove that the mod $2$ Betti numbers of double coverings of a complex hyperplane arrangement complement are combinatorially determined. The proof is based on a relation between the mod $2$ Aomoto complex and the transfer long exact sequence. Applying the above result to the icosidodecahedral arrangement ($16$ planes in the three dimensional space related to the icosidodecahedron), we conclude that the first homology of the Milnor fiber has non-trivial $2$-torsion.