Twisted Alexander Modules of Hyperplane Arrangement Complements

TL;DR

This paper investigates the torsion properties and divisibility of twisted Alexander modules of hyperplane arrangement complements, providing new computational methods and applications to distinguish non-homeomorphic arrangements.

Contribution

It introduces new divisibility results, computes specific twisted Alexander polynomials, and explores their roots and relations to homology jump loci in hyperplane arrangements.

Findings

Computed the first twisted Alexander polynomial of a punctured stratified tubular neighborhood.

Established divisibility properties between polynomials of the complement and neighborhood.

Applied results to distinguish non-homeomorphic arrangement complements.

Abstract

We study torsion properties of the twisted Alexander modules of the affine complement $M$ of a complex essential hyperplane arrangement, as well as those of punctured stratified tubular neighborhoods of complex essential hyperplane arrangements. We investigate divisibility properties between the twisted Alexander polynomials of the two spaces, compute the (first) twisted Alexander polynomial of a punctured stratified tubular neighborhood of an essential line arrangement, and study the possible roots of the twisted Alexander polynomials of both the complement and the punctured stratified tubular neighborhood of an essential hyperplane arrangement in higher dimensions. We apply our results to distinguish non-homeomorphic homotopy equivalent arrangement complements. We also relate the twisted Alexander polynomials of $M$ with the corresponding twisted homology jump loci.