On the topology of the Milnor fibration of a hyperplane arrangement

TL;DR

This paper surveys the topology of hyperplane arrangement complements, Milnor fibrations, and boundary structures, highlighting the role of resonance varieties and combinatorial formulas for Betti numbers, with examples of non-homotopy equivalent Milnor fibers.

Contribution

It provides a comprehensive overview of the topology of hyperplane arrangements and introduces new examples distinguishing Milnor fibers by torsion points in characteristic varieties.

Findings

Resonance and characteristic varieties relate to orbifold fibrations.

A combinatorial formula for the first Betti number of Milnor fibers.

Examples of arrangements with non-homotopy equivalent Milnor fibers.

Abstract

This note is mostly an expository survey, centered on the topology of complements of hyperplane arrangements, their Milnor fibrations, and their boundary structures. An important tool in this study is provided by the degree 1 resonance and characteristic varieties of the complement, and their tight relationship with orbifold fibrations and multinets on the underlying matroid. In favorable situations, this approach leads to a combinatorial formula for the first Betti number of the Milnor fiber and the algebraic monodromy. We also produce a pair of arrangements for which the respective Milnor fibers have the same Betti numbers, yet are not homotopy equivalent: the difference is picked up by isolated torsion points in the higher-depth characteristic varieties.