Milnor fibrations and oriented matroids

TL;DR

This paper develops a combinatorial model for Milnor fibrations of complexified real arrangements using oriented matroids, providing a new perspective that depends solely on combinatorial data and applies to all oriented matroids.

Contribution

It introduces a finite regular CW complex model for Milnor fibers based on oriented matroids, extending the concept to non-realizable cases and showing homotopy equivalence with the classical Milnor fiber.

Findings

The model is a poset quasi-fibration derived from the Salvetti complex.

Homotopy type of the Milnor fiber depends only on the oriented matroid.

The construction applies to all oriented matroids, real or not.

Abstract

We introduce a combinatorial model for the Milnor fibration of a complexified real arrangement using oriented matroids. It is a poset quasi-fibration, a notion recently introduced by the first author, whose domain is a subdivision of the Salvetti complex stemming from a natural subdivision of the dual oriented matroid complex. This yields a concrete finite regular CW complex which is homotopy equivalent to the Milnor fiber of the complexified real arrangement and implies that the homotopy type of the Milnor fiber of a complexified real arrangement only depends on the underlying combinatorial structure given by its oriented matroid. Moreover, our construction works for any oriented matroid, disregarding realizability, so we obtain a notion of a combinatorial Milnor fibration for any oriented matroid.