The Milnor fiber boundary of an arrangement determines its combinatorics

TL;DR

This paper proves that the boundary of the Milnor fiber uniquely determines the combinatorial structure of a complex line arrangement, establishing a key link between topology and combinatorics.

Contribution

It confirms the conjecture that the Milnor fiber boundary encodes the arrangement's combinatorics and provides a method to reconstruct the arrangement from its plumbing graph.

Findings

The boundary of the Milnor fiber is a combinatorial invariant.

The boundary determines the arrangement's combinatorics.

An explicit reconstruction method from plumbing graphs is provided.

Abstract

The boundary of the Milnor fiber associated with a complex line arrangement is a three dimensional plumbed manifold, and it is a combinatorial invariant. We prove the reverse implication, which was conjectured N\'emethi and Szil\'ard. That is, this boundary of the Milnor fiber determines the combinatorics of the arrangement. Furthermore, we give an explicit method which constructs the poset associated with the arrangement, given a plumbing graph in normal form for the boundary.