Euclidean matchings and minimality of hyperplane arrangements

TL;DR

This paper introduces a new class of acyclic matchings on hyperplane arrangement complexes, proving their minimality and providing geometric insights into Betti numbers, while also solving a conjecture related to finite reflection arrangements.

Contribution

It constructs explicit acyclic matchings on the Salvetti complex, proving minimality of the complement and solving a conjecture on characteristic polynomials.

Findings

Constructed maximal acyclic matchings on Salvetti complexes.

Proved the minimality of the hyperplane arrangement complement.

Solved a conjecture on the characteristic polynomial of finite reflection arrangements.

Abstract

We construct a new class of maximal acyclic matchings on the Salvetti complex of a locally finite hyperplane arrangement. Using discrete Morse theory, we then obtain an explicit proof of the minimality of the complement. Our construction provides interesting insights also in the well-studied case of finite arrangements, and gives a nice geometric description of the Betti numbers of the complement. In particular, we solve a conjecture of Drton and Klivans on the characteristic polynomial of finite reflection arrangements. The minimal complex is compatible with restrictions, and this allows us to prove the isomorphism of Brieskorn's Lemma by a simple bijection of the critical cells. Finally, in the case of line arrangements, we describe the algebraic Morse complex which computes the homology with coefficients in an abelian local system.