Magnitude homology of real hyperplane arrangements

TL;DR

This paper introduces the concepts of magnitude and magnitude homology for real hyperplane arrangements, revealing structural properties, combinatorial formulas, and conjectures about their algebraic and topological features.

Contribution

It defines magnitude homology for arrangements, provides structural and combinatorial formulas, and conjectures torsion-freeness and dependence on the intersection lattice.

Findings

Proves symmetry and palindromicity of magnitude

Provides combinatorial formulas for small-length magnitude homology

Shows magnitude homology detects Boolean arrangements

Abstract

We define and study the magnitude and magnitude homology of a real hyperplane arrangement by regarding its tope graph as a metric space. We prove several structural results for the magnitude of arrangements, including a symmetry formula, palindromicity of the numerator and denominator, a face decomposition formula, and results on the sign pattern of the magnitude power series. For the magnitude homology of arrangements, we obtain combinatorial formulas for small lengths and show that it detects Boolean arrangements. We also lift the face decomposition formula to a homological decomposition and derive explicit formulas for the diagonal magnitude Betti numbers. Another notable feature is that the magnitude Euler characteristic satisfies a reciprocity theorem analogous to Ehrhart--Macdonald reciprocity. We conclude by presenting several conjectures. In particular, we conjecture that the magnitude homology of an arrangement is torsion-free and is determined by the intersection lattice.