On the Falk invariant of hyperplane arrangements attached to gain graphs

TL;DR

This paper provides a combinatorial formula for the Falk invariant of hyperplane arrangements associated with certain gain graphs, linking topological invariants to combinatorial structures.

Contribution

It introduces a new combinatorial formula for the Falk invariant specifically for hyperplane arrangements derived from gain graphs.

Findings

Derived a combinatorial formula for the Falk invariant

Connected the Falk invariant to gain graph structures

Enhanced understanding of the topology of hyperplane arrangements

Abstract

The fundamental group of the complement of a hyperplane arrangement in a complex vector space is an important topological invariant. The third rank of successive quotients in the lower central series of the fundamental group was called Falk invariant of the arrangement since Falk gave the first formula and asked to give a combinatorial interpretation. In this article, we give a combinatorial formula for the Falk invariant of hyperplane arrangements attached to certain gain graphs.