Falk Invariants of Signed Graphic 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 combinatorial formula and asked to give a combinatorial interpretation. In this article we prove that the Falk invariant of an arrangement associated with signed graph G without loops is double of the sum of $k_3$, $k_4$ and $k^{\pm }_3,$ where $k_l$ is the number of subgraphs of G that are switching equivalent to the cliques with l vertices, $k_3^{\pm }$ is that of subgraphs which have 3 vertices and each two vertices are connected by double edges with different signs. This formula modifies the one given by Schenck and Suciu, and answers partially Falk’s question in the case of signed graphic arrangements.