On the Falk Invariant of Signed Graphic Arrangements
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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 a signed graphic arrangement that do not have a \(B_2\) as sub-arrangement.
KeywordsHyperplane arrangements Sign graph Falk invariant Lower central series
Mathematics Subject Classification52C35 05C22 20F14
The authors thank Professor Yoshinaga for the valuable discussions and the anonimous referee for suggesting a shorter proof for Lemma 4. The second authors also thanks Doctor Suyama and Doctor Tsujie for the valuable discussions on signed graphs.