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On the Falk Invariant of Signed Graphic Arrangements

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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 a signed graphic arrangement that do not have a \(B_2\) as sub-arrangement.

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Acknowledgements

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.

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Authors and Affiliations

  1. Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, 060-0810, Japan

    Weili Guo & Michele Torielli

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  1. Weili Guo
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  2. Michele Torielli
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Correspondence to Michele Torielli.

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The second author was supported by the MEXT grant for Tenure Tracking system.

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Guo, W., Torielli, M. On the Falk Invariant of Signed Graphic Arrangements. Graphs and Combinatorics 34, 477–488 (2018). https://doi.org/10.1007/s00373-018-1887-7

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  • DOI: https://doi.org/10.1007/s00373-018-1887-7

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