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Graphs and Combinatorics

, Volume 34, Issue 3, pp 477–488 | Cite as

On the Falk Invariant of Signed Graphic Arrangements

  • Weili Guo
  • Michele Torielli
Original Paper

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.

Keywords

Hyperplane arrangements Sign graph Falk invariant Lower central series 

Mathematics Subject Classification

52C35 05C22 20F14 

Notes

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.

References

  1. 1.
    Falk, M.: On the algebra associated with a geometric lattice. Adv. Math. 80(2), 152–163 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Falk, M.: Combinatorial and algebraic structure in Orlik-Solomon algebras. Eur. J. Comb. 22(5), 687–698 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Guo, Q., Guo, W., Hu, W., Jiang, G.: The global invariant of signed graphic hyperplane arrangements. Graphs Comb. 33, 527–535 (2017)Google Scholar
  4. 4.
    Guo, W., Guo, Q., Jiang, G.: Falk invariants of signed graphic arrangements. Graphs Comb (To appear)Google Scholar
  5. 5.
    Orlik, P., Terao, H.: Arrangements of Hyperplanes, vol. 300. Springer Science & Business Media, New York (2013)zbMATHGoogle Scholar
  6. 6.
    Schenck, H., Suciu, A.: Lower central series and free resolutions of hyperplane arrangements. Trans. Am. Math. Soc. 354(9), 3409–3433 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Zaslavsky, T.: Signed graphs. Discrete Appl. Math. 4(1), 47–74 (1982)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsHokkaido UniversitySapporoJapan

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