Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

The Global Invariant of Signed Graphic hyperplane Arrangements

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

The fundamental group of the complement of a hyperplane arrangement in a complex vector space is an interesting and complicated invariant. The third rank of successive quotients in the lower central series of the fundamental group was called the global invariant of the arrangement by Falk. Falk gave a general formula to compute the global invariant, and asked for a combinatorial interpretation of the global invariant. Schenck and Suciu proved that the global invariant of a graphic arrangement is double of the number of cliques with three or four vertices in the graph with which the arrangement associated. This solved Falk’s problem in the case of graphic arrangements. While in the case of signed graphic arrangements, we obtained a similar combinatorial formula. In this paper, we give a direct and simple proof for this combinatorial formula.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Falk, M.: Combinatorial and algebraic structure in Orlik–Solomon algebras. Eur. J. Combin. 22, 687–698 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Falk, M.: On the algebra associated with a geometric lattice. Adv. Math. 80, 152–163 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  3. Falk, M.: The minimal model of the complement of an arrangement of hyperplanes. Trans. Am. Math. Soc. 309, 543–556 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  4. Falk, M., Randell, R.: On the homotopy theory of arrangements, II. In: Arrangements-Tokyo, 1998, Adv. Stud. Pure Math., vol. 27, pp. 93–125 (2000)

  5. Guo, W., Li, Z., Guo, Q., Jiang, G.: Falk invariants of signed graphic arrangements (in preparation)

  6. Orlik, P., Terao, H.: Arrangements of Hyperplanes. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  7. Schenck, H., Suciu, A.: Lower central series and free resolutions of hyperplane arrangements. Trans. Am. Math. Soc. 354, 3409–3433 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Zaslavsky, T.: Characterizations of signed graphs. J. Graph Theory 5, 401–406 (1981)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Science, Beijing University of Chemical Technology, Beijing, 100029, People’s Republic of China

    Qiumin Guo, Weili Guo, Wentao Hu & Guangfeng Jiang

Authors
  1. Qiumin Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Weili Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wentao Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Guangfeng Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qiumin Guo.

Additional information

Supported by NSF of China under Grant 11071010.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guo, Q., Guo, W., Hu, W. et al. The Global Invariant of Signed Graphic hyperplane Arrangements. Graphs and Combinatorics 33, 527–535 (2017). https://doi.org/10.1007/s00373-017-1770-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-017-1770-y

Keywords

Mathematics Subject Classification

Search

Navigation