Higher holonomy maps for hyperplane arrangements


We develop a method to construct representations of the homotopy 2-groupoid of a manifold as a 2-category by means of Chen’s formal homology connections. As an application we describe 2-holonomy maps for hyperplane arrangements and discuss representations of the category of braid cobordisms.

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  1. 1.

    Baez, J.C., Huerta, J.: An invitation to higher gauge theory. Gen. Relativity Gravitation 43(9), 2335–2392 (2011)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Carter, J.S., Saito, M.: Knotted Surfaces and Their Diagrams. Mathematical Surveys and Monographs, vol. 55. American Mathematical Society, Providence (1998)

    MATH  Google Scholar 

  3. 3.

    Chen, K.: Iterated integrals of differential forms and loop space homology. Ann. Math. 97, 217–246 (1973)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Chen, K.-T.: Extension of \(C^{\infty }\) function algebra by integrals and Malcev completion of \(\pi _1\). Adv. Math. 23(2), 181–210 (1977)

    Article  Google Scholar 

  5. 5.

    Chen, K.T.: Iterated path integrals. Bull. Amer. Math. Soc. 83(5), 831–879 (1977)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Chmutov, S., Duzhin, S., Mostovoy, J.: Introduction to Vassiliev Knot Invariants. Cambridge University Press, Cambridge (2012)

    Book  Google Scholar 

  7. 7.

    Cirio, L.S., Faria Martins, J.: Categorifying the Knizhnik–Zamolodchikov connection. Differential Geom. Appl. 30(3), 238–261 (2012)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Cirio, L.S., Faria Martins, J.: Categorifying the \({{\mathfrak{s}}{\mathfrak{l}}}(2; {{\mathbb{C}}})\) Knizhnik–Zamolodchikov connection via an infinitesimal 2-Yang–Baxter operator in the string Lie-2-Algebra. Adv. Theor. Math. Phys. 21(1), 147–229 (2017)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Deligne, P., Griffiths, P., Morgan, J., Sullivan, D.: Real homotopy theory of Kähler manifolds. Invent. Math. 29(3), 245–274 (1975)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Kamada, S.: Braid and Knot Theory in Dimension Four. Mathematical Surveys and Monographs, vol. 95. American Mathematical Society, Providence (2002)

    Book  Google Scholar 

  11. 11.

    Kohno, T.: On the holonomy Lie algebra and the nilpotent completion of the fundamental group of the complement of hypersurfaces. Nagoya Math. J. 92, 21–37 (1983)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Kohno, T.: Monodromy representations of braid groups and Yang–Baxter equations. Ann. Inst. Fourier (Grenoble) 37(4), 139–160 (1987)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Kohno, T.: Vassiliev invariants of braids and iterated integrals. In: Falk, M., Terao, H. (eds.) Arrangements—Tokyo 1998. Advanced Studies in Pure Mathematics, vol. 27, pp. 157–168. Kinokuniya, Tokyo (2000)

    Google Scholar 

  14. 14.

    Kohno, T.: Bar complex of the Orlik–Solomon algebra. Topology Appl. 118(1–2), 147–157 (2002)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Kohno, T.: Higher holonomy of formal homology connections and braid cobordisms. J. Knot Theory Ramifications 26(12), # 1642007 (2016)

  16. 16.

    Kontsevich, M.: Vassiliev’s knot invariants. In: Gel’fand, S., Gindikin, S. (eds.) I.M. Gel’fand Seminar. Advances in Soviet Mathematics, pp. 137–150. American Mathematical Society, Providence (1993)

    Google Scholar 

  17. 17.

    Le, T.Q.T., Murakami, J.: Representation of the category of tangles by Kontsevich’s iterated integral. Comm. Math. Phys. 168(3), 535–562 (1995)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Orlik, P., Terao, H.: Arrangements of Hyperplanes. Grundlehren der Mathematischen Wissenschaften, vol. 300. Springer, Berlin (1992)

    MATH  Google Scholar 

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Correspondence to Toshitake Kohno.

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The author is partially supported by Grant-in-Aid for Scientific Research, KAKENHI 16H03931, Japan Society of Promotion of Science and by World Premier Research Center Initiative, MEXT, Japan.

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Kohno, T. Higher holonomy maps for hyperplane arrangements. European Journal of Mathematics 6, 905–927 (2020). https://doi.org/10.1007/s40879-019-00382-z

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  • Braid group
  • Iterated integral
  • Formal homology connection
  • Hyperplane arrangement
  • Higher holonomy
  • 2-Category
  • Braid cobordism

Mathematics Subject Classification

  • 20F36
  • 57M25
  • 55P62