Hurwitz equivalence in the Braid group B3

T. Ben-Itzhak; M. Teicher

doi:10.1142/S0218196703001389

Hurwitz equivalence in the Braid group B₃

T. Ben-Itzhak, M. Teicher

Department of Mathematics

Bar-Ilan University

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper we prove certain Hurwitz equivalence properties of B _n. In particular we prove that for n = 3 every two Artin factorizations of Δ₃² of the form H_i1 ⋯ H_i6, F_j1 ⋯ F_j6 (with i _k,j_k ∈ {1, 2}) where {H₁, H₂}, {F₁, F₂} are frames, are Hurwitz equivalent. The proof provided here is geometric, based on a newly defined frame type. The results will be applied to the classification of algebraic surfaces up to deformation. It is already known that there exist surfaces that are diffeomorphic to each other but are not deformations of each other (Manetti's example). We construct a new invariant based on Hurwitz equivalence of factorizations, to distinguish among diffeomorphic surfaces which are not deformation of each other. The main result of this paper will help us to compute the new invariant.

Original language	English
Pages (from-to)	277-286
Number of pages	10
Journal	International Journal of Algebra and Computation
Volume	13
Issue number	3
DOIs	https://doi.org/10.1142/S0218196703001389
State	Published - Jun 2003

Bibliographical note

Funding Information:
∗This work was partially supported by the Emmy Noether Research Institute for Mathematics (center of the Minerva Foundation of Germany), the Excellency Center “Group Theoretic Methods in the Study of Algebraic Varieties” of the Israel Science Foundation, and by EAGER (EU network, HPRN-CT-2009-00099).

Keywords

BMT
Braid

Access to Document

10.1142/S0218196703001389

Cite this

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title = "Hurwitz equivalence in the Braid group B3",

abstract = "In this paper we prove certain Hurwitz equivalence properties of B n. In particular we prove that for n = 3 every two Artin factorizations of Δ32 of the form Hi1 ⋯ Hi6, Fj1 ⋯ Fj6 (with i k,jk ∈ {1, 2}) where {H1, H2}, {F1, F2} are frames, are Hurwitz equivalent. The proof provided here is geometric, based on a newly defined frame type. The results will be applied to the classification of algebraic surfaces up to deformation. It is already known that there exist surfaces that are diffeomorphic to each other but are not deformations of each other (Manetti's example). We construct a new invariant based on Hurwitz equivalence of factorizations, to distinguish among diffeomorphic surfaces which are not deformation of each other. The main result of this paper will help us to compute the new invariant.",

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author = "T. Ben-Itzhak and M. Teicher",

note = "Funding Information: ∗This work was partially supported by the Emmy Noether Research Institute for Mathematics (center of the Minerva Foundation of Germany), the Excellency Center “Group Theoretic Methods in the Study of Algebraic Varieties” of the Israel Science Foundation, and by EAGER (EU network, HPRN-CT-2009-00099).",

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