Properties of Hurwitz equivalence in the Braid group of order n

In this paper we prove certain Hurwitz equivalence properties in Bn. Our main result is that every two Artinʹs factorizations of Δn2 of the form Hi1 Hin(n−1), Fj1 Fjn(n−1) (with ik,jk {1,…,n−1}), where {H1,…,Hn−1}, {F1,…,Fn−1} are frames, are Hurwitz equivalent. This theorem is a generalization of the theorem we have proved in [M. Teicher and T. Ben-Itzhak, Hurwitz equivalence in braid group B3, preprint], using an algebraic approach unlike the proof in loc. cit. which is geometric. 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 example). In our paper [M. Teicher, Braid monodromy type of 4 manifolds, preprint] we introduced a new invariant of surfaces of general type, the so-called BMT invariant. The BMT invariant can distinguish among diffeomorphic surfaces which are not deformations of each other. (More properties of BMT can be found at [V.S. Kulikov and M. Teicher, Izv. J. Russian Acad. Sci. 64 (2) (2000) 89].) To construct the BMT we look at the braid monodromy factorization of the branch curve of a generic projection of a given surface. The BMT invariant is an equivalent class of such factorizations induced by the Hurwitz equivalence relation. In this paper we establish properties of Hurwitz equivalence. These results will help us to compute the new invariant.