Hurwitz Equivalence in Braid Group B_3

Abstract

In this paper we prove certain Hurwitz equivalence properties of $B_n$. In particular we prove that for $n=3$ every two Artin's factorizations of $\Delta _3 ^2$ of the form $H_{i_1} ... H_{i_6}, \quad F_{j_1} ... F_{j_6}$ (with $i_k, j_k \in \{1,2 \}$) where $\{H_1, H_2 \}, \{F_1, F_2 \}$ 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 example). We are constructing a new invariant based on Hurwitz equivalence class of factorization, to distinguish among diffeomorphic surfaces which are not deformation of each other.