Properties of Hurwitz Equivalence in the Braid Group of Order n

Abstract

In this paper we prove certain Hurwitz equivalence properties in $B_n$. Our main result is that every two Artin's factorizations of $\Delta_n ^2$ of the form $H_{i_1} ... H_{i_{n(n-1)}}, \quad F_{j_1} ... F_{j_{n(n-1)}}$ (with $i_k, j_k \in \{1,...,n-1 \}$), where $\{H_1,...,H_{n-1} \}, \{F_1,...,F_{n-1} \}$ are frames, are Hurwitz equivalent. This theorem is a generalization of the theorem we have proved in \cite{Tz}, using an algebraic approach unlike the proof in \cite{Tz} which is geometric. The application of the result is 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 construct a new invariant based on a Hurwitz equivalence class of factorization, to distinguish among diffeomorphic surfaces which are not deformations of each other. The precise definition of the new invariant can be found in \cite{KuTe} or \cite{Te}. The main result of this paper will help us to compute the new invariant.