I have a very specific question. How does one check the following map ${\cal C}^n-\cup \{z_i=\pm z_j\ for\ i\neq j\}\to {({\cal C}^*)}^{n-1}-\cup\{z_i\neq z_j\ for\ i\neq j\}$ defined by $(z_1,z_2,\ldots , z_n)\mapsto (z_n^2-z_1^2,\ldots , z_n^2-z_{n-1}^2)$ is a locally trivial fibration? This is stated in the paper of E. Brieskorn `Sur les groupes de tresses' without proof.