For motivation I recall two classical examples of locally trivial fiber bundle projections, due to Brieskorn (in `Sur les groupes de tresses').
- ${\Bbb C}^n-\cup_{i\neq j}\{(z_1,z_2,\ldots ,z_n): z_i=\pm z_j\}\to ({\Bbb C}^*)^{n-1}-\cup_{i\neq j}\{(w_1,w_2,\ldots ,w_{n-1}): w_i=w_j\}$
defined by $w_i=z_n^2-z_i^2,$ for $i=1,2,\cdots, n-1$.
- $({\Bbb C}^*)^4-\cup_{i\neq j}\{(z_1,z_2,z_3,z_4): z_i=\pm z_j, z_1\pm z_2\pm z_3\pm z_4=0\}\to ({\Bbb C}^*)^{3}-\cup_{i\neq j}\{(w_1,w_2,w_3): w_i=w_j\}$
defined by $w_i=z_1z_2z_3z_4(z_4^2-z_i^2),$ for $i=1,2,3$.
I would like to define (if possible at all) a similar locally trivial fiber bundle projection map as follows. Let $M={\Bbb C}-\{\pm 1\}$ and $N={\Bbb C}^*-\{1\}$.
$M^n-\cup_{i\neq j}\{(z_1,z_2,\ldots ,z_n): z_i=\pm z_j\}\to $$N^{n-1}-\cup_{i\neq j}\{(w_1,w_2,\ldots ,w_{n-1}): w_i=w_j\}$.
Although, following the above examples, I chose the above particular codomain, instead, I am allowed to choose any fiber-type hyperplane arrangement complement in ${\Bbb C}^{n-1}$ as the target. My main interest is in fibering $M^n-\cup_{i\neq j}\{(z_1,z_2,\ldots ,z_n): z_i=\pm z_j\}$.
Here, I must admit that I am not able to check that the two maps in 1 and 2 are fibrations. Probably easy, however I do not see.
Any idea would be highly appreciated.
