  In this paper we give a very natural description of the bijections between
the minimal CW-complex homotopy equivalent to the complement of a supersolvable
arrangement $\mathcal{A}$, the $\textbf{nbc}$ basis of the Orlik-Solomon
algebra associated to $\mathcal{A}$ and the set of chambers of $\mathcal{A}$.
We use these bijections to get results on the first (co)homology group of the
Milnor fiber of $\mathcal{A}$ and to describe a bijection between the symmetric
group and the $\textbf{nbc}$ basis of the braid arrangement.
