our main topics follows.

Projection onto the first coordinates, ${\mathbb{C}}^n \to {\mathbb{C}}^{n-1}$, restricts to a topological fibration between the corresponding braid complements, with fiber ${\mathbb{C}}\setminus \{ n-1 \; \text{points} \}$. Arrangements whose complement may be obtained from the simplest arrangement, $\{ 0\}\subset {\mathbb{C}}$, by iterating a similar fibration construction, were introduced by Falk-Randell [@FR0] and called [*fiber-type*]{}. A remarkable result (due to Terao [@Te] and described in Section \[sect:lattice\]) says that this is equivalent to $\mathcal L({{\mathcal A}})$ being [*supersolvable*]{}, in the sense of Stanley [@St].

In Section \[sect:cohomology\], we examine the converse of the Orlik-Solomon theorem: the cohomology ring $H^*(M_{{\mathcal A}})$, together with its [*natural $1$-marking*]{} (in the sense of Choudary-Dimca-Papadima [@CDP]), determines the lattice $\mathcal L({{\mathcal A}})$.

Section \[sect:pi1\] treats the homotopy groups, $\{ \pi_i(M_{{{\mathcal A}}})\}_{i\ge 1}$. As shown by Rybnikov [@Ryb], the [*fundamental group*]{} $\pi_1(M_{{{\mathcal A}}})$ is [*not*]{} combinatorial (see also [@AAB0], [@AAB1]). Nevertheless, its rational [*associated graded Lie algebra*]{}, $\operatorname{gr}_*(\pi_1(M_{{{\mathcal A}}}))\otimes {\mathbb{Q}}$, is combinatorial, by a result of Kohno [@Ko]. The nilpotence of $\pi_1(M_{{{\mathcal A}}})$ is equivalent to the property of $\mathcal L({{\mathcal A}})$ of being in general position in codimension $2$; this fact is the first step in the proof of a famous conjecture of 