the $K(\pi,1)$ property (Fadell-Neuwirth [@FN]) – initiated typical directions of investigation in arrangement theory. Resolving a conjecture of Arnold, Brieskorn showed that the cohomology ring of the complement, $M_{{{\mathcal A}}}= V \setminus \bigcup_{i=1}^n H_i$, of an arbitrary complex arrangement ${{\mathcal A}}$ is generated by the classes of the closed $1$-forms $\frac{1}{2 \pi \sqrt{-1}}(\frac{d \alpha_H}{\alpha_H})$, where $\alpha_H=0$ are the defining equations for the hyperplanes $H \in {{\mathcal A}}$.

An emblematic result for the field is the theorem of Orlik and Solomon [@OS], that proves an isomorphism between the cohomology ring of the complement, $H^*(M_{{\mathcal A}})$, and the [*Orlik-Solomon algebra*]{} of the arrangement, $A^*({{\mathcal A}})$, which is determined by the lattice $\mathcal L({{\mathcal A}})=\{\bigcap_{H \in{{{\mathcal B}}}}H\;|\;{{\mathcal B}}\subset {{\mathcal A}}\}$, ordered by reverse inclusion.

We say that a topological invariant (property) of the complement $M_{{{\mathcal A}}}$ is [*combinatorial*]{} if it depends only on the intersection lattice $\mathcal L({{\mathcal A}})$.

We have chosen to survey here several natural, interesting topological facets of the combinatorial determination problem (based on personal taste, and with no intention of exhausting the subject). Furthermore, ample existing expository work covers various aspects of the field of hyperplane arrangements, see for instance [@FR1], [@FR2], [@Fa0], [@S1], [@Y1].

A brief discussion of 