recall another important topological invariant of a path-connected space $X$. Take the set of loops on $X$, starting and ending at some arbitrary fixed point, and identify two loops if one can be continuously deformed into another. This set, together with the obvious multiplication given by concatenation of loops, becomes a group, the [*fundamental group*]{} of the space $X$, denoted by $\pi_1(X)$.

\[wedgeS1\] The fundamental group of the bouquet of $n$ circles, $\bigvee_{n}S^1$, is the free group on $n$ generators, ${\mathbb{F}}_n$.

If ${{\mathcal A}}$ is an ordered arrangement, it has a [*natural*]{} $1$-marking of $\pi_1(M_{{{\mathcal A}}})$, since $(\pi_1(M_{{{\mathcal A}}}))_{ab}$ is the ${\mathbb{Z}}$-dual of $ H^1(M_{{{\mathcal A}}})$.

Let $M$ be a finite type connected CW-complex with fundamental group $G$. By a [*local system*]{} on $M$ we mean a right ${\mathbb{Z}}[G]$-module, over the group ring ${\mathbb{Z}}[G]$.

\[rk1\] The [*character torus*]{} ${\mathbb{T}}_{G}:=\operatorname{Hom}(G, {\mathbb{C}}^*)$ is an algebraic variety that encodes the so-called [*rank $1$ complex local systems*]{} on $M$. We shall denote by $_{\rho}{\mathbb{C}}$ the ${\mathbb{Z}}[G]$-module structure on ${\mathbb{C}}$ associated to an element $\rho \in {\mathbb{T}}_{G}$.

\[ex:exp\] When $M=M_{{{\mathcal A}}}$ is the complement of an arrangement ${{\mathcal A}}$ with $n$ hyperplanes, one has that ${\mathbb{T}}_{G}=({\mathbb{C}}^*)^n$, and the exponential map ${\mathbb{C}}\overset {\exp} \longrightarrow {\mathbb{C}}^*$ induces 