descending series of normal subgroups of $G$, called the [*lower central series*]{} associated to the group $G$, by $G_1:=G$ and $G_i:=[G_{i-1},G]$ for $i \geq 2$, where $[x,y]:= xyx^{-1}y^{-1}$ is the group commutator. Then, $\operatorname{gr}_*(G) =\bigoplus_{i \geq 1} (\operatorname{gr}_i(G):= \frac{G_i}{G_{i+1}})$ has a natural graded Lie algebra structure with the Lie bracket induced by the commutator. The [*rational associated graded Lie algebra*]{} is the graded Lie algebra over ${\mathbb{Q}},\; \operatorname{gr}(G) \otimes {\mathbb{Q}}$.

\[lie\]

1.  \[Zn\] If $G={\mathbb{Z}}^n$ is the free abelian group on $n$ generators, then $\operatorname{gr}_*(G)=\operatorname{gr}_1(G)$ is abelian, i.e., $[,]=0$.

2.  \[Fn\] If $G={\mathbb{F}}_n$ is the free group on $n$ generators, then $\operatorname{gr}(G)$ is the free Lie algebra on $n$ generators.

In the arrangement case, we point out that the rational associated graded Lie algebra of $\pi_1(M_{{{\mathcal A}}})$ is combinatorial, as a consequence of the formality of the complement space $M_{{{\mathcal A}}}$. See Kohno [@Ko]. [*It is an open question whether $\operatorname{gr}(\pi_1(M_{{{\mathcal A}}}))$ is determined by the lattice $\mathcal L({{\mathcal A}})$, since it is known to contain torsion elements.*]{}

Next, we recall the definition of other significant objects in algebraic topology, that play an important role in the homotopy classification of topological spaces: the [*higher homotopy groups*]{}. The $n$-th homotopy group 