  We study how the combinatorial structure of the Salvetti complexes of the
braid arrangements are related to homotopy theoretic properties of iterated
loop spaces. We prove the skeletal filtrations on the Salvetti complexes of the
braid arrangements give rise to the cobar-type Eilenberg-Moore spectral
sequence converging to the homology of $\Omega^2\Sigma^2 X$. We also construct
a new spectral sequence that computes the homology of
$\Omega^{\ell}\Sigma^{\ell} X$ for $\ell>2$ by using a higher order analogue of
the Salvetti complex. The $E^1$-term of the spectral sequence is described in
terms of the homology of $X$. The spectral sequence is different from known
spectral sequences that compute the homology of iterated loop spaces, such as
the Eilenberg-Moore spectral sequence and the spectral sequence studied by
Ahearn and Kuhn.
