The Salvetti complex and the little cubes

Abstract

For a real central arrangement $\mathcal{A}$, Salvetti introduced a construction of a finite complex Sal$(\mathcal{A})$ which is homotopy equivalent to the complement of the complexified arrangement in [Sal87]. For the braid arrangement ${\mathcal{A}}_{k-1}$, the Salvetti complex Sal$({\mathcal{A}}_{k-1})$ serves as a good combinatorial model for the homotopy type of the configuration space $F(\mathbb{C},k)$ of $k$ points in $\mathbb{C}$, which is homotopy equivalent to the space ${\mathcal{C}}_2(k)$ of $k$ little $2$-cubes. Motivated by the importance of little cubes in homotopy theory, especially in the study of iterated loop spaces, we study how the combinatorial structure of the Salvetti complexes of the braid arrangements are related to homotopy theoretic properties of iterated loop spaces.
As a consequence, 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 in [AK02].