We examine the behaviour of a complex oriented cohomology theory $G^{*}(-)$ on $D_p\left(X\right)$, the $C_p$-extended power of a space $X$, seeking a description of $G^{*}(D_p\left(X\right))$ in terms of the cohomology $G^{*}\left(X\right)$. We give descriptions for the particular cases of Morava $K$-theory $K\left(n\right)$ for any space $X$ and for complex cobordism $MU$, the Brown-Peterson theories BP and any Landweber exact theory for a wide class of spaces.

For a real central arrangement $𝒜$, Salvetti introduced a construction of a finite complex Sal$\left(𝒜\right)$ which is homotopy equivalent to the complement of the complexified arrangement in [Sal87]. For the braid arrangement ${𝒜}_{k-1}$, the Salvetti complex Sal$\left({𝒜}_{k-1}\right)$ serves as a good combinatorial model for the homotopy type of the configuration space $F(ℂ,k)$ of $k$ points in $C$, which is homotopy equivalent to the space $C_2\left(k\right)$ of k little $2$-cubes. Motivated by the importance of little cubes in homotopy theory, especially in the study of...