On the homology of configuration spaces associated to centers of mass

Abstract

The aim of this paper is to make sample computations with the Salvetti complex of the "center of mass" arrangement introduced in [CK07] by Cohen and Kamiyama. We compute the homology of the Salvetti complex of these arrangements with coefficients in the sign representation of the symmetric group on $\mathbb{F}_p$ in the case of four particles. We show, when $p$ is an odd prime, the homology is isomorphic to the homology of the configuration space $F(\mathbb{C}, 4)$ of distinct four points in $\mathbb{C}$ with the same coefficients. When $p = 2$, we show the homology is different from the equivariant homology of $F(\mathbb{C}, 4)$, hence we obtain an alternative and more direct proof of a theorem of Cohen and Kamiyama in [CK07].