On the Homology of Configuration Spaces Associated to Centers of Mass

TL;DR

This paper computes the homology of the Salvetti complex for a 'center of mass' arrangement with four particles, revealing differences based on prime coefficients and providing new insights into configuration space homology.

Contribution

It provides explicit homology computations for the Salvetti complex of the center of mass arrangement, extending understanding of configuration space homology with coefficients in sign representations.

Findings

Homology matches that of F(C,4) for odd primes p.

Homology differs from F(C,4) when p=2.

Offers a new proof of Cohen and Kamiyama's theorem.

Abstract

The aim of this paper is to make sample computations with the Salvetti complex of the "center of mass" arrangement introduced in [arXiv:math/0611732] by Cohen and Kamiyama. We compute the homology of the Salvetti complex of these arrangements with coefficients in the sign representation of symmetric groups on 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(C,4) of distinct four points in the complex plane with the same coefficients. When p=2, we show the homology is different from that of F(C,4), hence obtain an alternative and more direct proof of a theorem of Cohen and Kamiyama in [arXiv:math/0611732].