## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Arrangements of Hyperplanes — Sapporo 2009, H. Terao and S. Yuzvinsky, eds. (Tokyo: Mathematical Society of Japan, 2012), 417 - 457

### 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].

#### Article information

**Source***Arrangements of Hyperplanes — Sapporo 2009*, H. Terao and S. Yuzvinsky, eds. (Tokyo: Mathematical Society of Japan, 2012), 417-457

**Dates**

Received: 31 March 2010

Revised: 3 August 2010

First available in Project Euclid:
24 November 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1543085017

**Digital Object Identifier**

doi:10.2969/aspm/06210417

**Mathematical Reviews number (MathSciNet)**

MR2933805

**Zentralblatt MATH identifier**

1279.52021

**Subjects**

Primary: 52C35: Arrangements of points, flats, hyperplanes [See also 32S22]

Secondary: 55P35: Loop spaces

**Keywords**

Salvetti complex braid arrangement loop space

#### Citation

Tamaki, Dai. On the homology of configuration spaces associated to centers of mass. Arrangements of Hyperplanes — Sapporo 2009, 417--457, Mathematical Society of Japan, Tokyo, Japan, 2012. doi:10.2969/aspm/06210417. https://projecteuclid.org/euclid.aspm/1543085017