Advanced Studies in Pure Mathematics

On the homology of configuration spaces associated to centers of mass

Dai Tamaki

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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


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