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Configurations and parallelograms associated to centers of mass

F R Cohen and Yasuhiko Kamiyama

Geometry & Topology Monographs 11 (2007) 17–32

DOI: 10.2140/gtm.2007.11.17

arXiv: 0903.4867


The purpose of this article is to

1. define M(t,k) the t–fold center of mass arrangement for k points in the plane,
2. give elementary properties of M(t,k) and
3. give consequences concerning the space M(2,k) of k distinct points in the plane, no four of which are the vertices of a parallelogram.

The main result proven in this article is that the classical unordered configuration of k points in the plane is not a retract up to homotopy of the space of k unordered distinct points in the plane, no four of which are the vertices of a parallelogram. The proof below is homotopy theoretic without an explicit computation of the homology of these spaces.

In addition, a second, speculative part of this article arises from the failure of these methods in the case of odd primes p. This failure gives rise to a candidate for the localization at odd primes p of the double loop space of an odd sphere obtained from the p–fold center of mass arrangement. Potential consequences are listed.


braid group, configuration spaces, loop spaces

Mathematical Subject Classification

Primary: 20F36, 55N25


Received: 6 January 2005
Revised: 15 November 2005
Accepted: 12 December 2005
Published: 14 November 2007

F R Cohen
Department of Mathematics
University of Rochester
Rochester NY 14627
Yasuhiko Kamiyama
Department of Mathematics
University of the Ryukyus
Okinawa 903-0213