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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

Abstract

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.

Keywords

braid group, configuration spaces, loop spaces

Mathematical Subject Classification

Primary: 20F36, 55N25

References
Publication

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

Authors
F R Cohen
Department of Mathematics
University of Rochester
Rochester NY 14627
USA
Yasuhiko Kamiyama
Department of Mathematics
University of the Ryukyus
Nishihara-Cho
Okinawa 903-0213
Japan