Configurations, and parallelograms associated to centers of mass

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dc.creatorCohen, F. R.
dc.creatorKamiyama, Y.
dc.date2006-11-23
dc.date2006-11-27
dc.date.accessioned2026-07-07T07:33:16Z
dc.date.available2026-07-07T07:33:16Z
dc.descriptionThe purpose of this article is to \begin{enumerate} \item define $M(t,k)$ the $t$-fold center of mass arrangement for $k$ points in the plane, \item give elementary properties of $M(t,k)$ and \item 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. \end{enumerate} 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.
dc.description11 pages
dc.identifierhttps://arxiv.org/abs/math/0611732
dc.identifierhttp://arxiv.org/abs/math/0611732
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/119395
dc.subjectAlgebraic Topology
dc.subjectGeometric Topology
dc.subject20F35, 55N25
dc.titleConfigurations, and parallelograms associated to centers of mass
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