Title:Configurations and parallelograms associated to centers of mass

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.