Configurations and parallelograms associated to centers of mass

TL;DR

This paper introduces the concept of t-fold center of mass arrangements for points in the plane, explores their properties, and investigates the topological relationship between configuration spaces with and without parallelogram vertices, revealing homotopy-theoretic insights.

Contribution

It defines new arrangements M(t,k), analyzes their properties, and proves that the classical configuration space is not a homotopy retract of a related space, with implications for topology and sphere mappings.

Findings

Classical configuration space is not a homotopy retract of the space excluding parallelogram vertices.

Homotopy-theoretic methods show the structure of M(2,k) spaces.

Failure of methods at odd primes suggests new localization candidates for double loop spaces.

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.