与质心相关的平行四边形
Configurations, and parallelograms associated to centers of mass
摘要：
本文证明了平面上\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}点的经典无序构形不是平面上$k$无序相异点空间的上至同伦的收缩，其中没有四个点是一个点的顶点。下面的证明是同伦理论的，没有明确计算这些空间的同调。此外，本文的第二个推测部分来自于这些方法在奇素数$p$情况下的失败。这个失败引起的候选人的本地化在奇素数$p$的双闭路空间的奇数球从$p$倍的质心安排。列出了潜在后果。
英文摘要：
The 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
