Configuration Spaces of Points: A User's Guide

Abstract: This user's guide (updated version) consists of two parts. The first part is an extensive survey contributed to the Encyclopedia of Mathematical Physics, 2nd edition. It covers many of the main constructions, definitions, and applications of the classical configuration spaces of points. The second part delves into the geometry of chromatic configuration spaces, giving a detailed proof of the remarkable result that the Poincar\'e polynomial of the chromatic configuration spaces of $\mathbb R^N$, associated to a finite simple graph $\Gamma$, corresponds to the reciprocal of the chromatic polynomial of the graph (with signs). Further applications and a stable splitting are given.