Configuration Spaces of Points: A User's Guide

TL;DR

This paper provides a comprehensive survey of configuration spaces of points, including classical and chromatic variants, highlighting their mathematical structures, applications, and a key relation to graph chromatic polynomials.

Contribution

It offers an extensive survey of configuration spaces and proves a novel connection between chromatic configuration spaces and graph chromatic polynomials.

Findings

Poincaré polynomial of chromatic configuration spaces relates to the reciprocal of the chromatic polynomial.

Provides detailed proofs and applications of configuration space properties.

Introduces a stable splitting for these spaces.

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.