nLab
configuration space (mathematics)

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Idea

In mathematics typically by default the term “configuration space” of a topological space XX refers to the topological space of pairwise distinct points in XX, also called Fadell's configuration space, for emphasis.

In principle many other kinds of configurations and the spaces these form may be referred to by “configuration space”, notably in physics the usage is in a broader sense, see at configuration space (physics).

Examples

Classifying space of the symmetric group

Let X= X= \mathbb{R}^\infty. Then

  • the unordered configuration space of nn points in \mathbb{R}^\infty is a model for the classifying space BΣ(n)B \Sigma(n) of the symmetric group Σ(n)\Sigma(n);

  • the ordered configuration space of nn points, equipped with the canonical Σ(n)\Sigma(n)-action, is a model for the Σ(n)\Sigma(n)-universal principal bundle.

References

  • Edward Fadell, Lee Neuwirth, Configuration spaces Math. Scand. 10 (1962) 111-118, MR141126, pdf

  • Craig Westerland, Configuration spaces in geometry and topology, 2011, pdf

  • Graeme Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973), 213–221. MR 0331377 (pdf)

  • Edward R. Fadell, Sufian Y. Husseini, Geometry and topology of configuration spaces, Springer Monographs in Mathematics (2001), MR2002k:55038, xvi+313 pp.

  • F. R. Cohen, S. Gitler, On loop spaces of configuration spaces, Trans. Amer. Math. Soc. 354 (2002), no. 5, 1705–1748, MR2002m:55020

On the relation to Goodwillie calculus:

  • Michael Ching, Calculus of Functors and Configuration Spaces (pdf)
Revised on June 26, 2017 07:05:02 by Urs Schreiber (88.77.226.246)