# Contents

## Idea

In mathematics typically by default the term “configuration space” of a topological space $X$ refers to the topological space of pairwise distinct points in $X$, 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 $X= \mathbb{R}^\infty$. Then

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

• the ordered configuration space of $n$ points, equipped with the canonical $\Sigma(n)$-action, is a model for the $\Sigma(n)$-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)

Last revised on June 26, 2017 at 07:05:02. See the history of this page for a list of all contributions to it.