nLab
Fadell's configuration space

Contents

Idea

Given a manifold MM, the Fadell’s configuration space (in topology called simply configuration space) is the manifold of NN-tuples of pairwise distinct points in MM.

It is important in the study of topological fibrations, in the study of arrangements of hyperplanes, of Knizhnik-Zamolodchikov connections and in study of geometry of renormalization.

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

Revised on March 16, 2017 18:27:34 by Urs Schreiber (192.41.133.235)