Vipul at 18:52, 30 December 2010

2010-12-30T18:52:42Z

Vipul: Created page with '==Definition== Suppose $X$ is a topological space and $n$ is a natural number. The '''configuration space of unordered points''' $C_n(X)$
2010-12-30T18:44:44Z

Created page with '==Definition== Suppose <math>X</math> is a topological space and <math>n</math> is a natural number. The '''configuration space of unordered points''' <math>C_n(X)</math...'

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==Definition==

Suppose <math>X</math> is a [[topological space]] and <math>n</math> is a [[natural number]]. The '''configuration space of unordered points''' <math>C_n(X)</math> (often simply called the '''configuration space'''), sometimes also denoted <math>\binom{X}{n}</math>, is defined as follows:

* As a set, it is the set of <math>n</math>-element subsets of <math>X</math>.
* The topology is given as follows: A <math>n</math>-element subset of <math>X</math> can be thought of as an orbit under the action of the [[symmetric group]] <math>S_n</math> on the [[configuration space of ordered points]] <math>F_n(X)</math> (defined as the subspace of <math>X^n</math> comprising points which have pairwise distinct points). In other words, as a ''set'' <math>C_n(X) = F_n(X)/S_n</math>. We give this a topology as follows: first, we give <math>F_n(X)</math> the [[defining ingredient::subspace topology]] arising from the [[defining ingredient::product topology]] on <math>X^n</math>. Then, we give <math>C_n(X)</math> the [[defining ingredient::quotient topology]] under the equivalence relation induced by the action of <math>S_n</math>.

==Facts==

* The configuration space of unordered points is ''not'' a homotopy invariant. In other words, if <math>X</math> and <math>Y</math> are [[homotopy-equivalent spaces]], it does not necessarily follow that <math>C_n(X)</math> and <math>C_n(Y)</math> are homotopy-equivalent.

Configuration space of unordered points - Revision history

Vipul at 18:52, 30 December 2010