Configuration space of unordered points: Difference between revisions

Latest revision as of 18:52, 30 December 2010

Definition

Suppose $X$ is a topological space and $n$ is a natural number. The configuration space of unordered points $C_{n}(X)$ (often simply called the configuration space), sometimes also denoted ${\binom {X}{n}}$ , is defined as follows:

As a set, it is the set of $n$ -element subsets of $X$ .
The topology is given as follows: A $n$ -element subset of $X$ can be thought of as an orbit under the action of the symmetric group $S_{n}$ on the configuration space of ordered points $F_{n}(X)$ (defined as the subspace of $X^{n}$ comprising points which have pairwise distinct points). In other words, as a set $C_{n}(X)=F_{n}(X)/S_{n}$ . We give this a topology as follows: first, we give $F_{n}(X)$ the subspace topology arising from the product topology on $X^{n}$ . Then, we give $C_{n}(X)$ the quotient topology under the equivalence relation induced by the action of $S_{n}$ .

Facts

The configuration space of unordered points is not a homotopy invariant. In other words, if $X$ and $Y$ are homotopy-equivalent spaces, it does not necessarily follow that $C_{n}(X)$ and $C_{n}(Y)$ are homotopy-equivalent.

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 * 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>.
+* 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 [[defining ingredient::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.