classifying space of permutation groups

Question

Given a group $G$, how to compute the classifying space of $G$? In particular, how to compute the classifying space $B\Sigma_n$ for a permutation group $\Sigma_n$?

Answer 1

In general, the classifying space of a group is only well defined up to homotopy equivalence so 'computing' the classifying space isn't an exact science, especially if you want the space to be in some form you can easily recognise. It just so happens that a classifying space for $\Sigma_n$ can be written as the quotient of $$F_n(\mathbb{R}^{\infty})=\prod_{i=1,2,\ldots,n}\mathbb{R}^{\infty}\setminus\{(x_1,\ldots,x_n)\mid\exists i\neq j\mbox{ s.t. } x_i=x_j\}$$ by the equivalence relation $(x_1,\ldots,x_n)\sim(y_1,\ldots, y_n)$ if $\{x_1,\ldots, x_n\}=\{y_1\ldots, y_n\}$. The full space is usually called the ordered configuration space of $n$ points in $\mathbb{R}^{\infty}$ and the quotient $B_n(\mathbb{R}^{\infty})=F_n(\mathbb{R}^{\infty})/{\sim}$ (which is a covering space with fiber equal to $\Sigma_n$) is usually called the unordered configuration space of $n$ points in $\mathbb{R}^{\infty}$.

There is a general method of 'building' classifying spaces (at least for discrete groups) by starting with some CW complex with the correct fundamental group (which can be shown to exist) and then killing off higher homotopy groups by adding in cells of appropriate dimension. This is not usually a finite process.