Example of building a classifying space

Question

I'm reading some things about algebraic topology, and they mention the classifying space of a group $G$ as $BG$, but they doesn't build one, so I want to ask if someone knows where can I find the way to create one classifying space $BG$ from a given group $G$.

Answer 1

In his paper Construction of Universal Bundles, II, Milnor constructed a universal bundle $EG \to BG$ for any topological group $G$ using the notion of join. It is now called the Milnor (join) construction. Aside from the original paper, this construction can be found, for example, in Appendix B of Spin Geometry by Lawson & Michelsohn.

Answer 2

Note that space $X$ having a contractible (universal) cover is a $B\pi_1(X)$ i.e. a model for $K(\pi_1(X),1)$-space. This is because when $G$ is discrete a principal $G$-bundle is the same as a regular covering.

Thus, when $G$ is discrete other than Milnor construction you can construct $BG$ by the usual construction of $K(G,n)$-spaces (see Hatcher). Briefly, take a presentation of $G$ build a CW-complex $X$ with $\pi_1(X)=G$ and kill higher homotopy groups by attaching cells.