It looks like Ellis' papers that you're talking about (and Loday's too) are about classifying spaces of groups. Am I missing something?

Suppose $G=<t|t^m>$. Then $BG$ can be constructed inductively cell by cell by killing off higher homotopy and taking limit of the resulting spaces. It's (mostly) clear that the result is an infinite dimensional lens space. I would be happy seeing any example where someone says, here is a crossed complex, here is the construction, and here is the classifying space.

This is a really good book. Students in the class don't tend to like it at the time since most of the theorems really are given as problem. But I think they tend to learn the material much better than more "standard" texts like Munkres.

Should have listed that one! I've been fighting with some differentials from that spectral sequence lately.

The standard construction is actually in Hatcher section on $K(G,1)$'s on page 89. A more "hands on" construction exists if you have a finite presentation of your group. The only reference I know of is in an appendix in Knudson's "Homology of Linear Groups". I don't have the book handy and google books doesn't offer a preview of it so I can't give a more specific location, sorry. Basically, you construction a reduced CW complex dimension by dimension killing off higher homotopy as you go.

They are the same thing as long as $G$ has the discrete topology. If $G$ is a topological group, etc, $BG$ won't necessarily be $K(G,1)$. $G \to EG \to BG$ is a fibration so $\pi_n BG = \pi_{n-1} G$.

Right! The proof that the homology doesn't change (in the general construction) and that Quillen's K-groups agree with the classical ones (in the BGL case) are beautiful.

@Michael: My fault for posting before morning coffee. Vectornaut's link has a good picture. If it's done with a coffee cup it's easy to see when you've went around 360 degrees. Bredon's "Topology and Geometry" has a nice picture of this, but I can't find it online.

Is it possible (and convenient) to use non-standard packages?