Take G to be a group. I care about discrete groups, but the answer in general would be welcome too. There are the various ways to construct the classifying space of G, bar construction, cellular construction if G is finitely presented, etc.


What I'm wondering about, is there a notion of a smooth classifying space? That is, when can a classifying space for a group be given a smooth structure?



Smooth classifying spaces?
The answer to this does depend highly on the category in which you are prepared to work. If by "smooth structure" you mean "when is BG a **finite dimensional** manifold" then the answer is, as Andy says, "not many".


However if you are prepared to admit that there are more things that deserve the name "smooth" than just finite dimensional manifolds, then the answer ranges from "a few" to somewhere near "all".


To illustrate this with examples, the classifying space of ℤ is, of course, S1 whilst the classifying space of ℤ/2 is ℝℙ∞. Both are manifolds, but only the first is finite dimensional.


Here are some more details for the "somewhere near all". Take any topological model for BG. Then consider all continuous maps ℝ → BG. These correspond to G-bundles over ℝ. Amongst those will be certain bundles which deserve the name "smooth" bundles. By taking the corresponding curves, one determines a family of curves ℝ → BG which should be called "smooth". Using this one can define a Frölicher space structure on BG. (It is possible that you will get more smooth bundles than you bargained for this way. If that's a problem, you could work in the category of diffeological spaces but then you'd need to use all the ℝns).


In the middle, one can consider infinite dimensional manifolds. Then as your group is discrete it would be enough to ensure that you have a properly discontinuous action on an infinite sphere (there's a question somewhere around here about that being contractible). Some would say that your sphere "ought" to be the sphere in some Hilbert space. Failing that, if you have a faithful action on a Hilbert space (or more generally Banach space) with one or two topological conditions then you can quotient the general linear group by your group. Indeed, if your group is discrete then take the obvious action on ℓ2(G) (square summable sequences indexed by G).


A good example, but which is about as far from your situation for discrete groups as possible, is that of diffeomorphisms on a manifold. The classifying space of this group is the space of embeddings of that manifold in some suitable infinite dimensional space.


For more on the categories behind all this, see the nlab entries starting with [generalised smooth spaces](http://ncatlab.org/nlab/show/generalized+smooth+space) and the references therein. Also, anything by Kriegl, Michor, or Frolicher in the literature is worth a look.


