Suppose $G$ is a Topological group then classification theorem of Principal $G$ bundles says that
there is a Principal $G$ bundle $EG\rightarrow BG$ such that any principal $G$ bundle over a decent topological space $X$ has to be pullback of a continuous map $f:X\rightarrow BG$.
Can we replace Topological group by Lie group and Topolgical space by Smooth manifold. Do we get all Principal $G$ bundles over smooth manifold in this case? Is $BG$ a smooth manifold??
In his book Fiber bundles, Dale Husemoller does not say anything (I could not see anything) about smooth version of that classification result. Now I have a doubt if that Milnor constriction $BG$ for a Lie group $G$ gives a smooth manifold or is this classification only for topological Principal $G$ bundles.
In similar way, when doing classification of vector bundles we construct what is called Grassmannian $G_n$ for each $n$ and a topological vector bundle $E_n\rightarrow G_n$ and say that for a decent topological space $X$, any rank $n$ vector bundle (in Topological sense, not smooth sense) over $X$ should be pullback of a continuous map $X\rightarrow G_n$. Here also we are classifying only topological vector bundles, not smooth vector bundles, right? I was thinking $G_n$ is a manifold and it classifies all smooth vector bundles but then realise I am thinking wrong.
Is the classification only restricted to Topological set up?
Is there similar classification in smooth set up? Like classifying smooth Principal $G$ bundles and classifying smooth vector bundles?