mapping space between classifying spaces I wanted to ask a summary of known results and references about the homotopy type of the mapping space $\mathrm{Map}(BG,BK)$ (and specially the connected components) between the classifying spaces when G and K are general topological groups. 
Thank you. 
 A: As Neil Strickland points our, this is too difficult to have a good answer in general. In this post, I am only considering discrete groups. A very special case is the following


Definition Let $f : BG \to BK$ be a continuous map. We say that $f$ is a superposition, if for any ${\mathbb Q} K$-module $L$, the induced map on equivariant homology $$H^G_\ast(G,f^\ast(L)) \to H^K_\ast(K,L)$$
    is surjective.


Examples of superpositions include retractions, maps between oriented manifolds of non-zero degree, maps whose homotopy fibre is a finite complex with non-zero Euler characteristic. In the application below, one needs a much weaker condition which might be checked by hand for particular maps. The following result can be proved:


Theorem Let $BK$ be a finite complex. Let $f \colon BG \to BK$ be a continuous superposition. If $\chi(BK)  \neq 0$, then the mapping space $map(BG,BK,f)$ (that is the connected component of $f$) is contractible.


This is (a special case of) a consequence of results in [Daniel H. Gottlieb. Covering transformations and universal fibrations. Illinois J. Math., 13:432–437, 1969., Daniel H. Gottlieb. Self coincidence numbers and the fundamental group.
J. Fixed Point Theory Appl. 2 (2007), no. 1, 73–83.] and was proved in
[Thomas Schick and Andreas Thom. On a conjecture of Gottlieb. Algebr. Geom. Topol. 7 (2007), 779–784.]
A: The Sullivan conjecture, a beautiful theorem proved by Haynes Miller
(The Sullivan conjecture on maps from classifying spaces) has striking
consequences about maps between classifying spaces when the target is
completed at a prime $p$.  Some references are Dwyer and Zabrodsky
(Maps between classifying spaces), Jackowski, McClure, and Oliver
Homotopy classification of self-maps of $BG$ via $G$-actions. I, II;
Self-homotopy equivalences of classifying spaces of compact connected 
Lie groups); Notbohm (Maps between classifying spaces).  
The classical work of J.F. Adams on this topic is also well worth
remembering: Maps between classifying spaces (I), II, III and 
Maps between $p$-completed classifying spaces.
