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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.

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Any connected space $X$ is weakly equivalent to $BG$, where $G$ is a topological group weakly equivalent to $\Omega X$ that can be constructed using Kan's simplicial loop functor. Thus, you are effectively asking about $\text{Map}(X,Y)$ for arbitrary connected $X$ and $Y$, which is too general to say anything useful. –  Neil Strickland Feb 11 '13 at 9:57
sure, thank you, I wanted to know a summary of all well known cases... little bit more general than compact Lie groups. I was not enough precise. –  Fedotov Feb 11 '13 at 10:11

2 Answers 2

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.

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Don't the papers you mention treat compact Lie groups in which the OP doesn't seem to be interested (judging his/her comment above) ? –  Ralph Feb 12 '13 at 7:50
Thank you Peter for the references. If I'm not wrong the examples that you gave are more related to the case of (compact) Lie groups as Ralph has noticed. –  Fedotov Feb 12 '13 at 10:17
I thought anyone interested in the mathematics should know about those papers, even if they are not strictly germane. –  Peter May Feb 12 '13 at 14:36

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.]

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Thank you for the references Andreas! –  Fedotov Feb 12 '13 at 10:14

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