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Clarification of context
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Peter May
Peter May
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Edit: Maybe worth further clarifying the context of Charles' original question and his own answer below. In the early papers that Charles cites, I study general extensions of $G$ by $\Pi$, at first with no restrictions on the topological groups $G$ and $\Pi$. Charles is taking $G$ to act trivially on $\Pi$ and in that case he is proving that the natural map $ B_G \Pi = Map(EG,\Pi)/\Pi \to Map(EG,B\Pi)$ is a weak $G$-equivalence when $G$ and $\Pi$ are compact Lie groups with $\Pi$ a $1$-type, so a split extension of a torus by a finite group. The early work had only shown that when $\Pi$ is abelian or when $\Pi$ is discrete (in which case the map is a homeomorphism). The related questions (1) and (2) above are about categorical models for equivariant classifying spaces in the intermediate generality of split extensions of $G$ by $\Pi$ (so with $G$ acting non-trivially on $\Pi$), as studied with Guillou and Merling in http://arxiv.org/pdf/1201.5178.pdf.

Edit: Maybe worth further clarifying the context of Charles' original question and his own answer below. In the early papers that Charles cites, I study general extensions of $G$ by $\Pi$, at first with no restrictions on the topological groups $G$ and $\Pi$. Charles is taking $G$ to act trivially on $\Pi$ and in that case he is proving that the natural map $ B_G \Pi = Map(EG,\Pi)/\Pi \to Map(EG,B\Pi)$ is a weak $G$-equivalence when $G$ and $\Pi$ are compact Lie groups with $\Pi$ a $1$-type, so a split extension of a torus by a finite group. The early work had only shown that when $\Pi$ is abelian or when $\Pi$ is discrete (in which case the map is a homeomorphism). The related questions (1) and (2) above are about categorical models for equivariant classifying spaces in the intermediate generality of split extensions of $G$ by $\Pi$ (so with $G$ acting non-trivially on $\Pi$), as studied with Guillou and Merling in http://arxiv.org/pdf/1201.5178.pdf.

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Peter May
Peter May
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Charles, thanks for asking. This is not an answer, but it is too long for a comment. Like you, I encourage others to pursue the question and closely related ones I'll raise here. The paper of mine you cite is miserably written and hard to parse for what is proven in what generality. It is written more generally in terms of extensions $\Gamma$ of $G$ by $\Pi$ rather than just products $G\times \Pi$, and much of it does not restrict to compact Lie groups. Your (2) is its Theorem 10 (which has a typo $\Gamma$ for $G$), and that is just a reinterpretation of the result in the Lashof-May-Segal paper you cite. Its Theorem 5 specializes to say that your map $f$ is an equivalence for any topological group $G$ and discrete group $\Pi$. As you say, that is just an exercise in covering space theory.

More substantially, its Theorem 9 uses consequences of the Sullivan conjecture due to Dwyer and Zabrodsky and Notbohm to partially answer a kind of opposite question to the one you ask. When $G$ (not $\Pi$) is an extension of a finite $p$-group by a torus and $\Pi$ is a compact Lie group it specializes to show that your map $f$ induces an isomorphism on mod $p$ homology, with a stronger statement when $G$ is finite.

The related question goes as follows. One can ask for categorical models of equivariant classifying spaces. Let $G$ be a topological group and let $\tilde G$ denote the ``chaotic'' topological category with object space $G$ and morphism space $G\times G$, so that there is a unique morphism $g\to h$ for each pair of elements. Think of the topological group $\Pi$ as a topological category with a single object. (I'm actually more interested in the more general situation where $G$ acts on $\Pi$.) Then the classifying space of the topological category $Cat(\tilde G,\Pi)$ is another candidate for $B_G\Pi$, and there is a natural map $BCat(\tilde G,\Pi) \to Map(EG,B\Pi)$. At least if $G$ is discrete and $\Pi$ is either discrete or a compact Lie group, $B_G\Pi$ is equivalent to $BCat(\tilde G,\Pi)$ by a result of Guillou, Merling, and myself http://arxiv.org/pdf/1201.5178.pdf. The related questions then are

(1) How generally is $BCat(\tilde G,\Pi)$ equivalent to $B_G\Pi$?

(2) How generally is $BCat(\tilde G,\Pi)\longrightarrow Map(EG,B\Pi)$ a $G$-equivalence?

A key point in Charles' question is that $Map(EG,E\Pi)$ is a universal principal $(G,\Pi)$-bundle in complete generality (by Theorem 5 in my early paper he cites), so that $B_G\Pi$ is $Map(EG,E\Pi)/\Pi$. There is an evident natural map $$ Map(EG,E\Pi)/\Pi \longrightarrow Map(EG,B\Pi).$$ It is a $G$-equivalence if $\Pi$ is discrete, and a rephrasing of Charles' question is to ask ask how generally that map is a $G$-equivalence. An obvious diagram (Section 5 of the GMM paper) relates this question to my questions.