For which G is BLG weak homotopy equivalent to LBG?

Question

Let $G$ be a (Edit: path-)connected topological group. Under what additional hypotheses on $G$ is it true that $LBG$ is a classifying space for $LG$? (or, I guess equivalently, when is $LBG \sim BLG$?) Here I'm taking the free loop space, and the compact-open topology on it. I know it is true for strong hypotheses, such as $G$ a compact Lie group. If one can find a model of $BG$ that is locally contractible and paracompact, then by Atiyah and Bott's The Yang-Mills Equations over Riemann Surfaces (doi:10.1098/rsta.1983.0017), Proposition 2.4, I believe it is possible. So, alternatively, for what $G$ is it true that $BG$ can be thus chosen?

Answer 1

[UPDATE: There were some mistakes in the first version. Here is a more careful account.]

I'll work everywhere with CGWH spaces, so I have a Cartesian closed category.

Note that $BLG$ is always path-connected, but $\pi_0(LBG)=\pi_0(G)/\text{conjugacy}$, so we need to assume that $G$ is path-connected. (The question says connected, but this may be a bit stronger; I do not know whether there are connected topological groups that are not path-connected.)

For any space $X$ and any $t\in S^1$ we have an evaluation map $\epsilon_{X,t}\colon LX\to X$. For the special case of the basepoint $1\in S^1$ we write $p_X=\epsilon_{X,1}\colon LX\to X$. This is always a Hurewicz fibration. If $X$ is based we have a fibre sequence $\Omega X\xrightarrow{i_X}LX\xrightarrow{p_X}X$.

Now let $G$ be a topological group. We write $EG$ and $BG$ for the usual simplicial constructions, so $EG$ is contractible and has a free $G$-action with orbit space $BG$. Simplicial methods also give a natural commutative diagram $\require{AMScd}$ \begin{CD} G @>k_{G}>> EG @>r_{G}>> BG\\ @Vj_GVV @VV l_G V @VV 1 V\\ \Omega BG @>>> PBG @>>> BG \end{CD} Both $EG$ and $PBG$ are contractible, and the bottom row is a Hurewicz fibration. If the top row is also a Hurewicz fibration, we can conclude that $j_G\colon G\to\Omega BG$ is a homotopy equivalence. If the top row is merely a Serre fibration or quasifibration, we can still conclude that $j_G$ is a weak equivalence. I do not know what are the minimal conditions for the top row to be a quasifibration.

Next, given $u\in BLG$ and $t\in S^1$ we have a homomorphism $\epsilon_{G,t}\colon LG\to G$ and thus a map $B\epsilon_{G,t}\colon BLG\to BG$ and thus an element $(B\epsilon_{G,t})(u)\in BG$. We would like to define $f_G\colon BLG\to LBG$ by $(f_G(u))(t)=(B\epsilon_{G,t})(u)$. To justify this we need to check continuity in $t$ and then in $u$. This in turn needs some continuity properties of the functor $B$, which can be proved using some abstract nonsense with CGWH spaces and Cartesian closure. (The initial version of this answer referred to a natural map in the opposite direction, but I think that does not actually exist.) We now want to construct a diagram as follows: \begin{CD} B\Omega G @>Bi_G>> BLG @>Bp_G>> BG \\ @V f'_G VV @V f_G VV @VV 1 V \\ \Omega BG @>>i_{BG}> LBG @>>p_{BG}> BG \end{CD} Constructions that we have already discussed provide all spaces and maps except for $f'_G$. On the top row we note that $(Bp_G)\circ (Bi_G)$ is trivial, and on the bottom row we know that $i_{BG}$ is the fibre of $p_{BG}$, so there is a unique way to fill in $f'_G$. The bottom row is always a Hurewicz fibration. The top row is obtained by applying $B$ to a Hurewicz fibration of topological groups, but it is not clear exactly what we get from that. If the top row is at least a Serre fibration, we see that $f_G$ is a weak equivalence iff $f'_G$ is a weak equivalence.

Finally, define $\tau\colon S^1\wedge S^1\to S^1\wedge S^1$ by $\tau(s\wedge t)=t\wedge s$. From the definitions one can check that the following diagram commutes: \begin{CD} \Omega G @>j_{\Omega G}>> \Omega B\Omega G \\ @V \Omega j_G VV @VV \Omega f'_G V \\ \Omega^2 BG @>>\tau^*> \Omega^2 BG \end{CD}

If $j_G$ and $j_{\Omega G}$ are weak equivalences, we conclude that $f'_G$ is also a weak equivalence.

All this assumes that we start with the simplicial definition of $BG$. One could instead consider an axiomatic characterisation of $BG$, which might include the condition that the map $EG\to BG$ is a Serre fibration. The idea should be that $[X,BG]$ should biject with the set of isomorphism classes of principal $G$-bundles over $X$, but one would need to restrict attention to paracompact $X$, or to principal bundles over arbitrary $X$ that admit a numerable trivialising cover. I do not know how the technicalities would work out.