# For which G is BLG weak homotopy equivalent to LBG?

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?

[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.

• Why "connected" is not enough ? – GSM Mar 13 at 9:57
• @GSM Because $\pi _0(X)=[S^0,X]$ is the set of homotopy class of maps from $S^0$ to $X$, in other words, the set of path-connected components, and not the set of connected components. – user43326 Mar 13 at 10:44
• @Neil re connected vs path-connected. Yes, was being a bit sloppy there. Path connected is fine by me, I'm not considering topological groups with bad local connectedness properties. – David Roberts Mar 13 at 11:25
• Hmm, maybe I accepted this too quickly. How do we know the left and right maps are weak equivalences? Certainly the spaces involved are weakly equivalent, but it's not obvious to me the the right map even is a weak homotopy equivalence. – David Roberts Mar 14 at 7:41
• $\Omega LX\cong L\Omega X$. Therefore if $X$ is a delooping of $G$ then $LX$ is a delooping of $LG$. – Tom Goodwillie Mar 16 at 1:19