# Classifying spaces of topological groups whose underlying spaces are homotopy equivalent

Let $G$, $H$ be topological groups and $f:G\rightarrow H$ a continuous group homomorphism which happens to be a homotopy equivalence of the underlying topological spaces. Let us assume that $G$, $H$ are well-pointed compactly generated Hausdorff as topological spaces, where well-pointedness means that the inclusions of basepoints are closed cofibrations. (By well-pointedness, there is no difference between a homotopy equivalence and a based homotopy equivalence.) Let $B$ be the classifying space functor.

My question is: Is $Bf: BG \rightarrow BH$ a homotopy equivalence? (Again, there is no difference between based and non-based homotopy equivalence since $BG$ is well-pointed if $G$ is.)

I understand that $Bf$ is a weak homotopy equivalence even without the assumptions made above on the topologies of $G$ and $H$, by this post. I would like it to be a homotopy equivalence with those extra assumptions. Can we show $BG$ and $BH$ have the homotopy type of CW complexes or something?

• If $BG$ has the homotopy type of a CW complex then $G=\Omega BG$ has it too. – Denis Nardin Mar 29 '15 at 0:01
• You might try looking at Segal's paper Classifying Spaces and Spectral Sequences, where he explains that Milnor's infinite join construction of $BG$ can be built as the classifying space of a certain topological category $G_N$. Then induced functor $G_N \to H_N$ will induce a level-wise homotopy equivalence between the nerves of these cateogies (whose realizations give $BG$ and $BH$)... – Dan Ramras Mar 29 '15 at 0:46
• (cont'd) I think your conditions on $G$ and $H$ ought to imply that these nerves are good simplicial spaces. But I'm not certain if that's ensure that a level-wise homotopy equivalence induces a homotopy equivalence of realizations. Goerss and Jardine's book would be one place to look. – Dan Ramras Mar 29 '15 at 0:48
• @user46652: you need to specify which classifying space functor $B$ you are using for the question to make any sense. – John Klein Mar 29 '15 at 1:49
• $BG$ is fundamentally an object which only makes sense up to weak homotopy equivalence; asking questions like this means getting bogged down in the technicalities of spaces not having the homotopy type of a CW complex and there's just no reason to torture yourself like that. – Qiaochu Yuan Mar 29 '15 at 3:31

As John Klein remarked, the answer to this question will depend on the classifying space functor $B$ one uses.
We define $BG$ for a topological group $G$ to be the fat realization of the simplicial space obtained by applying the topological nerve construction to the topological category one obtains by regarding $G$ as a category in the usual way and including its topology. (See Segal's 'Classifying Spaces and Spectral Sequences' §3) A continuous group homomorphism $G\rightarrow H$ which is a homotopy equivalence will induce a morphism of the corresponding simplicial spaces which is a degreewise homotopy equivalence. (This is easy to check, right from the definitions.) Since we have chosen the fat realization the induced map $BG\rightarrow BH$ will be a homotopy equivalence by Proposition A.1 in Appendix A in Segal's 'Catgeories and Cohomology Theories'.
• This implies that $EG \to BG$ is always a principal bundle if I use the fat geometric realization for $EG$ and $BG$. Where can I find this statement in the literature? – Ulrich Pennig Jul 7 '15 at 15:39