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?
Best Answer
As John Klein remarked, the answer to this question will depend on the classifying space functor $B$ one uses.
Let me present one case for which the question can be answered positive which is basically the case Dan Ramras mentioned.
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' ยง3) 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'.
Up here, no point-set topological restrictions are needed, it even all works if the spaces are not compactly generated.
In the last paper cited, you'll also find information about the question, in which cases the fat realization is homotopy equivalent to the usual one. In the case of well pointed compactly generated groups, their simplicial nerves will be "good" in the sense of Segal. A survey on the results of that manner is given here.