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?

4more comments