Let $A\rightarrow B\rightarrow C$ be a short exact sequence of topological groups such that $B$ and $C$ are connected CW-complexes, can we prove that $A\rightarrow B\rightarrow C$ is a homotopy fiber sequence of underlying topological spaces ?
If $B\rightarrow C$ is a A-Principal Bundle then $A\rightarrow B\rightarrow C$ is clearly a fiber sequence. I could not find any reference to my question, most of the theory in the subject is around the locally compact groups. My question aims to find a sort of a generalization... I hope the question will be reopen.