# A short exact sequence of groups and their classifying spaces

Suppose that we have a short exact sequence of topological groups: $$1 \to H \to G \to K \to 1.$$ I have found some papers mentioning that the above sequence induces a fibration: $$BH \to BG \to BK.$$ Here $B$ assigns to each (topological) group its classifying space.

My question:

1. How can we show the fibration?
2. Are there any good books/papers explaining the categorical properties of the classifying space functor $B$?

Note:

• I found a relevant question at MO. But I can not see that $EK \times_K (EG/H)$ has the same homotopy type as $BG$.
• At the same page at MO, the paper "Cohomology of topological groups" (by Segal) is suggested. But it is not available for me. So I am looking for other papers.
• NB: To get the fibration, we need to assume several conditions on the subgroup $H$ of $G$. See Theorem 11.4 in the paper which Dylan introduced below. Commented May 23, 2013 at 14:22

I think there's a typo in that MO answer: $BG$ is modeled by $EG \times_G E(G/H)$ since this is just $BG \times E(G/H)$ and $E(G/H)$ is contractible.
Now we have a natural map $EG \times_G E(G/H) \rightarrow B(G/H)$ which is a fibration with fiber $(EG)/H \cong BH$.