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:
- How can we show the fibration?
- 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.
Best Answer
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 $(EG)/H \cong BH$.
For a great reference on all this stuff written in a very friendly style see:
http://www.math.washington.edu/~mitchell/Notes/prin.pdf