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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.
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  • $\begingroup$ 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. $\endgroup$
    – H. Shindoh
    May 23, 2013 at 14:22

1 Answer 1

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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

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