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Let $G$ be a topological group and $H$ be a normal subgroup of $G$ (I think $H$ is required to be admissible in the sense that the quotient map $G\to G/H$ is a principal $H$-bundle, am I right?). Then there exists a homotopy fibre sequence $BH\to BG\to B(G/H)$, where $BG$ denotes the classifying space of $G$.

My questions is: suppose that we already know the groups $G$ and $H$ and suppose that we know the classifying space of $G/H$ and the classifying space of $H$, to what extent can we decide the classifying space of $G$ from these information? How to find the classifying space of $G$ if we know the classifying space of $G/H$ and the classifying space of $H$?

Any reference will be greatly appreciated.

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If you already know the group $G$, don't you automatically know the classifying space of $G$? – S. Carnahan Jan 19 '12 at 9:08
What exactly do you want to know about $BG$? Its homotopy groups are determined by those of $G$. If you're after its (co)homology, use the Serre spectral sequence. – Mark Grant Jan 19 '12 at 9:13
That requires knowing how the transgression BH --> B G/H behaves. In many cases, the Eilenberg_Moore spectral seq is more efficient. For lots of examples, consult Borel or Greub-Hlapernin-VanStone vol III – Jim Stasheff Mar 6 '12 at 22:13
@jim stasheff, may I know which book you are refering to by "Borel"? It seems to me that Eilenbergāˆ’Moore spectral sequences require the base space B to be simply connected; what if this condition does not hold? Thanks in advance!! – Zuriel Apr 2 '12 at 0:51

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