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I'm looking for a reference (and precise hypothesis if more are needed) for the following facts (or a correction, if I'm just plain wrong):

Let $G$ and $H$ be topological groups and $f : G \to H$ be a continuous homomorphism. Applying the classifying space functor gives a map $Bf: BG \to BH$.

  1. The homotopy fiber of $Bf$ is the homotopy orbit space $H_{hG}$ where $G$ acts on $H$ via $g \cdot h = f(g)h$.

  2. The canonical map $H_{hG} \to H/G = H/\mathrm{im} f$ has homotopy fiber given by $BK$ where $K = \ker f$.

Maybe more well-known that those are the corollaries for when $f$ is either injective or surjective:

  1. If $G$ is a subgroup of $H$, the homotopy fiber of the map induced by the inclusion $BG \to BH$ is the coset space $H/G$.

  2. If $1 \to K \to G \to H \to 1$ is a short exact sequence of topological groups, applying the classifying space functor gives a fiber sequence $BK \to BG \to BH$.

EDIT: I sketched a proof here. I'm mostly looking for references, but would also appreciate alternate proofs (specially a proof that avoids using the generalized Mather cube property on a diagram whose shape is not a 1-category).

EDIT 2: Tyler Lawson's nice example shows that more hypothesis are needed for part 2 of the "fact" and for corollary 1. My current guess is that for corollary 1, it is enough that $H \to H/G$ locally have a section.

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    $\begingroup$ I'm interested in the answers as well but I imagine they all go back to at least Borel, perhaps earlier. $\endgroup$ – Ryan Budney Apr 6 '16 at 20:38
  • $\begingroup$ Borel sounds pretty likely, @RyanBudney. $\endgroup$ – Omar Antolín-Camarena Apr 6 '16 at 20:53
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    $\begingroup$ I don't have a reference, but doesn't an answer to the first question follow from the fact that the homotopy fiber of $Bf$ is the classifying space of a category whose objects are elements of $H$ and morphisms $h \to h'$ consist of $g \in G$ so that $f(g)h' = h$? $\endgroup$ – Vidit Nanda Apr 6 '16 at 20:58
  • $\begingroup$ @ViditNanda: Well, (1) it does follow from that, but you'd need to prove that :); (2) I think your characterization is easy to prove if G and H are discrete, is it also easy in general? $\endgroup$ – Omar Antolín-Camarena Apr 6 '16 at 21:08
  • $\begingroup$ @OmarAntolín-Camarena So just to confirm, what is your preferred definition of the homotopy fiber of $Bf$? $\endgroup$ – Vidit Nanda Apr 6 '16 at 21:39
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One source for some of this is section 8 of May's "Classifying spaces and fibrations"

He has G and H interchanged, unfortunately uses a coset notation for what turns out to be the homotopy fibre, and doesn't talk about the special cases you are after (his interest was more in the topological monoid case).

May is using the two-sided bar construction.

Your sketch doesn't say very much about the topologies involved, and I did wonder if, for example, you need $im f$ closed for your coset space identification.

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  • $\begingroup$ Great! Thanks Mike. This is a perfect reference for part 1 of the proposition (though possibly there are earlier ones). And I'm also worried about needing some topological conditions for things involving actual orbits rather than homotopy orbits. $\endgroup$ – Omar Antolín-Camarena Apr 7 '16 at 0:18
  • $\begingroup$ I think the only spot I might need some sort of hypothesis on the topology in that sketch is the two places where I use that if the action of $G$ on some $X$ is free, then $X_{hG}$ is (weakly) homotopy equivalent to $X/G$. I just realized I don't know exactly what's needed for that to hold. Maybe that needs that $X \to X/G$ is a principal $G$-bundle. $\endgroup$ – Omar Antolín-Camarena Apr 7 '16 at 1:44
  • $\begingroup$ Tyler Lawson has an example here showing you definitely need more hypothesis for the parts that relate to non-homotopy quotients. $\endgroup$ – Omar Antolín-Camarena Apr 7 '16 at 18:24

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