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There are several functorial constructions of the space BG for a topological group (meaning BG plus the universal G-bundle). First, there is the Milnor construction, treated in several textbooks. The Milnor construction is functorial and $EG \to BG$ is locally trivial for all topological groups. The Milnor construction is NOT monoidal in the sense that $B(G \times H) \cong BH \times BG$ ($B1$ is something like an infinite-dimensional simplex and not a point). On the other hand, there is the nerve-construction $BG:= |N_{\bullet} G|$ (plus a construction of $EG$). This is monoidal, but the map $EG \to BG$ is not always locally trivially (according to Graeme Segal, Classifying spaces and spectral sequences, p. 107). It is locally trivial if G is "locally well-behaved" (Segal gives a precise condition). Segal claims that if G is not locally well-behaved, then local triviality is not an appropriate concept. I would be happy to exclude groups like the p-adic integers from having a classiying space, but there are other groups which I do not like to throw away, like Homeo (X) for a manifold X (is this locally well-behaved??). Here is my question:

Is there a construction of $BG$, satisfying the following properties:

  1. functorial,
  2. monoidal,
  3. $EG \to BG$ is locally trivial,
  4. the class of groups to which it applies is "very large", including Homeo of reasonable spaces,
  5. Simple enough to be reasonably presented in a lecture course?
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I suppose that you want $EG\to BG$ to be a principal $G$-bundle. – Tom Goodwillie Oct 19 '10 at 19:24
Sure, but once it is locally trivial, this is clear for all constructions I have seen. – Johannes Ebert Oct 19 '10 at 19:34
It is for a lecture course in homotopy theory. Once I have monoidality, I can define Eilenberg-Mac-Lane spaces in 2 lines. Of course simplicial sets like EM-spaces are well-pointed and I can use the nerve construction. But I prefer to give \emph{one} construction that serves all purposes at once. – Johannes Ebert Oct 19 '10 at 22:43
You probably saw this, but this MO answer seems relevant to the discussion here. It shows how bad the usual nerve construction of BG can be when G is any old topological group.… In your question you say you're not worried about things like p-adic groups. Are the classes of locally contractible or well-pointed groups too restrictive for you? – Chris Schommer-Pries Oct 20 '10 at 14:21
The locally contractible groups are too weak for my purposes, because I wish to talk about groups as Homeo(X) for a space X as arbitrary as possible and Diff(M), without justifying in the lecture course why they are locally constactible. – Johannes Ebert Oct 20 '10 at 18:34

Segal wrote another paper in which this question came up.

Cohomology of topological groups In Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69), pages 377{387. Academic Press, London, (1970).

In the appendix to this paper he says "The following proposition replaces the vague remarks on the same subject in [Categories and Classifying Spaces]"

Then a proposition follows (Prop. A1) which states that G locally contractible is sufficient to guarantee that $EG \to BG$ admits local sections. Here we are using the geometric realization of the nerve.

So if the Homeomorphism group of a manifold is locally contractible, you are in business. I don't remember if this is the case.

Added Later (Based on comments below).

The paper that Jeremy Brazas cites below helps answer this question, and the answer seems to be that for both compact and non-compact manifolds, if they are reasonable, then the Homeomorphims group (with the compact-open topology) is locally contractible.

For general non-compact manifolds this statement fails, but the counter examples are things like this:

The paper in question is

Černavskiĭ, A. V. Local contractibility of the group of homeomorphisms of a manifold. (Russian) Mat. Sb. (N.S.) 79 (121) 1969 307–356.

More precisely Theorem 2 of this paper states:

Theorem: If the manifold Μ is the interior of a compact manifold N, then Homeo(M) is locally contractible.

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Thanks, Chris; I'll have a look at Segals paper. I doubt that it is easy to see that Hoemo (X) is locally contractible (if true at all). – Johannes Ebert Oct 19 '10 at 20:27
Homeo(X) seems to be locally contractible for compact manifolds but not always for non-compact manifolds. – Jeremy Brazas Oct 19 '10 at 20:30
Awesome embedded image, Chris. – Todd Trimble Oct 19 '10 at 22:02
To be fair, I stole the image from my mathematical grandfather: – Chris Schommer-Pries Oct 19 '10 at 22:07

Segal's classifying space $BG$ (the geometric realisation of the nerve of $G$ considered as a one-object topological groupoid) and the associated universal bundle (the geometric realisation of the nerve of the action groupoid of $G$ acting on itself by mulitplication, or equivalently, the codiscrete groupoid with objects $G$) was also studied by May, Milgram and Steenrod. These latter three only require that $G$ be well-pointed: the inclusion of the identity element is a closed cofibration. I note however that this is all done in the category of $k$-spaces.

The nice result of May is that $EG\to BG$ is not just a locally trivial bundle, but a numerable bundle, that is, there is a trivialising cover of $BG$ such that this cover admits a subordinate partition of unity. Thus $EG \to BG$ classifies numerable bundles (over paracompact spaces these are clearly the same as bundles).

Additionally, $EG$ is a topological group. Also since construction this uses ordinary geometric realisation, $E(-) \to B(-)$ preserves products (and indeed pullbacks) - this where the geometric realisation into k-spaces is used.

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