# Good functorial model for BG

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. mathoverflow.net/questions/41616/… 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.

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. iopscience.iop.org/0025-5734/8/3/A01 –  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: hausdorff-research-institute.uni-bonn.de/kreck-stratifolds –  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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