Good functorial model for BG

Question

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?

Answer 1

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.

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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:

(source: Wayback Machine)

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.

Answer 2

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.