Are all quotients of a weakly contractible space via a free group action classifying spaces of the group?

Question

I asked this question on math.stackexchange a week ago, but did not get an answer.

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First of all, I don't want to restrict to any kind of "nice spaces" since I am interested in the most general situation. Especially I do not work in any "convenient" category of spaces.

Wikipedia defines a classifying space of a topological group $G$ as the quotient space of a weakly contractible space $EG$ with respect to a free action of $G$ on $EG$, which confuses me.

For me, a classifying space $BG$ should be a space for which there exists a principal $G$-bundle $$G\rightarrow EG\rightarrow BG,$$ such that for sufficiently nice spaces $X$ the homotopy classes from $X$ to $BG$ are in bijection (via pullback of the bundle $EG\rightarrow BG$) to the set of isomorphism classes of principles $G$-bundles on $X$. I think it was Milnor (Can you give me a reference?) that showed that universality of the bundle $EG\rightarrow BG$ is equivalent to the (weak?) contractibility of $EG$.

From this viewpoint the definition of wikipedia suggests the quotient map of any weakly contractible space to the quotient via a free group action of $G$ being a principle $G$-bundle. This sounds strange to me as I would except, that I will need more topological restrictions on my spaces for this to work.

So here is the question: Which are the exact most general topological restrictions to each space, such that the situations I described work out? To be more specific:

1. When is the quotient map of a space with respect to a free action of a topological group $G$ a principal $G$-bundle?
2. Does the definition of wikipedia gives me always the right thing?
3. What is the canonical reference for the claim that a principal $G$-bundle is universal iff it's total space is (weakly?) contractible?
4. Which topological restrictions on $X$ do I need such that $[X,BG]$ classifies what it should classify?

Answer 1

I'll try to say as much as I can about each of the four questions above.

1. If $X$ is a space with an action of a topological group $G$, then the quotient map $X\to X/G$ is a principal bundle if and only if a) the action admits local slices and b) the map $X\times_{X/G} X\to G$, sending $(x, y)$ to the unique element of $G$ satisfying $xg = y$, is continuous. Husemoller's book on fiber bundles discusses this.

2. The Wikipedia article is a bit sloppy, as shown by the example of $X = G^{i}$, the set $G$ with the indiscrete topology.

3. I think Husemoller's book proves that if $E$ is weakly contractible and $E\to E/G$ is a principal $G$-bundle, then $E/G$ is universal in the sense that for every CW complex $X$, there is a bijection between $[X, E/G]$ and Prin$_G (X)$, the set of isomorphism classes of principal $G$-bundles over $X$. There is a discussion of this in my course notes: http://www.math.iupui.edu/~dramras/601.html

You're also interested in the converse statement, that if $P\to B$ is a principal $G$-bundle that is universal in an appropriate sense, then $P$ has to be weakly contractible. I don't recall seeing this sort of statement in the literature. Here's a simple version: if there exists a CW complex $B$ with a principal $G$-bundle $P\to B$ such that $P$ is contractible, then a Yoneda-type argument says that if $P'\to B'$ is a principal $G$-bundle over a CW complex which is universal in the sense that for all CW complexes $X$, $[X, B']$ is in bijection with Prin$_G (X)$, then there's a homotopy equivalence $f: B'\to B$ such that $f^*(P) = P'$. Then the LES in homotopy shows $\pi_* (P') = \pi_* (P) = 0$.

I don't know whether Milnor wrote anything about this. I don't see it in his 1956 Annals papers "Construction Universal Bundles, I and II."

4. One answer is "take $X$ to be a CW complex." But you probably want more than this. Segal discusses something more general in Section 4 of his paper Classifying Spaces and Spectral Sequences. He puts some conditions on the group, so as to obtain a classifying space $BG$ such that for all paracompact spaces $X$ he gets the desired bijection between $[X, BG]$ and the set of isomorphism classes of principal $G$-bundles over $X$. (Note that on a paracompact space all covers are numerable in the sense Segal discusses, which is just to say they admit subordinate partitions of unity.) There may be related ideas in old papers of Dold.