Two questions (more details below):

  1. Let $G$ be a compact Lie group and $X$ a $G$-space such that all stabilizer subgroups are conjugate to a fixed $H \leq G$. Denote by $\pi: X \to X/G$ the quotient map. Under which conditions on $X$ is $\pi$ a Serre fibration?

  2. Let $G$ be as above and $F$ a free $G$-space. Under which conditions is the canonical map $F_{hG} \to F/G$ from the homotopy quotient to the quotient a weak equivalence?

All spaces are assumed to be CGWH spaces.

Relation between 1. and 2.:

I am most interested in 2.. However, if $F$ is a free $G$-space, $EG$ denotes Bar construction, such that $(F \times EG)/G$ models $F_{hG}$, and both $F$ and $F \times EG$ are as in 1., i.e., the respective quotient maps are Serre fibrations, then it is not hard to see that $F_{hG} \to F/G$ is a weak equivalence, using the long exact sequence of homotopy groups.

So, any condition for 1. that holds for $EG$ and is stable under products provides a condition for 2.

Results known to me:

  • A sufficient condition for 1. is being a completely regular Hausdorff space (also known as a Tychonoff space) by a result in Bredon's "Introduction to Compact Transformation groups." He shows that the quotient map is a fiber bundle in this case. However, the Tychonoff property does not seem to be preserved by CGWH products, so it will not hold for many constructions which one would like to have results as in 1. or 2. for.

  • A sufficient condition for 2. should be that $F$ is a (retract of a) free $G$-CW-complex by using model category arguments. However, this also a rather severe condition that can be hard or impossible to check in practice.

Thus, I would like to know if there any other known results, preferably with a reference, that improve the sufficient conditions outlined above. Any results for more arbitrary topological groups will also be appreciated.