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What is a topological characterization of the class of spaces that have the form $G/H$ for a locally compact, Hausdorff group $G$ and a closed subgroup $H$ ?

Such a space $X=G/H$ necessarily satisfies the following conditions:

1) $X$ is locally compact, Hausdorff.

2) $X$ is (topologically) homogeneous (i.e., for any points $x,y\in X$ there exists a homeomorphism $\phi$ of $X$ such that $\phi(x)=y$).

3) $X$ satisfies the Suslin condition (i.e., every collection of non-empty, disjoint, open subsets of $X$ is countable; this follows from the Haar-measure on $G$).

Are these conditions sufficient? Maybe this is easier if one restricts attention to the class of separable, metric spaces.

Note that homogeneity for $X$ means exactly that the homeomorphism-group Homeo(X) acts transitively on $X$. If $X$ is also locally compact, separable and metric, then Homeo(X) is a separable, complete metric space (in the compact-open topology), and it follows that $X$ is homeomorphic to $G/G_x$, where $G_x$ is the stabilizer subgroup (also called isotopy group). See for instance the following articles for these results:

Effros (1965) # Transformation groups and C*-algebras [Ann. Math. (2) 81]

Ungar (1975) # On all kinds of homogeneous spaces [Trans. AMS 212]

The question now is under which conditions there exists a locally compact subgroup of Homeo(X) which still acts transitively on $X$.

The motivation for this question is to clarify the notion of "homogeneous space". Sometimes in the literature, by a homogeneous space it is not understood a (topological) homogeneous space but a coset space $G/H$ where $G$ is usually even assumed to be a Lie group.

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These conditions are not sufficient, the Sierpinski carpet, etc, are not homogeneous spaces of LC-groups (although compact, metrizable and homogeneous). The only 1-dimensional compact connected spaces occuring as homogeneous spaces of LC-groups are solenoids (projective limits of circles). Except the circle, these spaces are not path-connected. Also, all connected compact manifolds are homogeneous, but in dimension $\ge 2$ most of them are not homogeneous under a locally compact group (it is not hard to check that this implies homogeneous under a connected Lie group) –  YCor May 6 '13 at 17:25
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Condition 3) is not necessary unless you restrict to compact G´s. –  Ramiro de la Vega May 6 '13 at 19:23
    
@Ramiro: indeed, probably (3) should be replaced by: $X$ is a disjoint union of clopen subsets with the Suslin condition. –  YCor May 6 '13 at 23:07
    
@Yves: Thank you for the examples. I also agree with you (and Ramiro) that the Suslin condition is too strong and should be relaxed to what you suggest. –  Hannes Thiel May 7 '13 at 8:28

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