Talk:Homogeneous space

I've removed scheme (mathematics) from the introduction. It isn't a straightforward thing to explain what it means for a group action on a scheme S to be 'transitive'; nor what the 'orbits' of such an action are. It would be better to make it algebraic variety; but in any case one can regard that as a special case of a topological space without too much damage.

Charles Matthews 10:47, 21 Aug 2004 (UTC)

Homogeneous space without reference to a group
I've heard the definition of "homogeneous space" without reference to a group or group actions (this was in an introductory topology course), namely a space in which, for any points x, y in X, there exists a homeomorphism f from X to itself such that f(x) = y. Since the homeomorphisms are a group, this is just the action on X by evaluation which is transitive if it satisfies the preceding property. I think (but don't know for sure!) that this definition is pretty common (I'm guessing this from the entry on the Cantor set), so I inserted the "If X is simply called a homogeneous space without reference to a group, it is usually assumed that..." I don't know precisely how usually though... so anyone who knows better can edit away!

Choni 18:34, 27 Aug 2004 (UTC)

Note that the automorphisms do not always form a topological group. Thus, to speak over _continuous_ actions and their homogenous spaces, you should request additionally that $$G$$ is compact and Hausdorff. (Then, one can prove that the automorphisms form a _topological_ group). Mathelerner (talk) 18:50, 14 April 2021 (UTC)

Requirement of continuity
The requirement of continuity seems to me to be overly restrictive: a restriction of convenience for those not interested in other cases, but (or so it would seem to me) to be of no direct import to the concept of a homogeneous space.

An example Where the concept is of interest but where continuity does not apply is with geometries over finite fields. Essentially, the concept is one which says that every point of the space is indistinguishable from every other point: the group of symmetries maps every point to every point in the geometry. Thus, transitivity of the group action is the only requirement. I'm stripping out the continuity requirement in the lead, but feel free to correct me if I did it wrong. —Quondum 00:43, 18 September 2014 (UTC)

simplification
Could someone please add a simplified example for dummies? Some of us just want to get a quick idea of what you're talking about, without all the vocabulary. 23:30, 9 August 2021‎ User:Skysong263


 * Informally, a homogeneous space is a space that "looks the same, everywhere, however you move through it", with "movement" being defined by the action of a group G.


 * I'll try to stuff that into the intro. Not sure if it will be accepted. 67.198.37.16 (talk) 19:43, 11 June 2023 (UTC)

Quotient spaces vs. other spaces?
All of the examples given are quotient spaces of the classical Lie groups. (This matches up with how I learned them.) This is emphasized by the section that talks about cosets. All well and good. Except for two things.


 * (1) The lead paragraph seems to suggest that homogeneous spaces are almost any kind of space with some transitive group action. I don't think that's correct; I think its misleading. Are there examples of homogeneous spaces that are NOT quotient spaces?


 * (2) There is now a section on general relativity. So, in general, "spacetime" in general relativity is some solution to Einstein's equations, and these solutions are not, in general, quotient spaces of some Lie group. However, that paragraph seems to suggest that, sometimes, the "space part" (of spacetime) might be a quotient space of Lie groups. But I can't quite tell if that is what it's trying to say. (This is certainly true for, e.g. the Schwarzschild solution, where the "space part" is just some ordinary spheres and are thus homogeneous in the narrow sense.)

So: are there any examples of a homogeneous space that is not a quotient space of Lie groups? If not, the lead should be cleaned up to directly say "its just a quotient" instead of suggesting its some arbitrary action on some arbitrary space.

Hmm. We also have the article Principal homogeneous space which provides a more formal discussion. In that discussion, there is no call for continuity (an example given is flags on a polytope). That article makes no explicit mention of quotienting; however, it does talk of classifying spaces, which do have quotienting in the definition. I guess I can be happy with a definition that involves "any space with a transitive group action" which, I guess, is the same as saying "there might be a short exact sequence but not necessarily." I'd like an explicit example of the "not necessarily" part, so I can wrap my mind around it. 67.198.37.16 (talk) 20:28, 11 June 2023 (UTC)