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I would like to gain some intuition about homogeneous spaces. According to the definition, these spaces admit a transitive group action.

How I interpret what homogeneity tells us about a space (let´s call it $X$ and the group action given by a group $G$):

  1. The maps on X coming from elements of the group preserve the structure of X

  2. Transitivity says that we can get from any element to any other element (for any elements $x$, $y$ from the space, there is a $g$ from the group s.t. $x = gy$.)

Is this correct intuition for homogeneous spaces? And what else is "hidden" behind this definition? Since homogeneous spaces have interesting topological properties, I want to understand, how is homogeneity so important and what it tells about the space.

Tereza Tizkova
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    If you see $X$ as a "space" and $G$ as "symmetries" of this space, homogeneity tells you that the space $X$ looks the same at every point with regard to the symmetries of $G$. The symmetries can't detect a specific point or region in $X$. Think of $\mathbf S^2$ and $SO(3)$ for instance. – Adam Chalumeau Jul 29 '21 at 12:19
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    Welcome to MSE! <> Though "intuition" is subjective, I think an abstract answer to your question is difficult: The answer arguably depends on the action (and the invariant structure of the action) rather than on the space. (Offhand, at least six interesting actions come to mind on contractible open subsets of the plane.) What might help is a catalog of examples and non-examples, but that's open-ended and (again) subjective. – Andrew D. Hwang Jul 29 '21 at 12:31
  • @AdamChalumeau Thank you! However, what do you mean by "looks the same at every point"? For example if the $SO(3)$ is group of all rotations about the origin of 3D Euclidean space $\mathbb{R^3}$ under the composition... Could you please be more specific and describe on this example what you meant? – Tereza Tizkova Jul 29 '21 at 12:57
  • You should specify what you meany by a "space" and what structure the group action is supposed to preserve. Say, a "space" means a Riemannian manifold $M$ and the action preserves the Riemannian metric. Then $M\cong G/K$, where $K$ is a compact subgroup of a Lie group $G$. The metric comes from a left-$G$-invariant and right $K$-invariant metric on $G$. – Moishe Kohan Jul 30 '21 at 10:37
  • Maybe this answer can help you: https://math.stackexchange.com/a/4534678/601797 – A. J. Pan-Collantes Sep 20 '22 at 05:46

2 Answers2

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I do not know what kind of answer you are looking for but I can share my understanding.

Basically, homogeneous spaces are analytical versions of cosets.

If you have a group $G$ and a normal subgroup $N\leq G$, then you can also define the quotient space $G/N$ as the set of all cosets $\{gN : g\in G\}$. This space satisfies the following properties:

  1. $G/N$ is a group.

  2. There is a transitive action of $G$ on the space $G/N$ by $g.(hN) = ghN$.

The converse is also true. Suppose that $X$ is any group and there exists a surjective homomrphism $\varphi:G\rightarrow X$ (this defines an action of $G$ on $X$ by $g.x = \varphi(g)\cdot x$.). Then, there is an isomorphism $X\cong G/N$ where $N = \ker \varphi$.

If $N$ is no longer a normal subgroup, then $G/N$ need not be a group. In this case the following are equivalent: There exists a transitive action of $G$ on $X$ if and only if there is a bijection $X\cong G/N$ where $N=\{g\in G : gx_0=x_0\}$ is the stabilizer of some $x_0\in X$.

Usually, when we work with homogeneous spaces we also assume some topology on the groups. $G$ usually a Lie group and the lattice $N$ is often a discrete co-compact subgroup. Then the space $X=G/N$ is a compact manifold. Conversely, if a compact space $X$ admits a transitive (continuous) action of a Lie group, then it is isomorphic to the manifold $G/N$ where $N$ is the stabilizer of some element $x_0\in N$.

Point is, homogeneous spaces are cosets with analytic properties. The main reason they are so useful is because of the interplay between the algebraic and analytic aspects of this space. That is, on one hand it is a coset space while on the other hand it is a differentiable manifold.

Yanko
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What you wrote seems like a definition more than the intuition.

The basic intuition is pretty much what "homogeneous" means in plain English: the space looks the same everywhere.

Perhaps it would be more apparent if you took a different (but relatively easily shown to be equivalent) definition of homogeneity: a space $X$ is homogeneous if the pointed topological spaces $(X, x_0)$ are isomorphic (for all $x_0\in X$). (An isomorphism of pointed topological spaces $(X, x_0)$ and $(Y, y_0)$ is simply a homeomorphism of $X$ and $Y$ which maps $x_0$ to $y_0$.)

The group action formulation is just more convenient (from a certain point of view).

Note that the intuition can be quite different if you consider homogeneous $G$-spaces for a fixed group $G$: those spaces are still topologically homogeneous in the sense mentioned above, but here the focus is on a particular action of particular group (so it is more of a dynamical, rather than a purely topological object), so we can't really forget about the action. It can still be reformulated in the same spirit as I did above with plain homogeneity, but it's much less natural and more complicated.

tomasz
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