Are there such things as 'locally homogenous spaces'?

Question

A Euclidean space has the property that every point has a neighbourhood that is homeomorphic to some neighbourhood of any other point. I'm not sure what the name of this property is - I thought it might be a homogenous space - but looking at Wikipedia this seems to be a different but related idea.

So, the first question, what it is the standard name for this property?

For the purpose of this question I'll call a space with this property a homogenous space.

Now, fixing an open n-ball; a topological manifold is characterised as being locally homeomorphic to this open ball.

Is there a useful generalisation where one replaces the open n-ball by an arbitrary homogenous space? (Then such a 'generalised' manifold will also be homogenous).

A nice specific example which isn't a smooth manifold would be very helpful; for some reason I was thinking perhaps this might be more likely within Algebraic Geometry - if the above considerations even make sense there.

Answer 1

I answer to the second question

I think that the right name is $(X,G)$-space.

Let me be more precise.

Let $X$ be a topological space and $G$ be a group of homoeomorphisms of $X$, (for instance $G$ is the whole group of homeomorphism).

A space $Y$ is called an $(X,G)$-space if

1) it has an open covering $\{U_i\}$ with maps $\phi_i:U_i\to X$ so that $V_i=\phi_i(U_i)$ is open in $X$ and $\phi_i$ is an homeomorphism from $U_i$ to $V_i$;

2) for each $U_i\cap U_j\neq \emptyset$ the map $\phi_i\circ\phi_j^{-1}:\phi_j(U_i\cap U_j)\to \phi_i(U_i\cap U_j)$ is the restriction of a map in $G$.

Examples

1) $X=\mathbb R^2$ and $G$ the group of homeos, then you get topological surfaces;

2) $X=\mathbb R^2$ and $G$ the group of diffeomorphism, then you get differentiable surface;

3) $X=\mathbb R^2$ and $G$ the group of isometries, then you get Euclidean surfaces;

4) $X=\mathbb C$ and $G$ the group of biholomorphism, then you get Riemann surfaces;

Answer 2

The concept you mention might be called "locally modeled by", as in "a topological manifold is locally modeled by Euclidean space".

The usual definition of "homogeneous" (for a geometric structure) refers to the action of the group of automorphisms of the structure. In this sense, Euclidean space is "homogeneous" because if $p$ and $q$ are arbitrary points, there exists a Euclidean motion carrying $p$ to $q$. (That's a particularly strong assertion, since the group of Euclidean motions is finite-dimensional.)

In the same sense, a connected topological manifold is homogeneous under its homeomorphism group, i.e., if $p$ and $q$ are points, there exists a homeomorphism carrying $p$ to $q$. (That's somehow less impressive, since the homeomorphism group of a manifold is "large".)

It looks to me that the general property you're getting at is "homogeneity under the homeomorphism group". (If that's right, you don't strictly need to mention a "standard model space", though of course a standard model space is useful if you want to study the class of objects locally modeled by a particular space.)

Answer 3

Locally homogeneous spaces are indeed a common notion in geometry or in geometric topology, particularly starting with Thurston's description of the 8 geometries that are needed to describe all locally homogeneous compact 3-manifolds. See for example Goldman's article.