# Relation between homogeneous spaces and principal bundles

What is the relation between homogeneous spaces and principal bundles. I've been reading the two definitions and am left confused as to whether one is a subset of the other or whether no such relation exists.

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Both notions involve actions of groups, but there is a huge difference: In a homogeneous space, the group acts (transitively) on a (topological) space $X$. In a principal bundle $E\to X$, there is a base space $X$ and there are fibers $E_x$, one for each $x\in X$. The group acts transitively and freely on each of these fibers. Of course the actions on fibers give an action on the total space $E$, but this action is not transitive (unless $X$ has only one point) and fairly restricted by the property that points in $E$ are only mapped to other points in the same fiber.

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An homogeneous space is the base space of principal bundle (which has the acting group as total space and the isotropy group as fiber) In general, though, a principal bundle does not arise from this situation: for example, the principal $S^1$-bundle $S^{2n-1}\to \mathbb CP^{n-1}$ is not of that form for most $n$, because for most $n$ the $n$-sphere is not a group.

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So if the total space of a principle bundle has a group structure, does this mean we have a homogenous space? I mean is there a natural way to let the total space act on the base in a suitable way? –  MikhailMatrix Feb 3 '11 at 19:39
@Mikhail: not really. I just picked an example where the fact that the total space is not a group in any way makes it easy to see that the bundle does not arise from an homogeneous space. –  Mariano Suárez-Alvarez Feb 3 '11 at 19:53