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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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up vote 2 down vote accepted

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

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