Every manifold that I ever met in a differential geometry class was a homogeneous space: spheres, tori, Grassmannians, flag manifolds, Stiefel manifolds, etc. What is an example of a connected smooth manifold which is not a homogeneous space of any Lie group?


The only candidates for examples I can come up with are two-dimensional compact surfaces of genus at least two. They don't seem to be obviously homogeneous, but I don't know how to prove that they are not. And if there are two-dimensional examples then there should be tons of higher-dimensional ones.


The question can be trivially rephrased by asking for a manifold which does not carry a transitive action of a Lie group. Of course, the diffeomorphism group of a connected manifold acts transitively, but this is an infinite-dimensional group and so doesn't count as a Lie group for my purposes.


Orientable examples would be nice, but nonorientable would be ok too.



Example of a manifold which is not a homogeneous space of any Lie group
$\pi\_2$ of a Lie group is trivial, so $\pi\_2(G/H)$ is isomorphic to a subgroup of $\pi\_1(H)$, which is finitely generated (isomorphic to $\pi\_1$ of a maximal compact subgroup of the identity component of $H$). But $\pi\_2$ of a closed manifold is often not finitely generated. For example, the connected sum of two copies of $S^1\times S^2$ has as a retract a punctured $S^1\times S^2$, which is homotopy equivalent to $S^1\vee S^2$ and so has universal cover homotopy equivalent to an infinite wedge of copies of $S^2$.


EDIT This ad hoc answer can be extended as follows: All I really used was that $\pi\_2(G)$ and $\pi\_1(G)$ are finitely generated. But $\pi\_n(G)$ is finitely generated for all $n\ge 1$ (reduce to simply connected case and use homology), so $\pi\_n(G/H)$ is finitely generated for $n\ge 2$. That leads to a lot more higher-dimensional non simply connected examples. 


