TITLE: How special are homogeneous spaces?
QUESTION [8 upvotes]: Let $M$ be a smooth finite dimensional manifold, how restrictive is it to require $M$ to admit a smooth action by a finite dimensional Lie group $G$?
Related questions/approaches: Of course we need $\mathrm{dim}(G) \geq \mathrm{dim}(M)$, are there any results relating the minimal dimension of a Lie group acting transitively to that of $M$, perhaps in special cases?
Going in the other direction, any criteria which easily allow to say that a given smooth manifold does not admit a transitive group action?
EDIT: in the first question I meant to write transitive smooth group action.

REPLY [10 votes]: I suppose you want the action to be transitive as your title suggests. In this case, a classical theorem of Mostow (in 1950 for surfaces, in 2005 in general) says that for a compact homogeneous space $M = G/H$ the Euler characteristics is non-negative.
Mostow, G.D.: The extensibility of local Lie groups of transformations and groups on surfaces. Ann.
Math. (2) 52, 606–636 (1950)
Mostow, G.D.: A structure theorem for homogeneous spaces. Geom. Dedic. 114, 87–102 (2005)