The question as in the title:
Is there a simple example of a compact orientable smooth finite-dimensional manifold whose singular cohomology groups with integer coefficients are not isomorphic to those of some smooth homogeneous space? My heart tells me that the two holed torus is such an example but I left my algebraic topology skills in my other pair of pants.
(I would also probably satisfied if this were shown only for compact homogeneous spaces)
Similarly, is there a simple example of a nice manifold whose integral singular cohomology groups are not isomorphic to the singular cohomology of any double coset space of any finite dimensional Lie group? (I believe these double coset space do not have to be smooth manifolds right?)
I am of course assuming that such examples exist.