I have two questions:
- There is a nice theory to compute the cohomology of homogeneous spaces (like EMSS and other tools). Can we go back and tell from cohomology (or maybe homotopy groups) when space is NOT homogeneous for any group? I restrict to the case of manifolds and Lie groups here.
- Is it possible to answer the same questions about the actions of infinite-dimensional Lie groups? I consider a class of "nice" infinite-dimensional Lie groups, i.e. direct limits of finite-dimensional manifolds.
The last formulation is a bit vague, but maybe the question has a trivial answer, so I don't think it is worth elaborating at this point.
Edit: Ok, so some elaborations are necessary.
On the second question: Indeed, as pointed out by Jason if we take an arbitrary (even "nice") Lie group the question is trivial. What I'm really looking for is the case when I have a certain fixed infinite-dimensional Lie group. This group is a subgroup of the diffeomorphism group of the manifold.
The question is when I can reduce an action of all diffeomorphisms to the action of this particular subgroup.
It would be great to have an answer in terms of cohomology of this subgroup.