# cohomological obstruction for a manifold being homogeneous

I have two questions:

1. 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.
2. 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.

• I'm not exactly sure what you're looking for, but there are many obstructions to being a homogeneous space. For example, the Euler characteristic must be non-zero, and the homotopy groups must eventually (above a certain degree, which can be estimated solely in terms of the dimension of the manifold) be finite. As far as infinite dimensional Lie groups go, I know very little. But, are diffeomorphism groups infinite dimensional Lie groups? If so, then every connected manifold is homogeneous. Jun 13, 2020 at 14:28
• @JasonDeVito The spheres $S^n$ are all homogeneous no? So Euler characteristic doesn't need to be nonzero. Jun 13, 2020 at 17:14
• @JasonDeVito Nilmanifolds all have zero Euler characteristic Jun 13, 2020 at 17:19
• @Noel: Sorry, I meant non-negative Jun 13, 2020 at 18:45
• I do not even understand the 1st question: You already know that there are algebro-topological obstructions to homogeneity. Are you asking if a manifold homotopy equivalent to a homogeneous space is diffeomorphic to a homogeneous manifold? Jun 14, 2020 at 1:34