A GKM description of the equivariant cohomology ring of a homogeneous space

Let T be a torus of dimension n > 1 and M a compact T -manifold. M is a GKM manifold if the set of zero dimensional orbits in the orbit space M/T is zero dimensional and the set of one dimensional orbits in M/T is one dimensional. For such a manifold these sets of orbits have the structure of a labelled graph and it is known that a lot of topological information about M is encoded in this graph. In this paper we prove that every compact homogeneous space M of non-zero Euler characteristic is of GKM type and show that the graph associated with M encodes geometric information about M as well as topological information. For example, from this graph one can detect whether M admits an invariant complex structure or an invariant almost complex structure.