GKM Manifolds With Low Betti Numbers

Abstract: A GKM manifold is a symplectic manifold with a torus action such that the fixed points are isolated and the isotropy weights at the fixed points are linearly independent. Each GKM manifold has a GKM graph which contains much of the topological information of the manifold, in particular the equivariant cohomology and Chern classes. We will consider the case where the GKM graphs are complete. When the dimension of the torus action is sufficiently large, we can completely classify the complete GKM manifolds, and thus completely describe the cohomology and Chern classes of the associated "minimal" GKM manifolds. If we put some restrictions on the type of possible subgraph then we can completely classify the possible cohomology rings and Chern classes of minimal GKM manifolds. We can also classify some cases where the GKM graph is not complete.