Realizability of abstract GKM graphs by GKM manifolds

Determine which abstract GKM graphs arise as GKM graphs of GKM manifolds, equivalently, characterize the combinatorial conditions on an abstract GKM graph that are necessary and sufficient for realizability by a manifold.

Background

The paper studies abstract GKM graphs as purely combinatorial objects independently of their geometric realization by torus actions on manifolds. Although every GKM manifold determines an associated labeled GKM graph, the converse realization problem is unresolved: it is unknown which abstract labeled graphs can be obtained from GKM manifolds.

A complete solution would identify a list of combinatorial conditions equivalent to geometric realizability and would clarify whether abstract GKM graphs provide the appropriate combinatorial model for GKM geometry. The paper establishes realizability criteria in a specific setting—3-valent abstract T2-GKM graphs—by relating orientability to graph cohomology and rational GKM manifolds, but does not solve the general realization problem.

References

One possible viewpoint that makes the independent study of abstract GKM graphs interesting is the question of realizability: while any GKM manifold has a GKM graph associated to it, it is not known which abstract GKM graphs come from GKM manifolds.

— On orientability, Poincaré duality, and connectivity of GKM graphs  (2608.26999 - Goertsches et al., 27 Aug 2026) in Section 1, Introduction