Extensions of realizable Hamiltonian and complexity one GKM₄ graphs

2025/08/30 by Goertsches, Oliver, Solomadin, Grigory
#06A06 #13F55 #55N91 #57S12 #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.00392

Abstract

We prove that the GKM graphs of GKM₄ manifolds that are either Hamiltonian or of complexity one extend to torus graphs. The arguments are based on a reformulation of the extension problem in terms of a natural representation of the fundamental group of the GKM graph, using a coordinate-free version of the axial function group of Kuroki, as well as on covers of GKM graphs and acyclicity results for orbit spaces of GKM manifolds.

Citations

• The GKM correspondence in dimension 6
• On independent GKM-graphs without nontrivial extensions
• How is a graph not like a manifold?
• Orbit spaces of equivariantly formal torus actions of complexity one
• GKM sheaves and nonorientable surface group representations
• A GKM description of the equivariant cohomology ring of a homogeneous space
• One-skeleta, Betti numbers and equivariant cohomology

Related

• Monotone Symplectic Six-Manifolds that admit a Hamiltonian GKM Action are diffeomorphic to Smooth Fano Threefolds
• Hard Lefschetz Property for Hamiltonian torus actions on 6-dimensional GKM manifolds
• Combinatorial manifolds are Hamiltonian
• How is a graph not like a manifold?
• Realization of GKM fibrations and new examples of Hamiltonian non-Kähler actions