Extensions of realizable Hamiltonian and complexity one GKM${}_4$ graphs

Abstract

We prove that the GKM graphs of GKM${}_4$ 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.

16 pages, 4 figures; minor changes, published version