# Extensions of realizable Hamiltonian and complexity one GKM$_4$ graphs

## TL;DR

This paper demonstrates that certain GKM graphs associated with specific manifolds can be extended to torus graphs, using topological and combinatorial methods involving fundamental groups and covers.

## Contribution

It introduces a new approach to extending GKM graphs of GKM$_4$ manifolds of Hamiltonian or complexity one type to torus graphs, utilizing a reformulation in terms of fundamental groups.

## Key findings

- GKM graphs of GKM$_4$ manifolds extend to torus graphs under certain conditions
- The extension problem is reformulated using a natural representation of the fundamental group
- Acyclicity results support the extension of GKM 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.