On independent GKM-graphs without nontrivial extensions

TL;DR

This paper constructs specific examples of independent GKM-graphs that cannot be extended or realized by GKM-manifolds, highlighting limitations in the geometric realization of certain combinatorial structures.

Contribution

It provides explicit examples of independent GKM-graphs without nontrivial extensions for all relevant dimensions, and demonstrates their non-realizability as GKM-manifolds.

Findings

Constructed examples for all n ≥ k ≥ 3

Examples cannot be realized as GKM-manifolds for n=k=3 or n≥k≥4

Highlights limitations in GKM-graph realizability

Abstract

In this paper an example of a $k$-independent $(n,k)$-type GKM-graph without nontrivial extensions is constructed for any $n\geq k\geq 3$. It is shown that this example cannot be realized by a GKM-manifold for any $n=k=3$ or $n\geq k\geq 4$.