The GKM correspondence in dimension 6

TL;DR

This paper characterizes when abstract 3-valent GKM graphs in dimension 6 can be realized as GKM manifolds, establishing a correspondence and uniqueness under certain conditions, with counterexamples illustrating limitations.

Contribution

It proves that certain combinatorial conditions are sufficient for realizing 6-dimensional GKM manifolds from graphs and establishes a uniqueness result under specific local isotropy conditions.

Findings

Realization of GKM graphs as 6D GKM manifolds under specific conditions.

Uniqueness of realization when a fixed point has at most two neighboring finite isotropy groups.

Counterexamples showing the necessity of conditions for uniqueness.

Abstract

It follows from the GKM description of equivariant cohomology that the GKM graph of a GKM manifold has free equivariant graph cohomology, and satisfies a Poincar\'e duality condition. We prove that these conditions are sufficient for an abstract $3$-valent $T^2$-GKM graph to be realizable by a simply-connected $6$-dimensional GKM manifold. Our realization has the property that any closed stratum of a finite isotropy group contains a fixed point. Furthermore, we argue that in case there exists a fixed point in whose vicinity there occur at most two distinct finite nontrivial isotropy groups such a realization is unique up to equivariant homeomorphism, thus establishing a complexity one GKM correspondence in dimension $6$. We show that the statement on equivariant uniqueness is false without the two conditions on the finite isotropies by providing counterexamples in presence of a fixed point with three distinct neighbouring finite isotropy groups, as well as an example of a simply-connected integer GKM manifold with a closed stratum of a finite isotropy group which does not contain any fixed point.