Equivariant cohomology epimorphisms and face ring quotients for Hamiltonian and complexity one GKM$_4$ manifolds

TL;DR

This paper proves a surjectivity property of equivariant graph cohomology maps for GKM$_3$ actions and uses it to describe the cohomology rings of certain Hamiltonian GKM$_4$ manifolds.

Contribution

It establishes a new surjectivity result for GKM$_3$ actions and applies it to explicitly describe cohomology rings of Hamiltonian GKM$_4$ manifolds.

Findings

Surjectivity of restriction maps in equivariant graph cohomology for GKM$_3$ actions.

Failure of the analogous statement in GKM$_2$ setting.

Explicit description of cohomology rings for Hamiltonian and complexity one GKM$_4$ actions.

Abstract

Given a GKM$_3$ action of a torus $K$ on a manifold $M$ with GKM graph $\Gamma$, we show that for any extension of $\Gamma$ to an abstract GKM graph the corresponding restriction map in equivariant graph cohomology is surjective. While the corresponding statement for extensions of actions is well-known, we observe that this graph-theoretical statement is false in the GKM$_2$ setting. As a corollary, we obtain a description of the equivariant cohomology ring of Hamiltonian and complexity one GKM$_4$ actions in terms of generators and relations.