GKM graph locally modeled by $T^{n}\times S^{1}$-action on $T^{*}\mathbb{C}^{n}$ and its graph equivariant cohomology

TL;DR

This paper introduces a new class of labeled graphs representing certain GKM manifolds with $T^{n}\times S^{1}$-actions, computes their equivariant cohomology rings, and explores their algebraic structures.

Contribution

It defines a novel class of labeled graphs encompassing GKM graphs of specific manifolds, and provides methods to compute and analyze their equivariant cohomology.

Findings

Computed the graph equivariant cohomology rings for the new class of graphs.

Established a module basis using shelling structures.

Analyzed the multiplicative structure of the cohomology rings.

Abstract

We introduce a class of labeled graphs (with legs) which contains two classes of GKM graphs of $4n$-dimensional manifolds with $T^{n}\times S^{1}$-actions, i.e., GKM graphs of the toric hyperK${\rm\ddot{a}}$hler manifolds and of the cotangent bundles of toric manifolds. Under some conditions, the graph equivariant cohomology ring of such a labeled graph is computed. We also give a module basis of the graph equivariant cohomology by using a shelling structure of such a labeled graph and study their multiplicative structure.