The cohomology ring of the GKM graph of a flag manifold of classical type

Abstract

If a closed smooth manifold $M$ with an action of a torus $T$ satisfies certain conditions, then a labeled graph ${\mathcal{G}}_M$ with labeling in $H^2\left(BT\right)$ is associated with $M$, which encodes a lot of geometrical information on $M$. For instance, the “graph cohomology” ring $H_T^{\ast }\left({\mathcal{G}}_M\right)$ of ${\mathcal{G}}_M$ is defined to be a subring of ${\oplus }_{v\in V\left({\mathcal{G}}_M\right)}{H}^{\ast }\left(BT\right)$, where $V\left({\mathcal{G}}_M\right)$ is the set of vertices of ${\mathcal{G}}_M$, and is known to be often isomorphic to the equivariant cohomology $H_T^{\ast }\left(M\right)$ of $M$. In this paper, we determine the ring structure of $H_T^{\ast }\left({\mathcal{G}}_M\right)$ with $\mathbb{Z}$ (resp., $\mathbb{Z}\left[\frac{1}{2}\right]$) coefficients when $M$ is a flag manifold of type A, B, or D (resp., C) in an elementary way.