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

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

2011-04-11

arxiv.org/abs/1104.1832v1

Mathematics

Algebraic Topology

20 pages

Scientific paper

If a closed smooth manifold $M$ with an action of a torus $T$ satisfies certain conditions, then a labeled graph $\mG_M$ with labeling in $H^2(BT)$ is associated with $M$, which encodes a lot of geometrical information on $M$. For instance, the "cohomology" ring $\mHT^*(\mG_M)$ of $\mG_M$ is defined to be a subring of $\bigoplus_{v\in V(\mG_M)}H^*(BT)$, where $V(\mG_M)$ is the set of vertices of $\mG_M$, and is known to be often isomorphic to the equivariant cohomology $H^*_T(M)$ of $M$. In this paper, we determine the ring structure of $\mHT^*(\mG_M)$ when $M$ is a flag manifold of classical type directly without using the fact that $\mHT^*(\mG_M)$ is isomorphic to $H^*_T(M)$.

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