The graph cohomology ring of the GKM graph of a flag manifold of type $G_2$
The graph cohomology ring of the GKM graph of a flag manifold of type $G_2$
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$G_2$ 类型的标志流形的 GKM 图的图上同调环
DOI：
10.1016/b978-012252687-9/50070-x
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
Yukiko Fukukawa
中科院分区：
文献类型：
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作者：
Yukiko Fukukawa
假设一个紧化环面T作用于一个闭合光滑流形M。在一定条件下，Guillemin和Zara将$(M， T)$关联到一个标记图$\mG_M$，其中标签位于$H^2(BT)$中。他们还定义了$\bigoplus_{v\in v (\mG_M)}H^*(BT)$的子带$H_T^*(\mG_M)$，其中$ v (\mG_M)$是$\mG_M$的顶点集合，我们称$H_T^*(\mG_M)$为$\mG_M$的“图上同调”环。已知$M$的等变上同环可以用标记图的组合数据来描述。本文的主要成果是利用组合技术在图$\mG_M$上直接确定了$G_2$型标志流形的等变上同环的环结构。给出了G_2型标志流形的等变上同环的一种新的计算方法。
Suppose a compact torus $T$ acts on a closed smooth manifold $M$. Under certain conditions, Guillemin and Zara associate to $(M, T)$ a labeled graph $\mG_M$ where the labels lie in $H^2(BT)$. They also define the subring $H_T^*(\mG_M)$ of $\bigoplus_{v\in V(\mG_M)}H^*(BT)$, where $V(\mG_M)$ is the set of vertices of $\mG_M$ and we call $H_T^*(\mG_M)$ the "graph cohomology" ring of $\mG_M$. It is known that the equivariant cohomology ring of $M$ can be described by using combinatorial data of the labeled graph. The main result of this paper is to determine the ring structure of equivariant cohomology ring of a flag manifold of type $G_2$ directly, using combinatorial techniques on the graph $\mG_M$. This gives a new computation of the equivariant cohomology ring of a flag manifold of type $G_2$.