One-Beta数、贝蒂数与等变上同调
One-skeleta, Betti numbers and equivariant cohomology
摘要：
G-流形M的单骨架是M中的点p的集合，其中$\dim G_p \geq \dim G -1$ ;并且M是GKM流形，如果这个单骨架的维数是2。Goresky、Kottwitz和MacPherson证明了对于这样的流形，这个单骨架具有"标记”图$(Γ, α)$的结构，并且M的等变上同调环同构于这个图的“上同调环”。因此，如果M是辛的，则可以证明这个环是对称代数$\SS(\fg^*)$上的自由模，其中$b_{2i}(Γ)$生成元在2 i维，$b_{2i}(Γ)$是$Γ$的“组合”第2 i个贝蒂数。在这篇文章中，我们表明，这个“拓扑”的结果，事实上，是一个组合的结果图。
英文摘要：
The one-skeleton of a G-manifold M is the set of points p in M where $\dim G_p \geq \dim G -1$; and M is a GKM manifold if the dimension of this one-skeleton is 2. Goresky, Kottwitz and MacPherson show that for such a manifold this one-skeleton has the structure of a ``labeled" graph, $(Γ, α)$, and that the equivariant cohomology ring of M is isomorphic to the ``cohomology ring'' of this graph. Hence, if M is symplectic, one can show that this ring is a free module over the symmetric algebra $\SS(\fg^*)$, with $b_{2i}(Γ)$ generators in dimension 2i, $b_{2i}(Γ)$ being the ``combinatorial'' 2i-th Betti number of $Γ$. In this article we show that this ``topological'' result is , in fact, a combinatorial result about graphs.
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