图为何不像流形？
How is a graph not like a manifold?
AI总结：
研究了等变形式作用下流形的GKM图性质，揭示了其与GKM流形的拓扑条件及上同调代数的关联。
摘要：
对于一个紧致环面T的等变形式作用于光滑流形X上的情况，其中X有孤立固定点，我们研究了面子流形的分层格S(X)的全局同调性质。我们证明了在每个固定点处的切向量权重j-独立性条件，意味着对于r>j+1的骨架S(X)_r来说，(j+1)-acyclicity成立。这一结果为GKM图成为某种GKM流形的GKM图提供了必要的拓扑条件。我们利用特定的acyclicity论证，描述了具有(n-1)-独立作用的(n-1)维环面的等变上同调代数，其在某些颜色性假设下与GKM图的面代数相关联。这种观察强调了复杂度为一和环面流形作用之间的某些相似性。
英文摘要：
For an equivariantly formal action of a compact torus $T$ on a smooth manifold $X$ with isolated fixed points we investigate the global homological properties of the graded poset $S(X)$ of face submanifolds. We prove that the condition of $j$-independency of tangent weights at each fixed point implies $(j+1)$-acyclicity of the skeleta $S(X)_r$ for $r>j+1$. This result provides a necessary topological condition for a GKM graph to be a GKM graph of some GKM manifold. We use particular acyclicity arguments to describe the equivariant cohomology algebra of an equivariantly formal manifold of dimension $2n$ with an $(n-1)$-independent action of $(n-1)$-dimensional torus, under certain colorability assumptions on its GKM graph. This description relates the equivariant cohomology algebra to the face algebra of a simplicial poset. Such observation underlines certain similarity between actions of complexity one and torus manifolds.
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