arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1801.00296math.COmath.AT
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实环面流形的图不变量与贝蒂数

Graph invariants and Betti numbers of real toric manifolds

Boram Park, Hanchul Park, Seonjeong Park

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21 pages, 6 figures, 1 table
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摘要：

对于图$G$，图立方面体$\square_G$和图伴随面体$\triangle_G$，它们都是允许（实）环面流形的简单凸多面体.在本文中，我们引入了一个图的不变量，称为$b$ -数，我们表明，$b$ -数计算贝蒂数的实环面流形$X^\mathbb{R}(\square_G)$对应于一个图立方面体。$a$ -数是由S. Choi和第二位作者提出的方法，计算了对应于一个图伴随面体的实环面流形$X^\mathbb{R}(\triangle_G)$的贝蒂数.我们还从复曲面拓扑的角度研究了$a$ -数和$b$ -数之间的各种关系。有趣的是，对于森林$G$及其线图$L(G)$，实环面流形$X^\mathbb{R}(\triangle_G)$和$X^\mathbb{R}(\square_{L(G)})$具有相同的贝蒂数。

英文摘要：

For a graph $G$, a graph cubeahedron $\square_G$ and a graph associahedron $\triangle_G$ are simple convex polytopes which admit (real) toric manifolds. In this paper, we introduce a graph invariant, called the $b$-number, and we show that the $b$-numbers compute the Betti numbers of the real toric manifold $X^\mathbb{R}(\square_G)$ corresponding to a graph cubeahedron. The $b$-number is a counterpart of the notion of $a$-number, introduced by S. Choi and the second named author, which computes the Betti numbers of the real toric manifold $X^\mathbb{R}(\triangle_G)$ corresponding to a graph associahedron. We also study various relationships between $a$-numbers and $b$-numbers from a toric topological view. Interestingly, for a forest $G$ and its line graph $L(G)$, the real toric manifolds $X^\mathbb{R}(\triangle_G)$ and $X^\mathbb{R}(\square_{L(G)})$ have the same Betti numbers.

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