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April 2020 Graph invariants and Betti numbers of real toric manifolds
Boram Park, Hanchul Park, Seonjeong Park
Osaka J. Math. 57(2): 333-356 (April 2020).

Abstract

For a graph $G$, the graph cubeahedron $\square_G$ and the 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 show that the $b$-numbers compute the Betti numbers of the real toric manifold $X^\mathbb{R}(\square_G)$ corresponding to $\square_G$. 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 $\triangle_G$. We also study various relationships between $a$-numbers and $b$-numbers from the viewpoint of toric topology. 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.

Citation

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Boram Park. Hanchul Park. Seonjeong Park. "Graph invariants and Betti numbers of real toric manifolds." Osaka J. Math. 57 (2) 333 - 356, April 2020.

Information

Published: April 2020
First available in Project Euclid: 6 April 2020

zbMATH: 07196681
MathSciNet: MR4081735

Subjects:
Primary: 55U10 , 57N65
Secondary: 05C30

Rights: Copyright © 2020 Osaka University and Osaka City University, Departments of Mathematics

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Vol.57 • No. 2 • April 2020
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