Graph invariants and Betti numbers of real toric manifolds

Abstract

For a graph $G$, a graph cubeahedron ${\square }_G$ and a graph associahedron ${△}_G$ 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 $X^{\mathbb{R}}({△}_G)$ 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 $X^{\mathbb{R}}({△}_G)$ and $X^{\mathbb{R}}({\square }_{L(G)})$ have the same Betti numbers.

21 pages, 6 figures, 1 table

References in corpus (5)

• Compatibility fans for graphical nested complexes
• Integral cohomology groups of real toric manifolds and small covers
• Multiplication structure of the cohomology ring of real toric spaces
• The cohomology groups of real toric varieties associated to Weyl chambers of type $C$ and $D$
• On shellability for a poset of even subgraphs of a graph

Cited by in corpus (3)

• On shellability for a poset of even subgraphs of a graph
• Extended Nestohedra and their Face Numbers
• Toric Fano varieties associated to graph cubeahedra