• Honam Mathematical Journal
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• v.39 no.1
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• pp.41-47
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• 2017

The aim of this short paper is to give a new necessary and sufficient condition, topological in nature, for certain real toric manifolds to be orientable in terms of the connectedness of their particular submanifolds of codimension zero.

• Bulletin of the Korean Mathematical Society
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• v.52 no.2
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• pp.467-481
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• 2015

Choi and Park introduced an invariant of a finite simple graph, called signed a-number, arising from computing certain topological invariants of some specific kinds of real toric manifolds. They also found the signed a-numbers of path graphs, cycle graphs, complete graphs, and star graphs. We introduce a signed a-polynomial which is a generalization of the signed a-number and gives a-, b-, and c-numbers. The signed a-polynomial of a graph G is related to the $Poincar\acute{e}$ polynomial $P_{M(G)}(z)$, which is the generating function for the Betti numbers of the real toric manifold M(G). We give the generating functions for the signed a-polynomials of not only path graphs, cycle graphs, complete graphs, and star graphs, but also complete bipartite graphs and complete multipartite graphs. As a consequence, we find the Euler characteristic number and the Betti numbers of the real toric manifold M(G) for complete multipartite graphs G.

• Bulletin of the Korean Mathematical Society
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• v.59 no.3
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• pp.671-683
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• 2022

The aim of this paper is to deal with the realization problem of a given Lagrangian submanifold of a symplectic manifold as the fixed point set of an anti-symplectic involution. To be more precise, let (X, ω, µ) be a toric Hamiltonian T-space, and let ∆ = µ(X) denote the moment polytope. Let τ be an anti-symplectic involution of X such that τ maps the fibers of µ to (possibly different) fibers of µ, and let p₀ be a point in the interior of ∆. If the toric fiber µ^-1(p₀) is real Lagrangian with respect to τ, then we show that p₀ should be the origin and, furthermore, ∆ should be centrally symmetric.