Cohomology of a real toric variety and shellability of posets arising from a graph - Park, Boram; Park, Seonjeong (Proceedings of the Edinburgh Mathematical Society, 2023)
Given a graph G without loops, the pseudograph associahedron PG is a smooth polytope, so there is a projective smooth toric variety XG corresponding to PG. Taking the real locus of XG, we have the projective smooth real toric variety. The integral cohomology groups of can be computed by studying the topology of certain posets of even subgraphs of G; such a poset is neither pure nor shellable in general. We completely characterize the graphs whose posets of even subgraphs are always shellable. It follows that we get a family of projective smooth real toric varieties whose integral cohomology groups are torsion-free or have only 2-torsion.
Keywords: Betti number; chain-lexicographic-shellability; even subgraph; pseudograph associahedron; real smooth toric variety; shellable poset