Cohomology of a Real Toric Variety and Shellability of Posets Arising from a Graph

Abstract

Abstract Given a graph G without loops, the pseudograph associahedron P G is a smooth polytope, so there is a projective smooth toric variety X G corresponding to P G . Taking the real locus of X G , we have the projective smooth real toric variety $X^{\mathbb{R}}_G$ . The integral cohomology groups of $X^{\mathbb{R}}_G$ 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.