Multiplicative structure of the cohomology ring of real toric spaces

Abstract

A real toric space is a topological space which admits a wellbehaved Zk2-action. Real moment-angle complexes and real toric manifolds are typical examples of real toric spaces. A real toric space is determined by the pair of a simplicial complex K and a characteristic matrix Λ. In this paper, we provide an explicit R-cohomology ring formula of a real toric space in terms of K and Λ, where R is a commutative ring with unity in which 2 is a unit. Interestingly, it has a natural (Z ⊕ rowΛ)-grading. As corollaries, we compute the cohomology rings of (generalized) real Bott manifolds in terms of binary matroids, and we also provide a criterion for real toric spaces to be cohomologically symplectic.