Multiplication structure of the cohomology ring of real toric spaces

A real toric space is a topological space which admits a well-behaved ${\mathbb{Z}}_2^k$-action. Real moment-angle complexes and real toric varieties are typical examples of real toric spaces. A real toric space is determined by a pair of a simplicial complex $K$ and a characteristic matrix $\mathrm{\Lambda }$. In this paper, we provide an explicit $R$-cohomology ring formula of a real toric space in terms of $K$ and $\mathrm{\Lambda }$, where $R$ is a commutative ring with unity in which $2$ is a unit. Interestingly, it has a natural $($\mathbb{Z}$ \oplus \operatorname*{row} \Lambda)$-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 cohomology symplectic.