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.
The first author was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2019R1A2C2010989). The second named author was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2019R1G1A1007862). Received April 2, 2019; published on October 30, 2019. 2010 Mathematics Subject Classification: 57N65, 14M25 (Primary); 57S17, 55U10 (Secondary). Key words and phrases: real toric variety, small cover, real toric space, real moment-angle complex, real subspace arrangement, real Bott manifold, generalized real Bott manifold, cohomologically symplectic manifold. Article available at http://dx.doi.org/10.4310/HHA.2020.v22.n1.a7 Copyright \u00a9c 2019, International Press. Permission to copy for private use granted.
