A real toric space is a topological space which admits a well-behaved Z k 2 -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 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 ⊕ row Λ)-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.Date: November 15, 2017. 2010 Mathematics Subject Classification. 57N65, 14M25 (Primary) 57S17, 55U10 (Secondary).

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Given a root system, the Weyl chambers in the co-weight lattice give rise to a real toric variety, called the real toric variety associated to the Weyl chambers. We compute the integral cohomology groups of real toric varieties associated to the Weyl chambers of type Cn and Dn, completing the computation for all classical types.