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 $\Lambda$. In this paper, we provide an explicit $R$-cohomology ring formula of a real toric space in terms of $K$ and $\Lambda$, 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.

中文翻译：

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实复曲面空间上同调环的乘法结构

一个真正的复曲面空间是一个拓扑空间，它承认一个行为良好的 $\mathbb{Z}_2^k$-action。实矩角复合体和实复曲面变体是实复曲面空间的典型例子。实复曲面空间由一对单纯复形 $K$ 和特征矩阵 $\Lambda$ 确定。在本文中，我们提供了一个基于 $K$ 和 $\Lambda$ 的真实复曲面空间的显式 $R$-上同调环公式，其中 $R$ 是一个具有统一性的交换环，其中 $2$ 是一个单位。有趣的是，它有一个自然的 $(\mathbb{Z} \oplus \operatorname*{row} \Lambda)$-grading。作为推论，我们根据二元拟阵计算（广义）实 Bott 流形的上同调环，并且我们还提供了实复曲面空间是上同调辛的标准。