Multiplicative structure of the cohomology ring of real toric spaces
Multiplicative structure of the cohomology ring of real toric spaces
复制标题
DOI：
10.4310/hha.2020.v22.n1.a7
复制
发表时间：
2017-11
期刊：
影响因子：
--
通讯作者：
Suyoung Choi;Hanchul Park
中科院分区：
文献类型：
--
作者：
Suyoung Choi;Hanchul Park
一个真正的环面空间是一个拓扑空间，它允许一个表现良好的$\mathbb{Z}_2^k$ -作用。实矩角复形和实环变形是实环空间的典型例子。实环面空间由一对简单复矩阵$K$和特征矩阵$\Lambda$确定。本文给出了关于$K$和$\Lambda$的实环空间的一个显式$R$ -上同环公式，其中$R$是一个具有单位的交换环，$2$是一个单位。有趣的是，它有一个自然的$(\mathbb{Z} \oplus \operatorname*{row} \Lambda)$ -分级。作为推论，我们用二元拟阵计算了（广义）实伯特流形的上同调环，并给出了实环空间是上同调辛的一个判据。
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.