Multiplication structure of the cohomology ring of real toric spaces

Abstract

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 $(\mathbb{Z}\oplus \mathrm{row}\Lambda )$-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.

19 pages, 1 table

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