Multiplicative structure of the cohomology ring of real toric spaces

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 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 $\Lambda$. 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 cohomologically symplectic.

Keywords

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

2010 Mathematics Subject Classification

Primary 14M25, 57N65. Secondary 55U10, 57S17.

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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 2 April 2019

Accepted 4 July 2019

Published 30 October 2019