Multiplicative structure of the cohomology ring of real toric spaces

Suyoung Choi ; Hanchul Park

Homology, homotopy, and applications, Tome 22 (2020) no. 1, pp. 97-115.

Résumé

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.

Détail

DOI : 10.4310/HHA.2020.v22.n1.a7

Classification : 14M25, 57N65, 55U10, 57S17
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