Homology, Homotopy and Applications

Volume 22 (2020)

Number 1

Multiplicative structure of the cohomology ring of real toric spaces

Pages: 97 – 115

DOI: https://dx.doi.org/10.4310/HHA.2020.v22.n1.a7

Authors

Suyoung Choi (Department of Mathematics, Ajou University, Suwon, South Korea)

Hanchul Park (Department of Mathematics Education, Jeju National University, Jeju-si, Jeju-do, South Korea)

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 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.

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