Moscow Mathematical Journal

Volume 21, Issue 3, July–September 2021 pp. 467–492.

Integral Cohomology Groups of Real Toric Manifolds and Small Covers

Authors: Li Cai (1) and Suyoung Choi (2)
Author institution:(1) Department of Mathematical Sciences, Xi'an Jiaotong-Liverpool University, Suzhou 215123, Jiangsu, China
(2) Department of Mathematics, Ajou University, 206 Worldcup-ro, Suwon 16499, South Korea

Summary:

For a simplicial complex $K$ with $m$ vertices, there is a canonical $\mathbb{Z}_2^m$ -space known as a real moment angle complex $\mathbb{R}\mathcal{Z}_K$ . In this paper, we consider the quotient spaces $Y=\mathbb{R}\mathcal{Z}_K / \mathbb{Z}_2^{k}$ , where $K$ is a pure shellable complex and $\mathbb{Z}_2^k \subset \mathbb{Z}_2^m$ is a maximal free action on $\mathbb{R}\mathcal{Z}_K$ . A typical example of such spaces is a small cover, where a small cover is known as a topological analog of a real toric manifold. We compute the integral cohomology group of $Y$ by using the PL cell decomposition obtained from a shelling of $K$ . In addition, we compute the Bockstein spectral sequence of $Y$ explicitly.

2020 Math. Subj. Class. Primary: 57N65; Secondary: 55N10, 13H10.

Keywords: Real toric manifold, small cover, Bockstein homomorphisms, Cohomology groups.

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