Integral cohomology groups of real toric manifolds and small covers

Abstract

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.