Integral cohomology groups of real toric manifolds and small covers

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.