Integral cohomology groups of real toric manifolds and small covers

Abstract

For a simplicial complex K with m vertices, there is a canonical Zm2-space known as a real moment angle complex RZK. In this paper, we consider the quotient spaces Y = RZK/Zk2, where K is a pure shellable complex and Zk2 ⊂ Zm2 is a maximal free action on RZK. 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.