On the cohomology and their torsion of real toric objects

In this paper, we do the following two things: (i) We present a formula to compute the rational cohomology ring of a real topological toric manifold, and thus that of a small cover or a real toric manifold, which implies the formula of Suciu and Trevisan. Furthermore, the formula also works for an arbitrary coefficient ring G in which 2 is a unit. (ii) We construct infinitely many real toric manifolds and small covers whose integral cohomology rings have a q-torsion for any positive odd integer q.