Betti numbers of small covers and their two-fold coverings

We point out a gap in the proof of the Davis--Januszkiewicz theorem on cohomology of small covers of simple polytopes, and give a correction to this proof. We use this theorem to compute explicitly the Betti numbers for a wide class of two-fold coverings over small covers. We describe a series of examples in this class and explain why this class turns out to be special in the context of the general problem of computing the modulo two Betti numbers of two-fold coverings over small covers.