Betti numbers of small covers and their two-fold coverings

TL;DR

This paper corrects a proof related to the cohomology of small covers, and explicitly computes Betti numbers for a broad class of two-fold coverings, highlighting their unique properties.

Contribution

It identifies and corrects a gap in the Davis--Januszkiewicz theorem proof and provides explicit Betti number computations for specific two-fold coverings.

Findings

Corrected proof of the Davis--Januszkiewicz theorem

Explicit Betti number calculations for a class of two-fold coverings

Identification of a special class of small covers with unique properties

Abstract

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.