On small covers over Bier spheres

The Bier sphere of a simplicial complex $K$ is defined as the deleted join of $K$ and its combinatorial Alexander dual. We focus on the class of Bier spheres of the skeleta of a simplex. Since these Bier spheres are known to be polytopal, they give rise to small covers. We classify small covers over these Bier spheres up to Davis--Januszkiewicz equivalence. As applications, for all $m \geq 4$, we determine the homeomorphism types of small covers over the Bier spheres of the $0$-skeleton and the $(m-3)$-skeleton of an $(m-1)$-simplex. For the remaining cases $0<r<m-3$, we compute their rational Betti numbers.