2026/07/30 by Suyoung Choi, Younghan Yoon, Seonghyeon Yu
Mathematics · #math.AT #math.CO #msc:57S12 #msc:55U10 #msc:52B05 #msc:57N65 #msc:05E45
15 pages with 1 figure and 1 table
arxiv created 2026/07/30 · arxiv updated 2026/07/31
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 ≥ 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.