AI 中文总结
本文研究单形骨架的Bier球面上的小覆盖，按Davis–Januszkiewicz等价完成分类，确定了m≥4时两类Bier球面上小覆盖的同胚类型，还计算了其余情况的有理Betti数。
AI 中文摘要
单纯复形K的Bier球面定义为K与其组合Alexander对偶的删除并。本文聚焦于单形骨架的Bier球面类，由于这类Bier球面已知是多面体，故可生成小覆盖。我们对这些Bier球面上的小覆盖按Davis–Januszkiewicz等价进行分类；作为应用，对所有m≥4，确定了(m-1)单形的0-骨架和(m-3)-骨架的Bier球面上小覆盖的同胚类型；对其余0<r<m-3的情况，计算了它们的有理Betti数。
英文摘要
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.
Comments15 pages with 1 figure and 1 table
