AI 中文总结
该研究给出中心对称Bier球无法作为中心对称多面体边界的判据，得到新的不可几何实现的单纯多面体族，还证明顶点数≤12的Bier球均为多面体。
AI 中文摘要
Bier球由单纯复形与其组合Alexander对偶的删除并构造而成，是已知最大的单纯球族之一。我们研究中心对称的Bier球，给出其无法作为中心对称多面体边界的简单判据，由此得到一类带有组合自同构、无法几何实现的新单纯多面体族，此前仅Bokowski–Ewald–Kleinschmidt多面体是已知具备该性质的单纯例子。根据Smith理论，这些多面体的实现空间是非可缩的。最后，我们证明每个顶点数不超过12的Bier球都是多面体。
英文摘要
Bier spheres arise as deleted joins of simplicial complexes with their combinatorial Alexander duals and form one of the largest known families of simplicial spheres. We study centrally symmetric Bier spheres and give a simple criterion for when they cannot arise as boundaries of centrally symmetric polytopes. From this, we obtain a large new family of simplicial polytopes with combinatorial automorphisms that cannot be realized geometrically. Prior to our construction, the Bokowski--Ewald--Kleinschmidt polytope was the only known simplicial example exhibiting these properties. By Smith theory, these polytopes have noncontractible realization spaces. Finally, we establish that every Bier sphere with at most $12$ vertices is polytopal.
Comments13 pages. Comments are welcome
