Centrally symmetric Bier spheres on twelve vertices, and the Bokowski–Ewald–Kleinschmidt sphere

Abstract

The Bier sphere of a simplicial complex Δ on [n] is the deleted join of Δ with its combinatorial Alexander dual, a simplicial (n-2)-sphere on at most 2n vertices. Holleben and Yang recently gave a purely combinatorial criterion on Δ forcing Bier(Δ) to be centrally symmetric, polytopal, and yet not realizable by any centrally symmetric polytope, and exhibited one complex on [6] satisfying it. We classify the whole case: the complexes on [6] whose Bier spheres are the centrally symmetric neighborly 4-spheres on 12 vertices are exactly 1024 in number, they form 13 orbits under relabelling and Alexander duality, exactly 692 of them — exactly 7 of the orbits — satisfy the criterion, and the example of Holleben and Yang, read literally, is a family of 64 complexes realizing precisely those 7 orbits. All 13 orbits are polytopal, from explicit integer coordinates, so the 7 are polytopal but not centrally symmetric polytopal and the remaining 6 are the cases the criterion does not decide. We also settle the status of the sphere C of Bokowski, Ewald and Kleinschmidt, previously — up to two-point suspension — the only recorded simplicial sphere that is polytopal but not centrally symmetric polytopal: C is not the Bier sphere of any simplicial complex. Every statement this paper proves is machine-checked in Lean 4.

Open review

This founding-collection manuscript received AI review before publication. Independent human review is open. Submitted reviews enter editorial screening; submitting a review does not change this paper’s status. Contribute an assessment of specific claims, a reproduction, or a correction for editorial screening.

Archived files

1. Version 1 · current (opens in a new tab)

   Source snapshot 2026-08-30 15:34 UTC

   File fingerprint

   de28e91c6bfa12c3c940d4f1986de2554994d88bda3562fe0610fefcdbb6fabb