Bier spheres and toric topology

TL;DR

This paper computes Buchstaber numbers for Bier spheres, classifies all 13 types in dimension two, and explores their connections to nestohedra and permutohedra, with applications to toric varieties and small covers.

Contribution

It provides the first computation of Buchstaber numbers for Bier spheres and classifies all 13 types in dimension two, linking them to well-known polytopes.

Findings

12 of 13 Bier sphere types are nerve complexes of nestohedra

The remaining Bier sphere is a nerve complex of a generalized permutohedron

Constructed regular normal fans and computed cohomology rings for associated toric varieties

Abstract

We compute the real and complex Buchstaber numbers of an arbitrary Bier sphere. In dimension two, we identify all the 13 different combinatorial types of Bier spheres and show that 12 of them are nerve complexes of nestohedra, while the remaining one is a nerve complex of a generalized permutohedron. As an application of our results, we construct a regular normal fan for each of those 13 Delzant polytopes, compute the cohomology rings of the corresponding nonsingular projective toric varieties, and examine the orientability of the corresponding small covers.