Algebraic properties of Bier spheres (Q2904272)

This article deals with the Bier sphere of a simplicial complex. In an unpublished work of T. Bier, it is shown that, for any simplicial complex \(\Delta\) on the vertex set \([n]=\{1,2,\dots,n\}\) with \(\Delta \neq 2^{[n]}\), the deleted join of \(\Delta\) with its Alexander dual \(\Delta^{\star}\) is a combinatorial sphere, called the Bier sphere of \(\Delta\). In the article under review the authors prove that there are exactly eight simplicial complexes \(\Delta\) that are not cones, such that the Bier sphere of \(\Delta\) is flag. They define up to isomorphism four flag Bier spheres. Furthermore, the authors provide descriptions of the first and second Betti numbers of general Bier spheres. Finally, they compute the Betti numbers for a specific class of Bier spheres, constructed from skeletons of a full simplex.