Bier Spheres and Posets

In 1992 Thomas Bier presented a strikingly simple method to produce a huge number of simplicial (n− 2)-spheres on 2n vertices as deleted joins of a simplicial complex on n vertices with its combinatorial Alexander dual. Here we interpret his construction as giving the poset of all the intervals in a boolean algebra that “cut across an ideal.” Thus we arrive at a substantial generalization of Bier’s construction: the Bier posets Bier(P, I) of an arbitrary bounded poset P of finite length. In the case of face posets of PL spheres this yields cellular “generalized Bier spheres.” In the case of Eulerian or Cohen-Macaulay posets P we show that the Bier posets Bier(P, I) inherit these properties. In the boolean case originally considered by Bier, we show that all the spheres produced by his construction are shellable, which yields “many shellable spheres”, most of which lack convex realization. Finally, we present simple explicit formulas for the g-vectors of these simplicial spheres and verify that they satisfy a strong form of the g-conjecture for spheres. ∗Research partially supported by the European Commission’s IHRP Programme, grant HPRN-CT2001-00272, “Algebraic Combinatorics in Europe” ∗∗Research supported by the Deutsche Forschungsgemeinschaft within the European graduate program “Combinatorics, Geometry, and Computation” (GRK 588/2) ∗∗∗Partially supported by Deutsche Forschungs-Gemeinschaft, via the DFG Research Center “Mathematics in the Key Technologies” (FZT86), the Research Group “Algorithms, Structure, Randomness” (Project ZI 475/3), and a Leibniz grant (ZI 475/4), and by the German Israeli Foundation (G.I.F.)