Combinatorics of Bier spheres and toric topology

Abstract

Given an abstract simplicial complex K on [m] = {1, 2, . . . , m} different from 2[m], its Bier sphere Bier(K) is a deleted join of K and its Alexander dual simplicial complex K∨. This construction introduced by Thomas Bier in 1992 yields a wide class of PL-spheres with a rich combinatorial structure. In particular, almost all of them are non-polytopal, although no explicit example of a non-polytopal Bier sphere has been constructed so far. Moreover, it was shown that each Bier sphere admits a canonical starshaped realization and a wide class of polytopal Bier spheres was identified and studied in [1] and [2]. Finally, applications of the theory of Bier spheres in toric topology were found in [3]. In this talk we shall discuss the general formulae for the chromatic and Buchstaber numbers of Bier spheres, these combinatorial invariants play a significant role in toric topology. We shall also discuss a classification of two-dimensional combinatorial Bier spheres and show that the Nevo-Petersen conjecture on the γ-vectors of flag homology spheres holds for all flag Bier spheres. The talk is based on joint works with Matvey Sergeev (HSE University, Russia), Aleš Vavpetič (University of Ljubljana, Slovenia) and Rade Živaljević (MISANU, Serbia).