A Bier sphere is a deleted join of an abstract simplicial complex, different from a simplex, and its Alexander dual. This construction yields a wide class of PL-spheres with nice combinatorial properties. Almost all of them are non-polytopal, although no explicit example of a non-polytopal Bier sphere has been given so far. On the other hand, polytopal Bier spheres are generalized permutohedra due to Jevtić-Timotijević-Živaljević.

In this talk we shall see that moment-angle complexes over Bier spheres acquire equivariant smooth structures and discuss some basic topological properties of the corresponding moment-angle manifolds. We shall also discuss a classification of two-dimensional Bier spheres and show that the Gal and Nevo-Petersen conjectures hold for all flag Bier spheres.

The talk is based on joint works with Rade Živaljević and Matvei Sergeev.