Ivan Limončenko: Murai spheres, new results and open problems

In 1992, Thomas Bier introduced a beautiful and powerful construction, which yields a triangulated sphere starting from an abstract simplicial complex different from a simplex. Namely, a Bier sphere of such a simplicial complex equals the deleted join of the complex and its combinatorial Alexander dual. In 2011, Satoshi Murai generalized this construction from simplicial complexes to multicomplexes. In this talk, we are going to classify all neighborly Murai spheres and show that for any positive integer d, each triangulated d-sphere with no more than d+4 vertices is isomorphic to a Murai sphere. Finally, we shall discuss some open problems relating Murai spheres to toric geometry and topology. The talk is based on joint works with Aleš Vavpetič.