Abstracts

Rade Zivaljevic

Quasitoric manifolds associated with Bier spheres and generalized permutahedra

Quasitoric manifolds and polyhedral products are among the central objects studied by toric topology. Quasitoric manifolds are closely related to non-singular toric varieties, which are ubiquitous in algebraic geometry. These objects are closely related to moment-angle manifolds, which reveals their connection with classical mechanics and symplectic geometry.

Each simplicial complex K with n vertices is associated an (n-2)-dimensional, combinatorial sphere on (at most) 2n-vertices, called a Bier sphere Bier(K) (named after Thomas Bier).

Combinatorics and geometry of Bier spheres and their generalizations have been studied in numerous publications. One of the fundamental results (F. Jevtić, M. Timotijević, R. Živaljević, 2019, 2022, 2024) says that each Bier sphere Bier(K) admits a canonical starshaped embedding such that the corresponding (canonical) radial fan Fan(K) is a coarsening of the braid arrangement fan.

In turn, each simplicial complex K can be associated a canonical toric variety (quasitoric manifold) and this connection of the combinatorics of simplicial complexes and geometry/topology of the associated manifolds is the subject of current research. This is a report of some of more recent results in this area, obtained jointly with Ivan Limonchenko and Matvey Sergeev.