The role of polyhedral products in geometric and topological combinatorics听
The role of polyhedral products in geometric and topological combinatorics听
Online Talk听
The problem of deciding if a given triangulation of a sphere is realizable as the boundary sphere of a simplicial,听convex polytope is known as the 鈥淪implicial Steinitz problem鈥. This is an example of a problem of geometric combinatorics听which links together areas of mathematics as distant as toric topology, combinatorial optimization,听 convex polytopes,听algebraic geometry, topological combinatorics, discrete and computational geometry, etc.听It is known (by indirect and non-constructive arguments) that a vast majority of triangulated spheres are 鈥渘on-polytopal鈥,听in the sense that they are not combinatorially isomorphic to the boundary of a convex polytope. This holds, in particular,听for Bier spheres 听 Bier(K) (named after Thomas Bier),听 the (n-2)-dimensional, combinatorial spheres on 2n-vertices,听constructed with the aid of simplicial complexes听 K on听 n听 vertices.听Emphasizing connections with polyhedral products and toric topology,听 we review听 "hidden geometry鈥 of Bier spheres by听describing their natural geometric realizations, compute their volume, describe an effective criterion for their 鈥渟trong polytopality鈥,听and associate to Bier(K) a natural coarsening Fan(K) of the Braid fan.听 We also establish a connection of Bier spheres听of听maximal volume with recent generalizations of the classical Van Kampen-Flores theorem and clarify the role of听Bier听spheres in听the theory of generalized permutohedra.听