Toric topology of balanced simplicial complexes

Abstract

Stanley introduced in [4] an important class of simplicialcomplexes that arise often in combinatorics, topology andalgebra so called balanced simplicial complexes. An-dimensionalsimplicial complexKis called balanced if its set of verticescan be splitted intondisjoint subsets such that there is notwo vertices spanning the edge ofKand belonging to the samesubset. One of central questions in combinatorics is descriptionof all integer vectors that may appear as the face vectors ofconvex polytopes. The most celebrated result in this problematicis the famousg-theorem.In the paper [2] Klee and Novik conjectured a strongerbound concerning the face numbers of balanced simplicialn-polytopes. The conjecture known as the Balanced LowerBound Theorem is proved in [1, Theorem 1.3] (“if” part) andin [2, Theorem 5.8] (“only if” part)..