The University of Michigan Combinatorics Seminar


Abstract 

The fvector of an abstract simplicial complex records the number of its faces in each dimension. In 1970 McMullen conjectured a complete characterization of the possible fvectors of boundary complexes of simplicial polytopes. These numerical conditions were proved in 1980, their necessity by Stanley and their sufficiency by Billera and Lee; known as the gtheorem. The proof of necessity shows that the hardLefschetz property (decomposition) holds for an appropriate ring associated with the polytope, from which the numerical consequences follow. A major open problem in algebraic combinatorics, known as the gconjecture, is to extend these numerical and algebraic assertions to the larger family of simplicial sphere, and even more general families. We will indicate recent developments on this conjecture, focusing on the following results: 1. (Joint with Martina Kubitzke.) The (numerical) gconjecture holds for the barycentric subdivision of homology spheres. This follows from the following algebraic result: An `almost hardLefschetz' property holds for the barycentric subdivision of a shellable complex. 2. (Joint with Eric Babson.) If a homology sphere and one of its face links admit the hardLefschetz property, then its stellar subdivision at this face admits the hardLefschetz property, and hence satisfies the (numerical) gconjecture. One ingredient in the proof is showing that the hardLefschetz property is preserved under the join operation. 