The University of Michigan Combinatorics Seminar
The f-vector of an abstract simplicial complex records the number of its faces in each dimension.
In 1970 McMullen conjectured a complete characterization of the possible f-vectors 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 g-theorem. The proof of necessity shows that the hard-Lefschetz 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 g-conjecture, 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) g-conjecture holds for the barycentric subdivision of homology spheres. This follows from the following algebraic result: An `almost hard-Lefschetz' 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 hard-Lefschetz property, then its stellar subdivision at this face admits the hard-Lefschetz property, and hence satisfies the (numerical) g-conjecture. One ingredient in the proof is showing that the hard-Lefschetz property is preserved under the join operation.