g-elements, finite buildings and higher Cohen-Macaulay connectivity

The main result is a proof that the g-vector of a simplicial complex with a convex ear decomposition is an M-vector. This is a generalization of similar results for matroid complexes. We also show that a finite building has a convex ear decomposition. This leads to connections between higher Cohen-Macaulay connectivity and increasing h-vectors.Comment: To appear in JCT A. 20 page

Thirty-five years and counting

It has been 35 years since Stanley proved that f-vectors of boundaries of simplicial polytopes satisfy McMullen's conjectured g-conditions. Since then one of the outstanding questions in the realm of face enumeration is whether or not Stanley's proof could be extended to larger classes of spheres. Here we hope to give an overview of various attempts to accomplish this and why we feel this is so important. In particular, we will see a strong connection to f-vectors of manifolds and pseudomanifolds. Along the way we have included several previously unpublished results involving how the g-conjecture relates to bistellar moves and small g_2, the topology and combinatorics of stacked manifolds introduced independently by Bagchi and Datta, and Murai and Nevo, and counterexamples to over optimistic generalizations of the g-theorem.Comment: 29 page