Thirty-five years and counting

TL;DR

This paper reviews 35 years of research on the g-conjecture for simplicial spheres, exploring extensions to manifolds and pseudomanifolds, and discussing related topological and combinatorial results.

Contribution

It provides an overview of various approaches and unpublished results related to extending the g-theorem beyond simplicial polytopes, highlighting key challenges and developments.

Findings

Connections between g-conjecture and bistellar moves

Results on small g_2 and stacked manifolds

Counterexamples to generalized g-theorem

Abstract

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 Along the way we have included several previously unpublished results involving how the g-conjecture relates to bistellar moves and small g_2, the topology and combinatorics of stacked manifolds introduced independently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.