Thirty-five years and counting

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 Along the way we have included several previously unpublished results involving how the g-conjecture relates to bistellar moves and small g_2, the topology and combinatorics of stacked manifolds introduced independently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

29 pages

References in corpus (1)

• Algebraic Shifting and f-Vector Theory

Cited by in corpus (12)

• Combinatorial Lefschetz theorems beyond positivity
• Measures on polyhedral cones: characterizations and kinematic formulas
• A Duality in Buchsbaum rings and triangulated manifolds
• Three-dimensional normal pseudomanifolds with relatively few edges
• The characteristic 2 anisotropicity of simplicial spheres
• Stanley-Reisner rings for quasi-arithmetic matroids
• Weak Lefschetz Theorem for Simplicial PL-spheres
• $g$-vectors of manifolds with boundary
• Toric spaces and face enumeration on simplicial manifolds
• Sparse handlebody decompositions and non-finiteness of $g_3=0$
• The face numbers of homology spheres
• A note on Lefschetz spheres and their relatives