On the anisotropy theorem of Papadakis and Petrotou

TL;DR

This paper advances the understanding of the anisotropy theorem for Stanley-Reisner rings of homology spheres in characteristic 2, providing explicit descriptions, proving a key conjecture, and offering a new proof of the g-conjecture.

Contribution

It offers an explicit quadratic form description, proves a conjecture by Papadakis and Petrotou, and extends anisotropy results to certain spheres over the rationals.

Findings

Explicit quadratic form description for the anisotropy theorem.

Proof of the conjecture by Papadakis and Petrotou.

New proof of the g-conjecture for homology spheres in characteristic 2.

Abstract

We study the anisotropy theorem for Stanley-Reisner rings of simplicial homology spheres in characteristic 2 by Papadakis and Petrotou. This theorem implies the Hard Lefschetz theorem as well as McMullen's g-conjecture for such spheres. Our first result is an explicit description of the quadratic form. We use this description to prove a conjecture stated by Papadakis and Petrotou. All anisotropy theorems for homology spheres and pseudo-manifolds in characteristic 2 follow from this conjecture. Using a specialization argument, we prove anisotropy for certain homology spheres over the field $\mathbb{Q}$. These results provide another self-contained proof of the g-conjecture for homology spheres in characteristic 2.