The generic anisotropy of strongly edge decomposable spheres

TL;DR

This paper proves that all strongly edge decomposable spheres exhibit the property of generic anisotropy, which is crucial for understanding their algebraic and combinatorial characteristics.

Contribution

It establishes the conjecture that strongly edge decomposable spheres are generically anisotropic over any field, advancing the theory of homology spheres.

Findings

Proves the generic anisotropy conjecture for strongly edge decomposable spheres.

Supports the use of generic anisotropy in studying the algebraic properties of homology spheres.

Enhances understanding of the structure of Stanley-Reisner rings for these spheres.

Abstract

The generic anisotropy is an important property in the study of Stanley-Reisner rings of homology spheres, which was introduced by Papadakis and Petrotou. This property can be used to prove the strong Lefschetz property as well as McMullen's $g$-conjecture for homology spheres. It is conjectured that for an arbitrary field $\mathbb{F}$, any $\mathbb{F}$-homology sphere is generically anisotropic over $\mathbb{F}$. In this paper, we prove this conjecture for all strongly edge decomposable spheres.