Flag subdivisions and $\gamma$-vectors

TL;DR

This paper confirms conjectures about the nonnegativity and monotonicity of the $oldsymbol{eta}$-vector in flag simplicial homology spheres, specifically in dimensions 3 and 4, using existing theories and results.

Contribution

It proves the conjecture that the $oldsymbol{eta}$-vector is nonnegative and increases under subdivisions for flag simplicial homology spheres in dimensions 3 and 4.

Findings

Confirmed nonnegativity of the $oldsymbol{eta}$-vector in dimension 3.

Proved the increase of the $oldsymbol{eta}$-vector under subdivisions in dimensions 3 and 4.

Extended the understanding of $oldsymbol{eta}$-vector behavior in flag simplicial homology spheres.

Abstract

The $\gamma$-vector is an important enumerative invariant of a flag simplicial homology sphere. It has been conjectured by Gal that this vector is nonnegative for every such sphere $\Delta$ and by Reiner, Postnikov and Williams that it increases when $\Delta$ is replaced by any flag simplicial homology sphere which geometrically subdivides $\Delta$. Using the nonnegativity of the $\gamma$-vector in dimension 3, proved by Davis and Okun, as well as Stanley's theory of simplicial subdivisions and local $h$-vectors, the latter conjecture is confirmed in this paper in dimensions 3 and 4.