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Flag subdivisions and $γ$-vectors

The $γ$-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 $Δ$ and by Reiner, Postnikov and Williams that it increases when $Δ$ is replaced by any flag simplicial homology sphere which geometrically subdivides $Δ$. Using the nonnegativity of the $γ$-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.