  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.
