  The well known $g$-conjecture for homology spheres follows from the stronger
conjecture that the face ring over the reals of a homology sphere, modulo a
linear system of parameters, admits the strong-Lefschetz property. We prove
that the strong-Lefschetz property is preserved under the following
constructions on homology spheres: join, connected sum, and stellar
subdivisions. The last construction is a step towards proving the
$g$-conjecture for piecewise-linear spheres.
