  We have investigated the reduction of the glass transition temperature,
$T_g$, for thin supported films of particularly small molecular weight ($M_W$ =
2 kg/mol) polystyrene, and found good agreement with earlier studies on larger
molecules. By combining the eigenmodes of the films with mode coupling theory,
we arrive at a simple and predictive model which is in quantitative agreement
with the data. Its only fitting parameter is the shear modulus governing the
relevant fluctuations. The weak dependence of the latter upon the molecular
weight of the polymer is in accordance with the general observation that the
reduction of $T_g$ is largely independent of chain length at sufficiently low
molecular weight.
