This Monte Carlo simulation was undertaken to provide some qualitative insight into the dependence of the glass transition of polymer melts on the spatial dimension. To this end, two- and three-dimensional systems were simulated, in which we kept the external conditions, such as the density or the cooling schedule, the same for both dimensions. The melts, simulated by a lattice model (bond-fluctuation model), undergo a kinetic freezing, while being continuously cooled from the equilibrium liquid to the nonequilibrium glassy phase. The resulting glass transition as well as the other simulation results indicate that the spatial constraints are stronger in three than in two dimensions. This finding is reminiscent of the influence of the spatial dimension on the ability of a liquid to crystallize.