We investigate the collapse of nonrotating gas spheres with a polytropic equation of state: n = 3, corresponding to y = 4/3. Such polytropes provide a reasonable approximation to collapsing stellar cores during the early phase before nuclear density is reached. We find a family of exact homologously collapsing configurations. Homologous collapse of the entire core is possible if the pressure at a given density is reduced by up to 3% from the value for a marginally stable static core. For a greater pressure reduction, an inner core can collapse homologously, the mass of which varies as the 3/2 power of the reduced pressure at the onset of collapse. Linear perturbations of these homologously collapsing solutions are separable in space and time. Low order radial and nonradial modes are calculated, and it is found that all modes are essentially stable.