The impact of topology on the structure of a smectic monolayer confined to a sphere is explored by particle-resolved computer simulations of hard rods. The orientations of the particles are tangential to the sphere and either free or restricted to a prescribed director field with a latitude or longitude orderings. Depending on the imprinted topology, a wealth of different states are found including equatorial smectic with isotropic poles, equatorial smectic with empty poles, a broken egg-shell like modulated smectic, a capped nematic with equatorial bald patches, equatorial nematic with empty poles, and a situation with 4 or 8 half-strength topological defects. Potentially these states could be verified in experiments with Pickering emulsions of droplets with colloidal rods. The unique nature of dipolar structures consisting of positive and negative half-strength disclinations is revealed. These structures, classified by their density and interaction with other defects in the system, relieve the strain of the poles by separating closely positioned half-strength defects. The proximity of these structures to the half-strength defects might enhance the structural diffusion of the defects across the system.