  We propose an experimental scheme to construct an optical lattice where the
atoms are confined to the surface of a torus. This construction can be realized
with spatially shaped laser beams which could be realized with recently
developed high resolution imaging techniques. We numerically study the
feasibility of this proposal by calculating the tunneling strengths for atoms
in the torus lattice. To illustrate the non-trivial role of topology in atomic
dynamics on the torus, we study the quantized superfluid currents and
fractional quantum Hall (FQH) states on such a structure. For FQH states, we
numerically investigate the robustness of the topological degeneracy and
propose an experimental way to detect such a degeneracy. Our scheme for torus
construction can be generalized to Riemann surfaces with higher genus for
exploration of richer topological physics.
