  We propose a simple scheme for the realization of a topological quasienergy
band structure with ultracold atoms in a periodically driven optical square
lattice. It is based on a circular lattice shaking in the presence of a
superlattice that lowers the energy on every other site. The topological band
gap, which separates the two bands with Chern numbers $\pm 1$, is opened in a
way characteristic to Floquet topological insulators, namely, by terms of the
effective Hamiltonian that appear in subleading order of a high-frequency
expansion. These terms correspond to processes where a particle tunnels several
times during one driving period. The interplay of such processes with particle
interactions also gives rise to new interaction terms of several distinct
types. For bosonic atoms with on-site interactions, they include nearest
neighbor density-density interactions introduced at the cost of weakened
on-site repulsion as well as density-assisted tunneling. Using exact
diagonalization, we investigate the impact of the individual induced
interaction terms on the stability of a bosonic fractional Chern insulator
state at half filling of the lowest band.
