  Stimulated by the recent progress in engineering topological band structures
in cold atomic gases, we study the dynamic topological phenomena for atoms
loaded in a periodically driven optical lattice. When the frequency of the
periodic modulation is low, the time-dependent Hamiltonian can be mapped to a
two-dimensional topological insulator, with the discretized frequency
components playing the role of an additional, synthetic dimension. In the
high-frequency limit, we derive the effective Floquet Hamiltonian of the
system, and reveal the occurrence of Floquet dynamic quantum phase transitions
-- an emergent topological phenomenon in the micromotion of the Floquet
dynamics. Addressing the relation between the topology of the effective Floquet
Hamiltonian and the presence of dynamic topological phenomena, we demonstrate
that the topologically non-trivial nature of the Floquet Hamiltonian is a
sufficient but not necessary condition for the onset of the Floquet dynamic
quantum phase transition. We further discuss the relation of the topology of
the Floquet Hamiltonian with the existence of dynamic skyrmion structures in
the emergent momentum-time manifold of the micromotion, as well as the fate of
these dynamic topological phenomena when the modulation frequency decreases
away from the high-frequency limit. Finally, making use of the rich level
structures of $^{171}$Yb atoms, we show that the system under study can be
implemented in a one-dimensional Raman lattice where states in the $^1S_0$
ground-state manifold are coupled by Raman beams with periodically modulated
amplitudes.
