  To define the topology of driven systems, recent works have proposed
synthetic dimensions as a way to uncover the underlying parameter space of
topological invariants. Using time as a synthetic dimension, together with a
momentum dimension, gives access to a synthetic 2D Chern number. It is,
however, still unclear how the synthetic 2D Chern number is related to the
Chern number that is defined from a parametric variable that evolves with time.
Here we show that in periodically driven systems in the adiabatic limit, the
synthetic 2D Chern number is a multiple of the Chern number defined from the
parametric variable. The synthetic 2D Chern number can thus be engineered via
how the parametric variable evolves in its own space. We justify our claims by
investigating Thouless pumping in two 1D tight-binding models, a three-site
chain model and a two-1D-sliding-chains model. The present findings could be
extended to higher dimensions and other periodically driven configurations.
