  The Bott index is an index that discerns among pairs of unitary matrices that
can or cannot be approximated by a pair of commuting unitary matrices. It has
been successfully employed to describe the approximate integer quantization of
the transverse conductance of a system described by a short-range, bounded and
spectrally gapped Hamiltonian on a finite two dimensional lattice on a torus
and to describe the invariant of the Bernevig- Hughes-Zhang model even with
disorder. This paper shows the constancy in time of the Bott index and the
Chern number related to the time-evolved Fermi projection of a
thermodynamically large system described by a short-range and time-dependent
Hamiltonian that is initially gapped. The general situation of a ramp of a
time-dependent perturbation is considered, a section is dedicated to
time-periodic perturbations.
