  By analyzing an exactly solvable model in the second quantized formulation
which allows a unified treatment of adiabatic and non-adiabatic geometric
phases, it is shown that the topology of the adiabatic Berry's phase, which is
characterized by the singularity associated with possible level crossing, is
trivial in a precise sense. This topology of the geometric phase is quite
different from the topology of the Aharonov-Bohm effect, where the topology is
specified by the external local gauge field and it is exact for the slow as
well as for the fast motion of the electron.
