  Using the Laughlin's argument on a torus with two pin-holes, we numerically
demonstrate that the discontinuities of the center-of-mass work well as an
invariant of the pumping phenomena during the process of the flux-attachment,
trading the magnetic flux for the statistical one. This is consistent with the
bulk-edge correspondence of the fractional quantum Hall effect of anyons. We
also confirm that the general feature of the edge states remains unchanged
during the process while the topological degeneracy is discretely changed. This
supports the stability of the quantum Hall edge states in the adiabatic
heuristic principle.
