  Block-spin transformations from a fine lattice to a coarse one are shown to
give rise to a one-to-one correspondence between the zero-modes of the
Ginsparg-Wilson Dirac operators. The index is then preserved under the blocking
process. Such a one-to-one correspondence is violated and the block-spin
transformation becomes necessarily ill-defined when the absolute value of the
index is larger than 2rN, where N is the number of the sites on the coarse
lattice and r is the dimension of the gauge group representation of the fermion
variables.
