  We show that even if a lattice Dirac operator satisfies the conditions
consisting of locality, free of species doublings, correct continuum behavior,
$\gm5$-hermiticity and the Ginsparg-Wilson relation, it does not necessarily
have exact zero modes in nontrivial gauge backgrounds. This implies that each
lattice Dirac operator has its own topological characteristics which cannot be
fixed by these conditions. The role of topological characteristics in the axial
anomaly is derived explicitly.
