  Qualitative features of Ginsparg-Wilson fermions, as formulated by Neuberger,
coupled to two dimensional U(1) gauge theory are studied. The role of the
Wilson mass parameter in changing the number of massless flavors in the theory
and its connection with the index of the Dirac operator is studied. Although
the index of the Dirac operator is not related to the geometric definition of
the topological charge for strong couplings, the two start to agree as soon as
one goes to moderately weak couplings. This produces the desired singularity in
the quenched chiral condensate which appears to be very difficult to reproduce
with staggered fermions. The fermion determinant removes the singularity and
reproduces the known chiral condensate and the meson mass within understandable
errors.
