  The fixed point Dirac operator on the lattice has exact chiral zero modes on
topologically non-trivial gauge field configurations independently whether
these configurations are smooth, or coarse. The relation $n_L-n_R = Q^{FP}$,
where $n_L$ $(n_R)$ is the number of left (right)-handed zero modes and
$Q^{FP}$ is the fixed point topological charge holds not only in the continuum
limit, but also at finite cut-off values. The fixed point action, which is
determined by classical equations, is local, has no doublers and complies with
the no-go theorems by being chirally non-symmetric. The index theorem is
reproduced exactly, nevertheless. In addition, the fixed point Dirac operator
has no small real eigenvalues except those at zero, i.e. there are no
'exceptional configurations'.
