  We classify SU(3) gauge field configurations in different topological sectors
by the smearing technique. In each sector we compute the distribution of low
lying eigenvalues of the staggered Dirac operator. In all sectors we find
perfect agreement with the predictions for the sector of topological charge
zero. The smallest Dirac operator eigenvalues of staggered fermions at
presently realistic lattice couplings are thus insensitive to gauge field
topology. On the smeared configurations, $4\nu$ eigenvalues go to zero in
agreement with the index theorem.
