  We investigate topological charge and the index theorem on finite lattices
numerically. Using mean field improved gauge field configurations we calculate
the topological charge Q using the gluon field definition with ${\cal
O}(a^4)$-improved cooling and an ${\cal O}(a^4)$-improved field strength tensor
$F_{\mu\nu}$. We also calculate the index of the massless overlap fermion
operator by directly measuring the differences of the numbers of zero modes
with left- and right--handed chiralities. For sufficiently smooth field
configurations we find that the gluon field definition of the topological
charge is integer to better than 1% and furthermore that this agrees with the
index of the overlap Dirac operator, i.e., the Atiyah-Singer index theorem is
satisfied. This establishes a benchmark for reliability when calculating
lattice quantities which are very sensitive to topology.
