  Motivated by the connection between gauge field topology and the axial
anomaly in fermion currents, I use the fourth power of the naive Dirac operator
to define a local lattice measure of topological charge. For smooth gauge
fields this reduces to the usual topological density. For typical gauge field
configurations in a numerical simulation, however, quantum fluctuations
dominate, and the sum of this density over the system does not generally give
an integer winding. On cooling with respect to the Wilson gauge action,
instanton like structures do emerge. As cooling proceeds, these objects tend
shrink and finally "fall through the lattice." Modifying the action can block
the shrinking at the expense of a loss of reflection positivity. The cooling
procedure is highly sensitive to the details of the initial steps, suggesting
that quantum fluctuations induce a small but fundamental ambiguity in the
definition of topological susceptibility.
