  We investigate and clarify the role of topology and the issues surrounding
the epsilon regime for staggered quarks. We study unimproved and improved
staggered quark Dirac operators on quenched lattice QCD gluon backgrounds
generated using a Symanzik-improved gluon action. For the improved Dirac
operators we find a clear separation of the spectrum into would-be zero modes
and others. The number of would-be zero modes depends on the topological charge
as predicted by the continuum Index Theorem, and the expectation values of
their chirality are large for the most improved actions (approx 0.7). The
remaining modes have low chirality and show clear signs of clustering into
quartets that become degenerate in the continuum limit. We demonstrate that the
lattice spacing and volume dependence of the eigenvalues follow expectations.
Furthermore, the non-zero modes follow the random matrix theory predictions for
all topological charge sectors. The values of the chiral condensate extracted
from fits to the theoretical distributions are consistent with each other, and
with the results obtained from the total density of eigenvalues using the
Banks-Casher relation. We conclude that staggered quarks respond correctly to
QCD topology when both fermion and gauge actions are improved.
