  When the density parameter is close to unity, the universe has a large
curvature radius independently of its being hyperbolic, flat, or spherical.
Whatever the curvature, the universe may have either a simply connected or a
multiply connected topology. In the flat case, the topology scale is arbitrary,
and there is no a priori reason for this scale to be of the same order as the
size of the observable universe. In the hyperbolic case any nontrivial topology
would almost surely be on a length scale too large to detect. In the spherical
case, by contrast, the topology could easily occur on a detectable scale. The
present paper shows how, in the spherical case, the assumption of a nearly flat
universe simplifies the algorithms for detecting a multiply connected topology,
but also reduces the amount of topology that can be seen. This is of primary
importance for the upcoming cosmic microwave background data analysis.
  This article shows that for spherical spaces one may restrict the search to
diametrically opposite pairs of circles in the circles-in-the-sky method and
still detect the cyclic factor in the standard factorization of the holonomy
group. This vastly decreases the algorithm's run time. If the search is widened
to include pairs of candidate circles whose centers are almost opposite and
whose relative twist varies slightly, then the cyclic factor along with a
cyclic subgroup of the general factor may also be detected. Unfortunately the
full holonomy group is, in general, unobservable in a nearly flat spherical
universe, and so a full 6-parameter search is unnecessary. Crystallographic
methods could also potentially detect the cyclic factor and a cyclic subgroup
of the general factor, but nothing else.
