  [Abridged] In a Universe with a detectable nontrivial spatial topology the
last scattering surface contains pairs of matching circles with the same
distribution of temperature fluctuations - the so-called circles-in-the-sky.
Searches for nearly antipodal circles in maps of cosmic microwave background
have so far been unsuccessful. This negative outcome along with recent
theoretical results concerning the detectability of nearly flat compact
topologies is sufficient to exclude a detectable nontrivial topology for most
observers in very nearly flat positively and negatively curved Universes
($0<|\Omega_{tot}-1| \lesssim 10^{-5}$). Here we investigate the consequences
of these searches for observable nontrivial topologies if the Universe turns
out to be exactly flat ($\Omega_{tot}=1$). We demonstrate that in this case the
conclusions deduced from such searches can be radically different. We show that
for all multiply-connected orientable flat manifolds it is possible to directly
study the action of the holonomies in order to obtain a general upper bound on
the angle that characterizes the deviation from antipodicity of pairs of
matching circles associated with the shortest closed geodesic. This bound is
valid for all observers and all possible values of the compactification length
parameters. We also show that in a flat Universe there are observers for whom
the circles-in-the-sky searches already undertaken are insufficient to exclude
the possibility of a detectable nontrivial spatial topology. It is remarkable
how such small variations in the spatial curvature of the Universe, which are
effectively indistinguishable geometrically, can have such a drastic effect on
the detectability of cosmic topology.
