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-the-sky in maps of cosmic microwave background radiation 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 , whose total matter-energy density satisfies 0 < \mid \Omega _ { \text { tot } } -1 \mid \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 _ { \text { tot } } = 1 ) .
We demonstrate that in this case the conclusions deduced from such searches can be radically different .
We show that , although there is no characteristic topological scale in the flat manifolds , 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 .
Another important outcome of our results is that they offer a framework with which to make statistical inferences from future circles-in-the-sky searches on whether the Universe is exactly flat .