  The immediate observational consequence of a non-trivial spatial topology of
the Universe is that an observer could potentially detect multiple images of
radiating sources. In particular, a non-trivial topology will generate pairs of
correlated circles of temperature fluctuations in the anisotropies maps of the
cosmic microwave background (CMB), the so-called circles-in-the-sky. In this
way, a detectable non-trivial spatial topology may be seen as an observable
attribute, which can be probed through the circles-in-the-sky for all locally
homogeneous and isotropic universes with no assumptions on the cosmological
dark energy (DE) equation of state (EOS) parameters. We show that the knowledge
of the spatial topology through the circles-in-the-sky offers an effective way
of reducing the degeneracies in the DE EOS parameters. We concretely illustrate
the topological role by assuming, as an exanple, a Poincar\'{e} dodecahedral
space topology and reanalyzing the constraints on the parameters of a specific
EOS which arise from the supernovae type Ia, baryon acoustic oscillations and
the CMB plus the statistical topological contribution.
