Cosmological Parameters and Cosmic Topology

Geometry constrains but does not dictate the topology of the 3--dimensional space. In a locally spatially homogeneous and isotropic universe, however, the topology of its spatial section dictates its geometry. We show that, besides determining the geometry, the knowledge of the spatial topology through the circles--in--the--sky offers an effective way of setting constraints on the density parameters associated with dark matter ($\Omega_m$) and dark energy ($\Omega_{\Lambda}$). By assuming the Poincar\'e dodecahedral space as the circles--in--the--sky detectable topology of the spatial sections of the Universe, we re-analyze the constraints on the density parametric plane $\Omega_{m}$--$\Omega_{\Lambda}$ from the current type Ia supenovae (SNe Ia) plus X-ray gas mass fraction data, and show that a circles--in--the sky detection of the dodecahedral space topology give rise to strong and complementary constraints on the region of the density parameter plane currently allowed by these observational data sets.