Cosmological parameters and cosmic topology

Geometry constrains but does not dictate the topology of the three‐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 (Ω m ) and dark energy (Ω Λ ) . By assuming the Poincaré dodecahedral space as the circles‐in‐the‐sky detectable topology of the spatial sections of the Universe, we re‐analyse the constraints on the density parametric plane Ω m –Ω Λ from the current Type Ia supernova plus X‐ray gas mass fraction data, and show that a circles‐in‐the sky detection of the dodecahedral space topology gives rise to strong and complementary constraints on the region of the density parameter plane currently allowed by these observational data sets.