## Detectability of cosmic topology in almost flat universes |

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3-manifold 3-sphere astronomical survey bounds on cosmological catalogue with zmax closed geodesic CMBR compact manifold compact topologies concrete examples corresponding solution curve cosmic topology cosmological constant covering group covering space critical density curvature radius cyclic groups density parameters depth zmax detectability of cosmic exclude certain families families of possible families of topologies fimo FÍSICA fixed given manifold globally homogeneous horizon radius infinite number injectivity radius isometry lens space topologies Lens spaces L(p manifolds that remain maximal inradius r™ax multiple images multiply connected nearly flat FLRW nearly flat hyperbolic Nearly Flat Universes non-trivial topologies non-zero curvature non-zero lower bound observational bounds orientable hyperbolic manifolds possible topologies quasars question of detectability r,nj recent observations redshift cut-off remain undetectable rinj rmax shape smallest closed geodesies SnapPea spherical manifolds subset of lens Table topology of nearly total density undetectable and families undetectable topologies vacuum energy values of Xobs Xhor Xoba