## Spikes in cosmic crystallography II: topological signature of compact flat universes |

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3-manifold 3-torus that covers amplitude approximately 240 sources Área de Publicações ball of radius bins of width Brasileiro de Pesquisas built with 50 Centro Brasileiro Clifford translations comparable catalogs compute Cosmic Crystallography cosmic sources covering group distance correlations EPSH evince the topological Fagundes and Gausmann Figure Flat Universes given catalog histogram PSH individual PSH's isometries in PSH's isometry g manifold masked by statistical minimal 3-torus MPSH technique non-translational isometries number of pairs number of sources observational limitations observed universe corresponding odd integer positions pair separations histogram pairs of sources PSH's corresponding real catalogs Reducing the noise result of Ref Rio de Janeiro sections of spacetime selection rules separations histogram EPSH shown signature of non-translational simply connected simulated catalogs spatial sections spike spectra spikes in PSH's statistical fluctuations statistical noise statistical spikes studying the topological subdivided in bins subintervals topological and statistical topological origin topological spikes universal covering universe models Xavier Sigaud