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The Cosmic Microwave Background CMB originates from an early stage in thehistory of the universe. Observed low multipole contributions of CMBfluctuations have motivated the search for selection rules from the underlyingtopology of 3-space. Everitt (2004) has generated all homotopies for Platonicspherical 3-manifolds by face gluings. We transform the glue generators intoisomorphic deck transformations. The deck transformations act on a sphericalPlatonic 3-manifold as prototile and tile the 3-sphere by its images. Acomplete set of orthonormal functions on the 3-sphere is spanned by the Wignerharmonic polynomials. For a tetrahedral, two cubic and three octahedralmanifolds we construct algebraically linear combinations of Wigner polynomials,invariant under deck transformations and with domain the manifold. We proveboundary conditions on polyhedral faces from homotopy. By algebraic means wepass to a multipole expansion. Assuming random models of the CMB radiation, wederive multipole selection rules, depending on the point symmetry of themanifold.

Platonic topology and CMB fluctuations: Homotopy, anisotropy, and multipole selection rules