  The Cosmic Microwave Background CMB originates from an early stage in the
history of the universe. Observed low multipole contributions of CMB
fluctuations have motivated the search for selection rules from the underlying
topology of 3-space. Everitt (2004) has generated all homotopies for Platonic
spherical 3-manifolds by face gluings. We transform the glue generators into
isomorphic deck transformations. The deck transformations act on a spherical
Platonic 3-manifold as prototile and tile the 3-sphere by its images. A
complete set of orthonormal functions on the 3-sphere is spanned by the Wigner
harmonic polynomials. For a tetrahedral, two cubic and three octahedral
manifolds we construct algebraically linear combinations of Wigner polynomials,
invariant under deck transformations and with domain the manifold. We prove
boundary conditions on polyhedral faces from homotopy. By algebraic means we
pass to a multipole expansion. Assuming random models of the CMB radiation, we
derive multipole selection rules, depending on the point symmetry of the
manifold.
