  From the homotopy groups of three distinct octahedral spherical 3-manifolds
we construct the isomorphic groups H of deck transformations acting on the
3-sphere. The H-invariant polynomials on the 3-sphere constructed by
representation theory span the bases for the harmonic analysis on three
spherical manifolds. Analysis of the Cosmic Microwave Background in terms of
these new bases can reveal a non-simple topology of the space part of
space-time.
