Harmonic polynomials for expanding the fluctuations of the Cosmic Microwave Background: The Poincare and the 3-sphere model

Fluctuations of the Cosmic Microwave Background CMB are observed by the WMAP. When expanded into the harmonic eigenmodes of the space part of a cosmological model, they provide insight into the large-scale topology of space. All harmonic polynomials on the multiply connected dodecahedral Poincare space are constructed. Strong and specific selection rules are given by comparing the polynomials to those on the 3-sphere, its simply connected cover.