The following post is contributed by Peter Kramer.
The new Planck data on the cosmic microwave background (CMB) has come in. For cosmic topology, the data sets contain interesting information related to the size and shape of the universe. The curvature of the three-dimensional space leads to a classification into hyperbolic, flat, or spherical cases. Sometimes in popular literature, the three cases are said to imply an inifinite (hyperbolic, flat) or finite (spherical) size of the universe. This statement is not correct. Topology supports a much wider zoo of possible universes. For instance, there are finite hyperbolic spaces, as depicted in the figure (taken from Group actions on compact hyperbolic manifolds and closed geodesics, arxiv version). The figure also shows the resulting geodesics, which is the path of light through such a hyperbolic finite sized universe. The start and end-points must be identified and lead to smooth connection.
Recent observational data seem to suggest a spherical space. Still, it does not resolve the issue of the size of the universe.
Instead of a fully filled three-sphere, already smaller parts of the sphere can be closed topologically and thus lead to a smaller sized universe. A systematic exploration of such smaller but still spherical universes is given in my recent article
Topology of Platonic Spherical Manifolds: From Homotopy to Harmonic Analysis.
In physics, it is important to give specific predictions for observations of the topology, for instance by predicting the ratio of the different angular modes of the cosmic microwave background. It is shown that this is indeed the case and for instance in a cubic (still spherical!) universe, the ratio of 4th and 6th multipole order squared are tied together in the proportion 7 : 4, see Table 5. On p. 35 of ( the Planck collaboration article) the authors call for models yielding such predictions as possible explanations for the observed anisotropy and the ratio of high and low multipole moments.