§ 3. Cosmology and topology.
If the spatial part of the cosmos is modelled by a spherical 3-manifold, this implies homotopic boundary conditions for the description of the CMBR.
The large-scale distribution of matter in the cosmos usually is treated by Einstein's field equations. These differential equations link the metric to the stress-energy tensor. To model the large-scale features one replaces the observed discrete by a smoothed continuous mass distribution. Moreover one models the space part of space-time by a 3-manifold. The present observations are compatible with positive spatial average curvature. This suggests for the topological cover the 3-sphere of positive average curvature.
One outstanding and well observed feature of cosmology is the Cosmic Microwave Background Radiation CMBR. It is believed to originate from an early state of the universe. Low multipole data and anisotropy from the CMBR compared to computations from standard models have motivated the search for alternative models of 3-space and their implications. It is here that cosmic topology comes into play. If 3-space is a topological 3-manifold , the observed CMBR must be described by functional analysis on . Given the algebraic basis construction on Platonic spherical 3-manifolds, we can look into the harmonic basis for observable signatures of the given topology. Such signatures can be the homotopic boundary conditions, the anisotropy, or selection rules for the multipole order of the CMBR. These questions are addressed in .